Base field \(\Q(\sqrt{161}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 40 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-40, -1, 1]))
gp: K = nfinit(Polrev([-40, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-40, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([1,0]),K([-137480,20093]),K([10787904,-1576182])])
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([1,0]),Polrev([-137480,20093]),Polrev([10787904,-1576182])], K);
magma: E := EllipticCurve([K![0,1],K![0,0],K![1,0],K![-137480,20093],K![10787904,-1576182]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-5a-29)\) | = | \((-a+7)\cdot(-32a+219)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a-3)\) | = | \((-a+7)^{2}\cdot(-32a+219)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -28 \) | = | \(-2^{2}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3389}{28} a + \frac{9710}{7} \) | ||
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 | $\left(-\frac{21141}{361} a + \frac{143286}{361} : -\frac{9536562}{6859} a + \frac{65359176}{6859} : 1\right)$ |
| Height | \(2.6945850653786047148313932070280977634\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-52 a + 352 : 778 a - 5312 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 2.6945850653786047148313932070280977634 \) | ||
| Period: | \( 27.300496340440277365541874410316728391 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.5767185567531219512004070367965454233 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+7)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-32a+219)\) | \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
14.1-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.