Base field \(\Q(\sqrt{3}, \sqrt{5})\)
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 8, -7, -2, 1]))
gp: K = nfinit(Polrev([1, 8, -7, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -7, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-13/7,31/7,6/7,-4/7]),K([-2/7,-20/7,2/7,1/7]),K([-2/7,-13/7,2/7,1/7]),K([13/7,74/7,-41/7,-17/7]),K([-1,-9,6,2])])
gp: E = ellinit([Polrev([-13/7,31/7,6/7,-4/7]),Polrev([-2/7,-20/7,2/7,1/7]),Polrev([-2/7,-13/7,2/7,1/7]),Polrev([13/7,74/7,-41/7,-17/7]),Polrev([-1,-9,6,2])], K);
magma: E := EllipticCurve([K![-13/7,31/7,6/7,-4/7],K![-2/7,-20/7,2/7,1/7],K![-2/7,-13/7,2/7,1/7],K![13/7,74/7,-41/7,-17/7],K![-1,-9,6,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((5/7a^3-4/7a^2-37/7a+11/7)\) | = | \((5/7a^3-4/7a^2-37/7a+11/7)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 59 \) | = | \(59\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((113/7a^3-75/7a^2-111/7a-541/7)\) | = | \((5/7a^3-4/7a^2-37/7a+11/7)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -714924299 \) | = | \(-59^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{812095893390}{5004470093} a^{3} - \frac{2396259161650}{5004470093} a^{2} + \frac{1748065265525}{5004470093} a - \frac{653835118135}{5004470093} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-\frac{5}{7} a^{3} + \frac{4}{7} a^{2} + \frac{37}{7} a + \frac{3}{7} : -\frac{6}{7} a^{3} - \frac{5}{7} a^{2} + \frac{43}{7} a + \frac{5}{7} : 1\right)$ |
| Height | \(0.020973006448587162977468189008028429003\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.020973006448587162977468189008028429003 \) | ||
| Period: | \( 329.30956782135054087063619922912416878 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.30220389649954 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((5/7a^3-4/7a^2-37/7a+11/7)\) | \(59\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
59.3-a
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.