Euler Stratification of Binary Octics

Each stratum is labeled by a decorated partition as follows: a label is a tuple \((\lambda, m_0, m_{\infty})\). Here, \(\lambda\) is a partition of \(8-m_0-m_{\infty}\). The corresponding stratum consists of binary forms with a root of multiplicity \(m_0\) at zero, a root of multiplicity \(m_{\infty}\) at infinity, and \(|\lambda|\) roots in the torus. The first root in the torus has multiplicity \(\lambda_1\), the second has multiplicity \(\lambda_2\), and so on. The equations for strata of binary forms in \(\mathbb{P}^1\) correspond to the decorated partitions with \(m_0=m_{\infty}=0\).

The .json files can be imported into Oscar at version 0.14 with the following command:

Oscar.load("stratum.json")
Euler Stratification of Binary Octics

Stratum

Generators of ideal

Ideals as Oscar object

([0], (0, 8))

([0], (0, 8)).txt

([0], (0, 8)).json

([0], (1, 7))

([0], (1, 7)).txt

([0], (1, 7)).json

([0], (2, 6))

([0], (2, 6)).txt

([0], (2, 6)).json

([0], (3, 5))

([0], (3, 5)).txt

([0], (3, 5)).json

([0], (4, 4))

([0], (4, 4)).txt

([0], (4, 4)).json

([0], (5, 3))

([0], (5, 3)).txt

([0], (5, 3)).json

([0], (6, 2))

([0], (6, 2)).txt

([0], (6, 2)).json

([0], (7, 1))

([0], (7, 1)).txt

([0], (7, 1)).json

([0], (8, 0))

([0], (8, 0)).txt

([0], (8, 0)).json

([1], (0, 7))

([1], (0, 7)).txt

([1], (0, 7)).json

([1], (1, 6))

([1], (1, 6)).txt

([1], (1, 6)).json

([1], (2, 5))

([1], (2, 5)).txt

([1], (2, 5)).json

([1], (3, 4))

([1], (3, 4)).txt

([1], (3, 4)).json

([1], (4, 3))

([1], (4, 3)).txt

([1], (4, 3)).json

([1], (5, 2))

([1], (5, 2)).txt

([1], (5, 2)).json

([1], (6, 1))

([1], (6, 1)).txt

([1], (6, 1)).json

([1], (7, 0))

([1], (7, 0)).txt

([1], (7, 0)).json

([1, 1], (0, 6))

([1, 1], (0, 6)).txt

([1, 1], (0, 6)).json

([1, 1], (1, 5))

([1, 1], (1, 5)).txt

([1, 1], (1, 5)).json

([1, 1], (2, 4))

([1, 1], (2, 4)).txt

([1, 1], (2, 4)).json

([1, 1], (3, 3))

([1, 1], (3, 3)).txt

([1, 1], (3, 3)).json

([1, 1], (4, 2))

([1, 1], (4, 2)).txt

([1, 1], (4, 2)).json

([1, 1], (5, 1))

([1, 1], (5, 1)).txt

([1, 1], (5, 1)).json

([1, 1], (6, 0))

([1, 1], (6, 0)).txt

([1, 1], (6, 0)).json

([1, 1, 1], (0, 5))

([1, 1, 1], (0, 5)).txt

([1, 1, 1], (0, 5)).json

([1, 1, 1], (1, 4))

([1, 1, 1], (1, 4)).txt

([1, 1, 1], (1, 4)).json

([1, 1, 1], (2, 3))

([1, 1, 1], (2, 3)).txt

([1, 1, 1], (2, 3)).json

([1, 1, 1], (3, 2))

([1, 1, 1], (3, 2)).txt

([1, 1, 1], (3, 2)).json

([1, 1, 1], (4, 1))

([1, 1, 1], (4, 1)).txt

([1, 1, 1], (4, 1)).json

([1, 1, 1], (5, 0))

([1, 1, 1], (5, 0)).txt

([1, 1, 1], (5, 0)).json

([1, 1, 1, 1], (0, 4))

([1, 1, 1, 1], (0, 4)).txt

([1, 1, 1, 1], (0, 4)).json

([1, 1, 1, 1], (1, 3))

