Base field \(\Q(\sqrt{3}, \sqrt{7})\)
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([1, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,1,1,0]),K([-3,4,1,-1]),K([1,0,0,0]),K([-47,106,11,-22]),K([213,-467,-43,98])])
gp: E = ellinit([Polrev([-3,1,1,0]),Polrev([-3,4,1,-1]),Polrev([1,0,0,0]),Polrev([-47,106,11,-22]),Polrev([213,-467,-43,98])], K);
magma: E := EllipticCurve([K![-3,1,1,0],K![-3,4,1,-1],K![1,0,0,0],K![-47,106,11,-22],K![213,-467,-43,98]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-4a)\) | = | \((-a-1)\cdot(-a^3+5a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-9a^2+9a+18)\) | = | \((-a-1)^{6}\cdot(-a^3+5a-1)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 59049 \) | = | \(3^{6}\cdot3^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4301312}{27} a^{3} - \frac{8602624}{9} a + \frac{11388736}{27} \) | ||
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 | $\left(6 a^{3} - 3 a^{2} - 29 a + 13 : 37 a^{3} - 16 a^{2} - 176 a + 79 : 1\right)$ |
| Height | \(0.077414024668015782569919519312503639504\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a + 1 : 4 a^{3} - 4 a^{2} - 23 a + 10 : 1\right)$ |
| 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: | \( 0.077414024668015782569919519312503639504 \) | ||
| Period: | \( 1179.9840110213103214505835189896997572 \) | ||
| Tamagawa product: | \( 12 \) = \(( 2 \cdot 3 )\cdot2\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.44995732281060 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((-a^3+5a-1)\) | \(3\) | \(2\) | \(I_{4}\) | Non-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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.3-a
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{7}) \) | 2.2.28.1-27.2-a4 |
| \(\Q(\sqrt{7}) \) | 2.2.28.1-27.1-b4 |