Base field 4.4.9248.1
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([-4,-5,1,1]),K([1,0,0,0]),K([-275,-427,43,92]),K([-1064,-1654,162,353])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([-4,-5,1,1]),Polrev([1,0,0,0]),Polrev([-275,-427,43,92]),Polrev([-1064,-1654,162,353])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![-4,-5,1,1],K![1,0,0,0],K![-275,-427,43,92],K![-1064,-1654,162,353]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-3a)\) | = | \((a)\cdot(a^2+a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 26 \) | = | \(2\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-24a^3-75a^2+64a+194)\) | = | \((a)^{6}\cdot(a^2+a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -308915776 \) | = | \(-2^{6}\cdot13^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{13179827674651809978387}{38614472} a^{3} + \frac{28149201874182484162433}{38614472} a^{2} - \frac{5778658371471464319445}{38614472} a - \frac{12341938383390744470007}{38614472} \) | ||
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 | $\left(\frac{11}{2} a^{3} + 7 a^{2} - 12 a - 9 : -\frac{137}{4} a^{3} - \frac{239}{4} a^{2} + \frac{105}{2} a + \frac{101}{2} : 1\right)$ |
| Height | \(1.0485833041334571638199253867261092402\) |
| 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: | $\left(\frac{3}{4} a^{2} - \frac{7}{2} a - \frac{9}{4} : -\frac{3}{8} a^{3} + \frac{11}{8} a^{2} + \frac{23}{8} a + \frac{5}{8} : 1\right)$ |
| 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: | \( 1.0485833041334571638199253867261092402 \) | ||
| Period: | \( 14.243818376744087536939896535227502902 \) | ||
| Tamagawa product: | \( 12 \) = \(2\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.86374590102889 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((a^2+a-3)\) | \(13\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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
26.4-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.