Base field \(\Q(\sqrt{-31}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 8 \); class number \(3\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -1, 1]))
gp: K = nfinit(Polrev([8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,1]),K([-11,1]),K([-9,0])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,1]),Polrev([-11,1]),Polrev([-9,0])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,1],K![-11,1],K![-9,0]]);
This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a-6)\) | = | \((2,a)\cdot(5,a+1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 50 \) | = | \(2\cdot5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((-1796a+36324)\) | = | \((2,a)^{2}\cdot(2,a+1)^{12}\cdot(5,a+1)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1280000000 \) | = | \(2^{2}\cdot2^{12}\cdot5^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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Minimal discriminant: | \((151a-444)\) | = | \((2,a)^{2}\cdot(5,a+1)^{7}\) |
Minimal discriminant norm: | \( 312500 \) | = | \(2^{2}\cdot5^{7}\) |
j-invariant: | \( \frac{2911}{20} a - \frac{193027}{20} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(0\) |
Torsion structure: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-1 : 0 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 0 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(0\) | ||
Regulator: | \( 1 \) | ||
Period: | \( 4.3378671445693905396920407035935352107 \) | ||
Tamagawa product: | \( 4 \) = \(2\cdot1\cdot2\) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.5582078773048180198833341099963290398 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
Primes of good reduction for the curve but which divide the
discriminant of the model above (if any) are included.
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2,a)\) | \(2\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
\((2,a+1)\) | \(2\) | \(1\) | \(I_0\) | Good | \(1\) | \(0\) | \(0\) | \(0\) |
\((5,a+1)\) | \(5\) | \(2\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(7\) | 7B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
50.1-a
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.