Base field \(\Q(\zeta_{13})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,3,-3,-4,1,1]),K([2,5,-4,-5,1,1]),K([1,5,-3,-5,1,1]),K([1051,4023,-4471,-7561,962,1801]),K([46396,170108,-170680,-301077,36003,70970])])
gp: E = ellinit([Polrev([0,3,-3,-4,1,1]),Polrev([2,5,-4,-5,1,1]),Polrev([1,5,-3,-5,1,1]),Polrev([1051,4023,-4471,-7561,962,1801]),Polrev([46396,170108,-170680,-301077,36003,70970])], K);
magma: E := EllipticCurve([K![0,3,-3,-4,1,1],K![2,5,-4,-5,1,1],K![1,5,-3,-5,1,1],K![1051,4023,-4471,-7561,962,1801],K![46396,170108,-170680,-301077,36003,70970]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 64 \) | = | \(64\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-16384)\) | = | \((2)^{14}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 19342813113834066795298816 \) | = | \(64^{14}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{38575685889}{16384} \) | ||
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 | $\left(-\frac{264}{13} a^{5} - \frac{161}{13} a^{4} + \frac{1056}{13} a^{3} + 53 a^{2} - \frac{409}{13} a + \frac{16}{13} : \frac{1490}{13} a^{5} + \frac{645}{13} a^{4} - \frac{6527}{13} a^{3} - 254 a^{2} + \frac{4412}{13} a + 112 : 1\right)$ |
| Height | \(1.1887004589488550673030416174013041568\) |
| 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: | \( 1.1887004589488550673030416174013041568 \) | ||
| Period: | \( 0.31322469557395309030833421932615461695 \) | ||
| Tamagawa product: | \( 14 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.51505 \) | ||
| Analytic order of Ш: | \( 49 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(64\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
64.1-b
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 338.c1 |
| \(\Q\) | 338.e1 |
| \(\Q(\sqrt{13}) \) | 2.2.13.1-676.1-b1 |
| 3.3.169.1 | a curve with conductor norm 1352 (not in the database) |
| 3.3.169.1 | 3.3.169.1-8.1-a1 |