([1, 1, 1, 1], (1, 3)).txt

([1, 1, 1, 1], (1, 3)).json

([1, 1, 1, 1], (2, 2))

([1, 1, 1, 1], (2, 2)).txt

([1, 1, 1, 1], (2, 2)).json

([1, 1, 1, 1], (3, 1))

([1, 1, 1, 1], (3, 1)).txt

([1, 1, 1, 1], (3, 1)).json

([1, 1, 1, 1], (4, 0))

([1, 1, 1, 1], (4, 0)).txt

([1, 1, 1, 1], (4, 0)).json

([1, 1, 1, 1, 1], (0, 3))

([1, 1, 1, 1, 1], (0, 3)).txt

([1, 1, 1, 1, 1], (0, 3)).json

([1, 1, 1, 1, 1], (1, 2))

([1, 1, 1, 1, 1], (1, 2)).txt

([1, 1, 1, 1, 1], (1, 2)).json

([1, 1, 1, 1, 1], (2, 1))

([1, 1, 1, 1, 1], (2, 1)).txt

([1, 1, 1, 1, 1], (2, 1)).json

([1, 1, 1, 1, 1], (3, 0))

([1, 1, 1, 1, 1], (3, 0)).txt

([1, 1, 1, 1, 1], (3, 0)).json

([1, 1, 1, 1, 1, 1], (0, 2))

([1, 1, 1, 1, 1, 1], (0, 2)).txt

([1, 1, 1, 1, 1, 1], (0, 2)).json

([1, 1, 1, 1, 1, 1], (1, 1))

([1, 1, 1, 1, 1, 1], (1, 1)).txt

([1, 1, 1, 1, 1, 1], (1, 1)).json

([1, 1, 1, 1, 1, 1], (2, 0))

([1, 1, 1, 1, 1, 1], (2, 0)).txt

([1, 1, 1, 1, 1, 1], (2, 0)).json

([1, 1, 1, 1, 1, 1, 1], (0, 1))

([1, 1, 1, 1, 1, 1, 1], (0, 1)).txt

([1, 1, 1, 1, 1, 1, 1], (0, 1)).json

([1, 1, 1, 1, 1, 1, 1], (1, 0))

([1, 1, 1, 1, 1, 1, 1], (1, 0)).txt

([1, 1, 1, 1, 1, 1, 1], (1, 0)).json

([2], (0, 6))

([2], (0, 6)).txt

([2], (0, 6)).json

([2], (1, 5))

([2], (1, 5)).txt

([2], (1, 5)).json

([2], (2, 4))

([2], (2, 4)).txt

([2], (2, 4)).json

([2], (3, 3))

([2], (3, 3)).txt

([2], (3, 3)).json

([2], (4, 2))

([2], (4, 2)).txt

([2], (4, 2)).json

([2], (5, 1))

([2], (5, 1)).txt

([2], (5, 1)).json

([2], (6, 0))

([2], (6, 0)).txt

([2], (6, 0)).json

([2, 1], (0, 5))

([2, 1], (0, 5)).txt

([2, 1], (0, 5)).json

([2, 1], (1, 4))

([2, 1], (1, 4)).txt

([2, 1], (1, 4)).json

([2, 1], (2, 3))

([2, 1], (2, 3)).txt

([2, 1], (2, 3)).json

([2, 1], (3, 2))

([2, 1], (3, 2)).txt

([2, 1], (3, 2)).json

([2, 1], (4, 1))

([2, 1], (4, 1)).txt

([2, 1], (4, 1)).json

([2, 1], (5, 0))

([2, 1], (5, 0)).txt

([2, 1], (5, 0)).json

([2, 1, 1], (0, 4))

([2, 1, 1], (0, 4)).txt

([2, 1, 1], (0, 4)).json

([2, 1, 1], (1, 3))

([2, 1, 1], (1, 3)).txt

([2, 1, 1], (1, 3)).json

([2, 1, 1], (2, 2))

([2, 1, 1], (2, 2)).txt

([2, 1, 1], (2, 2)).json

([2, 1, 1], (3, 1))

([2, 1, 1], (3, 1)).txt

([2, 1, 1], (3, 1)).json

([2, 1, 1], (4, 0))

