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(Pol(Vecrev([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([-1,0,0,0]),K([0,-3,0,1]),K([-2,0,0,0]),K([0,0,0,0])])
gp: E = ellinit([Pol(Vecrev([0,-3,0,1])),Pol(Vecrev([-1,0,0,0])),Pol(Vecrev([0,-3,0,1])),Pol(Vecrev([-2,0,0,0])),Pol(Vecrev([0,0,0,0]))], K);
magma: E := EllipticCurve([K![0,-3,0,1],K![-1,0,0,0],K![0,-3,0,1],K![-2,0,0,0],K![0,0,0,0]]);
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: | \( \frac{35152}{9} \) | ||
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: | \(1\) | |
| Generator | $\left(a + 1 : -a^{3} + 4 a + 1 : 1\right)$ | |
| Height | \(0.269818466169295\) | |
| Torsion structure: | \(\Z/2\Z\times\Z/8\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{3} - a^{2} - 7 a + 4 : 3 a^{3} - 2 a^{2} - 12 a + 7 : 1\right)$ | $\left(\frac{1}{2} : -\frac{3}{4} a^{3} + \frac{9}{4} a : 1\right)$ |
| 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: | \( 0.269818466169295 \) | ||
| Period: | \( 1384.17317605613 \) | ||
| Tamagawa product: | \( 16 \) = \(2^{2}\cdot2^{2}\) | ||
| Torsion order: | \(16\) | ||
| Leading coefficient: | \( 1.94518480872993 \) | ||
| 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 and 8.
Its isogeny class
72.1-b
consists of curves linked by isogenies of
degrees dividing 32.
Base change
This curve is the base change of elliptic curves 24.a4, 144.b4, 192.d4, 576.b4, defined over \(\Q\), so it is also a \(\Q\)-curve.