Base field 4.4.14272.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 5 x^{2} + 2 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 2, -5, -2, 1]))
gp: K = nfinit(Polrev([3, 2, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 2, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,-5,-5,2]),K([5,-6,-5,2]),K([5,-6,-5,2]),K([145,-13,-230,68]),K([1132,37,-1968,568])])
gp: E = ellinit([Polrev([5,-5,-5,2]),Polrev([5,-6,-5,2]),Polrev([5,-6,-5,2]),Polrev([145,-13,-230,68]),Polrev([1132,37,-1968,568])], K);
magma: E := EllipticCurve([K![5,-5,-5,2],K![5,-6,-5,2],K![5,-6,-5,2],K![145,-13,-230,68],K![1132,37,-1968,568]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-2a-3)\) | = | \((-a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^3+7a^2+4a-12)\) | = | \((-a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 729 \) | = | \(3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -74976000 a^{3} + 87518656 a^{2} + 436386432 a + 243770624 \) | ||
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 | $\left(\frac{9011}{729} a^{3} - \frac{51269}{1458} a^{2} - \frac{23764}{729} a + \frac{35623}{729} : -\frac{2188504}{19683} a^{3} + \frac{23606897}{78732} a^{2} + \frac{27604583}{78732} a - \frac{37148987}{78732} : 1\right)$ |
| Height | \(2.3836941905054172637479801010189183469\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(2 a^{3} - 7 a^{2} - a + \frac{11}{2} : \frac{3}{2} a^{3} - \frac{25}{4} a^{2} + \frac{25}{4} a + \frac{1}{4} : 1\right)$ |
| 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: | \( 2.3836941905054172637479801010189183469 \) | ||
| Period: | \( 85.715109933751011930852567450826509090 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.42054652294683 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-b
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.