([2, 1, 1], (4, 0)).txt

([2, 1, 1], (4, 0)).json

([2, 1, 1, 1], (0, 3))

([2, 1, 1, 1], (0, 3)).txt

([2, 1, 1, 1], (0, 3)).json

([2, 1, 1, 1], (1, 2))

([2, 1, 1, 1], (1, 2)).txt

([2, 1, 1, 1], (1, 2)).json

([2, 1, 1, 1], (2, 1))

([2, 1, 1, 1], (2, 1)).txt

([2, 1, 1, 1], (2, 1)).json

([2, 1, 1, 1], (3, 0))

([2, 1, 1, 1], (3, 0)).txt

([2, 1, 1, 1], (3, 0)).json

([2, 1, 1, 1, 1], (0, 2))

([2, 1, 1, 1, 1], (0, 2)).txt

([2, 1, 1, 1, 1], (0, 2)).json

([2, 1, 1, 1, 1], (1, 1))

([2, 1, 1, 1, 1], (1, 1)).txt

([2, 1, 1, 1, 1], (1, 1)).json

([2, 1, 1, 1, 1], (2, 0))

([2, 1, 1, 1, 1], (2, 0)).txt

([2, 1, 1, 1, 1], (2, 0)).json

([2, 1, 1, 1, 1, 1], (0, 1))

([2, 1, 1, 1, 1, 1], (0, 1)).txt

([2, 1, 1, 1, 1, 1], (0, 1)).json

([2, 1, 1, 1, 1, 1], (1, 0))

([2, 1, 1, 1, 1, 1], (1, 0)).txt

([2, 1, 1, 1, 1, 1], (1, 0)).json

([2, 1, 1, 1, 1, 1, 1], (0, 0))

([2, 1, 1, 1, 1, 1, 1], (0, 0)).txt

([2, 1, 1, 1, 1, 1, 1], (0, 0)).json

([2, 2], (0, 4))

([2, 2], (0, 4)).txt

([2, 2], (0, 4)).json

([2, 2], (1, 3))

([2, 2], (1, 3)).txt

([2, 2], (1, 3)).json

([2, 2], (2, 2))

([2, 2], (2, 2)).txt

([2, 2], (2, 2)).json

([2, 2], (3, 1))

([2, 2], (3, 1)).txt

([2, 2], (3, 1)).json

([2, 2], (4, 0))

([2, 2], (4, 0)).txt

([2, 2], (4, 0)).json

([2, 2, 1], (0, 3))

([2, 2, 1], (0, 3)).txt

([2, 2, 1], (0, 3)).json

([2, 2, 1], (1, 2))

([2, 2, 1], (1, 2)).txt

([2, 2, 1], (1, 2)).json

([2, 2, 1], (2, 1))

([2, 2, 1], (2, 1)).txt

([2, 2, 1], (2, 1)).json

([2, 2, 1], (3, 0))

([2, 2, 1], (3, 0)).txt

([2, 2, 1], (3, 0)).json

([2, 2, 1, 1], (0, 2))

([2, 2, 1, 1], (0, 2)).txt

([2, 2, 1, 1], (0, 2)).json

([2, 2, 1, 1], (1, 1))

([2, 2, 1, 1], (1, 1)).txt

([2, 2, 1, 1], (1, 1)).json

([2, 2, 1, 1], (2, 0))

([2, 2, 1, 1], (2, 0)).txt

([2, 2, 1, 1], (2, 0)).json

([2, 2, 1, 1, 1], (0, 1))

([2, 2, 1, 1, 1], (0, 1)).txt

([2, 2, 1, 1, 1], (0, 1)).json

([2, 2, 1, 1, 1], (1, 0))

([2, 2, 1, 1, 1], (1, 0)).txt

([2, 2, 1, 1, 1], (1, 0)).json

([2, 2, 1, 1, 1, 1], (0, 0))

([2, 2, 1, 1, 1, 1], (0, 0)).txt

([2, 2, 1, 1, 1, 1], (0, 0)).json

([2, 2, 2], (0, 2))

([2, 2, 2], (0, 2)).txt

([2, 2, 2], (0, 2)).json

([2, 2, 2], (1, 1))

