Base field 4.4.12400.1
Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 0, -12, 0, 1]))
gp: K = nfinit(Polrev([31, 0, -12, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5/2,-5/2,1/2,1/2]),K([-5/2,7/2,1/2,-1/2]),K([1,-5/2,0,1/2]),K([9/2,-1,-1/2,0]),K([-11/2,0,1/2,0])])
gp: E = ellinit([Polrev([-5/2,-5/2,1/2,1/2]),Polrev([-5/2,7/2,1/2,-1/2]),Polrev([1,-5/2,0,1/2]),Polrev([9/2,-1,-1/2,0]),Polrev([-11/2,0,1/2,0])], K);
magma: E := EllipticCurve([K![-5/2,-5/2,1/2,1/2],K![-5/2,7/2,1/2,-1/2],K![1,-5/2,0,1/2],K![9/2,-1,-1/2,0],K![-11/2,0,1/2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1/2a^3-7/2a-1)\) | = | \((1/2a^3-7/2a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 19 \) | = | \(19\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^3+9/2a^2-12a-41/2)\) | = | \((1/2a^3-7/2a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -6859 \) | = | \(-19^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{43567442560}{6859} a^{3} + \frac{129020350240}{6859} a^{2} - \frac{163951874560}{6859} a - \frac{485555172640}{6859} \) | ||
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 | $\left(-\frac{1}{2} a^{3} - 2 a^{2} + \frac{7}{2} a + 12 : -\frac{3}{2} a^{3} - \frac{5}{2} a^{2} + \frac{29}{2} a + \frac{49}{2} : 1\right)$ |
| Height | \(0.18803588306485390521930469202551597644\) |
| 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: | $\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{7}{4} a + \frac{1}{4} : \frac{1}{8} a^{3} + \frac{1}{8} a^{2} - \frac{5}{8} a - \frac{17}{8} : 1\right)$ |
| 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: | \( 0.18803588306485390521930469202551597644 \) | ||
| Period: | \( 453.85755379731274289934979975150364768 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.29916804176374 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/2a^3-7/2a-1)\) | \(19\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
19.1-a
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.