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([1,1]),K([-1,1]),K([0,1]),K([137,36]),K([-2411,-619])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([0,1]),Polrev([137,36]),Polrev([-2411,-619])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![0,1],K![137,36],K![-2411,-619]]);
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: | \((-41503)\) | = | \((a+3)^{6}\cdot(a-6)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1722499009 \) | = | \(7^{6}\cdot11^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4657463}{41503} \) | ||
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: | \(2\) | |
| Generators | $\left(\frac{27}{16} a + \frac{31}{4} : \frac{57}{16} a + \frac{1113}{64} : 1\right)$ | $\left(11 a + 44 : -152 a - 587 : 1\right)$ |
| Heights | \(1.4803018111773844322220459505544877205\) | \(0.29598217557507840705821871917140265422\) |
| 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: | $\left(a + 5 : -4 a - 12 : 1\right)$ | |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 2 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(2\) | ||
| Regulator: | \( 0.43814295058001116259750755253776824996 \) | ||
| Period: | \( 3.2101873408696119145423660433455397356 \) | ||
| Tamagawa product: | \( 24 \) = \(( 2 \cdot 3 )\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.8469105277667522036274761400779848577 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(7\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((a-6)\) | \(11\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
77.1-d
consists of curves linked by isogenies of
degree 2.
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.d2 |
| \(\Q\) | 847.a2 |