Base field \(\Q(\sqrt{77}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, -1, 1]))
gp: K = nfinit(Polrev([-19, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,0]),K([15430,3967]),K([4933819,1269148])])
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([15430,3967]),Polrev([4933819,1269148])], K);
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![1,0],K![15430,3967],K![4933819,1269148]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a+1)\) | = | \((a+3)\cdot(a-6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 77 \) | = | \(7\cdot11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-115539436859)\) | = | \((a+3)^{4}\cdot(a-6)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 13349361469694847785881 \) | = | \(7^{4}\cdot11^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{9463555063808}{115539436859} \) | ||
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: | \(0\) |
| 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.60288154826947420090237316500130575727 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.0992756604426931422383433677754557934 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(7\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((a-6)\) | \(11\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3 and 9.
Its isogeny class
77.1-j
consists of curves linked by isogenies of
degrees dividing 9.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 539.b3 |
| \(\Q\) | 847.c3 |