Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, -4, 0, 1]))
gp: K = nfinit(Polrev([1, 0, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-3,0,1]),K([0,-3,0,1]),K([1,-3,0,1]),K([-243,486,0,-162]),K([-2130,4485,0,-1495])])
gp: E = ellinit([Polrev([0,-3,0,1]),Polrev([0,-3,0,1]),Polrev([1,-3,0,1]),Polrev([-243,486,0,-162]),Polrev([-2130,4485,0,-1495])], K);
magma: E := EllipticCurve([K![0,-3,0,1],K![0,-3,0,1],K![1,-3,0,1],K![-243,486,0,-162],K![-2130,4485,0,-1495]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2)\) | = | \((a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((243)\) | = | \((a^2-2)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 3486784401 \) | = | \(9^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{58591911104}{243} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(6 a^{3} - \frac{9}{2} a^{2} - 27 a + \frac{25}{2} : -\frac{9}{2} a^{3} + \frac{9}{2} a^{2} + 18 a - 11 : 1\right)$ | $\left(-3 a^{3} + \frac{9}{2} a^{2} + 18 a - \frac{11}{2} : -\frac{9}{2} a^{2} - \frac{9}{2} a + 7 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 6.4203790231163829729727767338864631532 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.417993425984140 \) | ||
| Analytic order of Ш: | \( 25 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-2)\) | \(9\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 20.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q(\sqrt{2}) \) | 2.2.8.1-9.1-a4 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-1296.1-c4 |