Base field \(\Q(\sqrt{7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
gp: K = nfinit(Polrev([-7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-320,-115]),K([-3392,-1198])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-320,-115]),Polrev([-3392,-1198])], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-320,-115],K![-3392,-1198]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((3a)\) | = | \((-a+2)\cdot(-a-2)\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 63 \) | = | \(3\cdot3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((77193a+131544)\) | = | \((-a+2)^{4}\cdot(-a-2)^{16}\cdot(a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -24407490807 \) | = | \(-3^{4}\cdot3^{16}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1153486390269896663}{301327047} a + \frac{435976874792639720}{43046721} \) | ||
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 | $\left(\frac{22}{9} a + \frac{391}{18} : -\frac{2951}{108} a + \frac{1157}{108} : 1\right)$ |
| Height | \(2.9851968347780548356377504145262127076\) |
| 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: | $\left(-4 a - \frac{19}{4} : \frac{15}{8} a + 14 : 1\right)$ |
| 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: | \( 2.9851968347780548356377504145262127076 \) | ||
| Period: | \( 0.91297048553448105560065052163650303928 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.0301030910447432370746231996231031830 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(3\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-a-2)\) | \(3\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
| \((a)\) | \(7\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
63.1-b
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.