Base field 4.4.4205.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,4,2,-1]),K([1,9,3,-2]),K([1,0,0,0]),K([27,-617,-773,345]),K([2678,-7188,-13899,5642])])
gp: E = ellinit([Polrev([0,4,2,-1]),Polrev([1,9,3,-2]),Polrev([1,0,0,0]),Polrev([27,-617,-773,345]),Polrev([2678,-7188,-13899,5642])], K);
magma: E := EllipticCurve([K![0,4,2,-1],K![1,9,3,-2],K![1,0,0,0],K![27,-617,-773,345],K![2678,-7188,-13899,5642]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^3+3a^2+8a)\) | = | \((-2a^3+3a^2+8a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(25\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-2a^3+2a^2+12a-1)\) | = | \((-2a^3+3a^2+8a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -625 \) | = | \(-25^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{246199023533429793802832076808}{25} a^{3} + \frac{453138213764112525371698488272}{25} a^{2} + \frac{56160379457348258811003590942}{25} a - \frac{86673433010725415101151339157}{25} \) | ||
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 | $\left(-\frac{2879209}{41405} a^{3} + \frac{6522676}{41405} a^{2} + \frac{4661296}{41405} a - \frac{48947}{8281} : -\frac{24227978807}{18839275} a^{3} + \frac{2372212099}{753571} a^{2} + \frac{6233538173}{3767855} a - \frac{11388321673}{18839275} : 1\right)$ |
| Height | \(2.9002291376893457920392594114680327588\) |
| 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: | $\left(-10 a^{3} + \frac{37}{2} a^{2} + \frac{91}{4} a + \frac{13}{2} : \frac{95}{8} a^{3} - \frac{177}{8} a^{2} - \frac{201}{8} a - \frac{21}{8} : 1\right)$ |
| 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: | \( 2.9002291376893457920392594114680327588 \) | ||
| Period: | \( 2.3611206100625969270973820858952378634 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.90081560465370 \) | ||
| Analytic order of Ш: | \( 9 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^3+3a^2+8a)\) | \(25\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 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
25.2-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.