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([1,1,0,0]),K([-2,0,1,0]),K([-1,0,1,0]),K([1,16,-40,16]),K([-42,-16,249,-119])])
gp: E = ellinit([Pol(Vecrev([1,1,0,0])),Pol(Vecrev([-2,0,1,0])),Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([1,16,-40,16])),Pol(Vecrev([-42,-16,249,-119]))], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-2,0,1,0],K![-1,0,1,0],K![1,16,-40,16],K![-42,-16,249,-119]]);
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: | \((6)\) | = | \((a^3-4a+1)^{4}\cdot(a^2-2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1296 \) | = | \(2^{4}\cdot9^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{28756228}{3} \) | ||
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 | $\left(-13 a^{3} + 26 a^{2} + a - 5 : -193 a^{3} + 372 a^{2} + 52 a - 100 : 1\right)$ | |
| Height | \(0.539636932338590\) | |
| 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: | $\left(2 a^{3} - 4 a^{2} - 3 a + 3 : a^{3} - a^{2} : 1\right)$ | $\left(3 a^{2} - 8 a + 4 : 4 a^{3} - 17 a^{2} + 21 a - 7 : 1\right)$ |
| 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: | \( 0.539636932338590 \) | ||
| Period: | \( 2768.34635211226 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-4a+1)\) | \(2\) | \(2\) | \(III\) | Additive | \(1\) | \(3\) | \(4\) | \(0\) |
| \((a^2-2)\) | \(9\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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-b
consists of curves linked by isogenies of
degrees dividing 32.
Base change
This curve is the base change of 24.a2, 144.b2, 192.d2, 576.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.