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([-2,-4,1,1]),K([3,-3,-1,1]),K([-2,1,1,0]),K([6,-97,-242,-96]),K([-897,-190,4017,2001])])
gp: E = ellinit([Polrev([-2,-4,1,1]),Polrev([3,-3,-1,1]),Polrev([-2,1,1,0]),Polrev([6,-97,-242,-96]),Polrev([-897,-190,4017,2001])], K);
magma: E := EllipticCurve([K![-2,-4,1,1],K![3,-3,-1,1],K![-2,1,1,0],K![6,-97,-242,-96],K![-897,-190,4017,2001]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+a^2+5a-5)\) | = | \((a^3-4a+1)^{3}\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 72 \) | = | \(2^{3}\cdot9\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((36)\) | = | \((a^3-4a+1)^{8}\cdot(a^2-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1679616 \) | = | \(2^{8}\cdot9^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3065617154}{9} \) | ||
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: | \(\Z/2\Z\oplus\Z/8\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(6 a^{3} + 5 a^{2} - 15 a - 9 : 6 a^{3} + 2 a^{2} - 28 a - 13 : 1\right)$ | $\left(3 a^{2} + 6 a + 1 : -4 a^{3} - 3 a^{2} + 10 a + 6 : 1\right)$ |
| 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: | \( 1033.6726101924187563959078007327281429 \) | ||
| Tamagawa product: | \( 16 \) = \(2^{2}\cdot2^{2}\) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 1.34592787785471 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a+1)\) | \(2\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(3\) | \(8\) | \(0\) |
| \((a^2-2)\) | \(9\) | \(4\) | \(I_{4}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
72.1-a
consists of curves linked by isogenies of
degrees dividing 32.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 10 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 48.a1 |
| \(\Q\) | 72.a1 |
| \(\Q\) | 192.b1 |
| \(\Q\) | 576.d1 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-24.1-b7 |
| \(\Q(\sqrt{3}) \) | 2.2.12.1-768.1-e7 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-48.1-b6 |
| \(\Q(\sqrt{6}) \) | 2.2.24.1-24.1-a6 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-144.1-b7 |
| \(\Q(\sqrt{2}) \) | 2.2.8.1-648.1-a7 |