([2, 2, 2], (1, 1)).txt

([2, 2, 2], (1, 1)).json

([2, 2, 2], (2, 0))

([2, 2, 2], (2, 0)).txt

([2, 2, 2], (2, 0)).json

([2, 2, 2, 1], (0, 1))

([2, 2, 2, 1], (0, 1)).txt

([2, 2, 2, 1], (0, 1)).json

([2, 2, 2, 1], (1, 0))

([2, 2, 2, 1], (1, 0)).txt

([2, 2, 2, 1], (1, 0)).json

([3], (0, 5))

([3], (0, 5)).txt

([3], (0, 5)).json

([3], (1, 4))

([3], (1, 4)).txt

([3], (1, 4)).json

([3], (2, 3))

([3], (2, 3)).txt

([3], (2, 3)).json

([3], (3, 2))

([3], (3, 2)).txt

([3], (3, 2)).json

([3], (4, 1))

([3], (4, 1)).txt

([3], (4, 1)).json

([3], (5, 0))

([3], (5, 0)).txt

([3], (5, 0)).json

([3, 1], (0, 4))

([3, 1], (0, 4)).txt

([3, 1], (0, 4)).json

([3, 1], (1, 3))

([3, 1], (1, 3)).txt

([3, 1], (1, 3)).json

([3, 1], (2, 2))

([3, 1], (2, 2)).txt

([3, 1], (2, 2)).json

([3, 1], (3, 1))

([3, 1], (3, 1)).txt

([3, 1], (3, 1)).json

([3, 1], (4, 0))

([3, 1], (4, 0)).txt

([3, 1], (4, 0)).json

([3, 1, 1], (0, 3))

([3, 1, 1], (0, 3)).txt

([3, 1, 1], (0, 3)).json

([3, 1, 1], (1, 2))

([3, 1, 1], (1, 2)).txt

([3, 1, 1], (1, 2)).json

([3, 1, 1], (2, 1))

([3, 1, 1], (2, 1)).txt

([3, 1, 1], (2, 1)).json

([3, 1, 1], (3, 0))

([3, 1, 1], (3, 0)).txt

([3, 1, 1], (3, 0)).json

([3, 1, 1, 1], (0, 2))

([3, 1, 1, 1], (0, 2)).txt

([3, 1, 1, 1], (0, 2)).json

([3, 1, 1, 1], (1, 1))

([3, 1, 1, 1], (1, 1)).txt

([3, 1, 1, 1], (1, 1)).json

([3, 1, 1, 1], (2, 0))

([3, 1, 1, 1], (2, 0)).txt

([3, 1, 1, 1], (2, 0)).json

([3, 1, 1, 1, 1], (0, 1))

([3, 1, 1, 1, 1], (0, 1)).txt

([3, 1, 1, 1, 1], (0, 1)).json

([3, 1, 1, 1, 1], (1, 0))

([3, 1, 1, 1, 1], (1, 0)).txt

([3, 1, 1, 1, 1], (1, 0)).json

([3, 2], (0, 3))

([3, 2], (0, 3)).txt

([3, 2], (0, 3)).json

([3, 2], (1, 2))

([3, 2], (1, 2)).txt

([3, 2], (1, 2)).json

([3, 2], (2, 1))

([3, 2], (2, 1)).txt

([3, 2], (2, 1)).json

([3, 2], (3, 0))

([3, 2], (3, 0)).txt

([3, 2], (3, 0)).json

([3, 2, 1], (0, 2))

([3, 2, 1], (0, 2)).txt

([3, 2, 1], (0, 2)).json

([3, 2, 1], (1, 1))

([3, 2, 1], (1, 1)).txt

([3, 2, 1], (1, 1)).json

([3, 2, 1], (2, 0))

([3, 2, 1], (2, 0)).txt

([3, 2, 1], (2, 0)).json

([3, 2, 1, 1], (0, 1))

([3, 2, 1, 1], (0, 1)).txt

([3, 2, 1, 1], (0, 1)).json

([3, 2, 1, 1], (1, 0))

([3, 2, 1, 1], (1, 0)).txt

([3, 2, 1, 1], (1, 0)).json

([3, 2, 2], (0, 1))

([3, 2, 2], (0, 1)).txt

([3, 2, 2], (0, 1)).json

([3, 2, 2], (1, 0))

([3, 2, 2], (1, 0)).txt

([3, 2, 2], (1, 0)).json

([3, 2, 2, 1], (0, 0))

([3, 2, 2, 1], (0, 0)).txt

([3, 2, 2, 1], (0, 0)).json

([3, 3], (0, 2))

([3, 3], (0, 2)).txt

([3, 3], (0, 2)).json

([3, 3], (1, 1))

([3, 3], (1, 1)).txt

([3, 3], (1, 1)).json

([3, 3], (2, 0))

([3, 3], (2, 0)).txt

([3, 3], (2, 0)).json

([3, 3, 1], (0, 1))

([3, 3, 1], (0, 1)).txt

([3, 3, 1], (0, 1)).json

([3, 3, 1], (1, 0))

([3, 3, 1], (1, 0)).txt

([3, 3, 1], (1, 0)).json

([3, 3, 1, 1], (0, 0))

([3, 3, 1, 1], (0, 0)).txt

([3, 3, 1, 1], (0, 0)).json

([3, 3, 2], (0, 0))

([3, 3, 2], (0, 0)).txt

([3, 3, 2], (0, 0)).json

([4], (0, 4))

([4], (0, 4)).txt

([4], (0, 4)).json

([4], (1, 3))

([4], (1, 3)).txt

([4], (1, 3)).json

([4], (2, 2))

([4], (2, 2)).txt

([4], (2, 2)).json

([4], (3, 1))

([4], (3, 1)).txt

([4], (3, 1)).json

([4], (4, 0))

([4], (4, 0)).txt

([4], (4, 0)).json

([4, 1], (0, 3))

([4, 1], (0, 3)).txt

([4, 1], (0, 3)).json

([4, 1], (1, 2))

([4, 1], (1, 2)).txt

([4, 1], (1, 2)).json

([4, 1], (2, 1))

([4, 1], (2, 1)).txt

([4, 1], (2, 1)).json

([4, 1], (3, 0))

([4, 1], (3, 0)).txt

([4, 1], (3, 0)).json

([4, 1, 1], (0, 2))

([4, 1, 1], (0, 2)).txt

([4, 1, 1], (0, 2)).json

([4, 1, 1], (1, 1))

([4, 1, 1], (1, 1)).txt

([4, 1, 1], (1, 1)).json

([4, 1, 1], (2, 0))

([4, 1, 1], (2, 0)).txt

([4, 1, 1], (2, 0)).json

([4, 1, 1, 1], (0, 1))

([4, 1, 1, 1], (0, 1)).txt

([4, 1, 1, 1], (0, 1)).json

([4, 1, 1, 1], (1, 0))

([4, 1, 1, 1], (1, 0)).txt

([4, 1, 1, 1], (1, 0)).json

([4, 1, 1, 1, 1], (0, 0))

([4, 1, 1, 1, 1], (0, 0)).txt

([4, 1, 1, 1, 1], (0, 0)).json

([4, 2], (0, 2))

([4, 2], (0, 2)).txt

([4, 2], (0, 2)).json

([4, 2], (1, 1))

([4, 2], (1, 1)).txt

([4, 2], (1, 1)).json

([4, 2], (2, 0))

([4, 2], (2, 0)).txt

([4, 2], (2, 0)).json

([4, 2, 1], (0, 1))

([4, 2, 1], (0, 1)).txt

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([4, 2, 1], (1, 0))

([4, 2, 1], (1, 0)).txt

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([4, 2, 2], (0, 0))

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([4, 3], (0, 1))

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([4, 4], (0, 0))

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([5], (0, 3))

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([5, 1], (0, 2))

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([5, 2], (0, 1))

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([5, 3], (0, 0))

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([6], (0, 2))

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([6, 1], (0, 1))

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([6, 1, 1], (0, 0))

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([6, 2], (0, 0))

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([7], (0, 1))

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([7], (1, 0))

([7], (1, 0)).txt

([7], (1, 0)).json

([7, 1], (0, 0))

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([8], (0, 0))

([8], (0, 0)).txt

([8], (0, 0)).json