Base field 4.4.10025.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 11 x^{2} + 10 x + 20 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([20, 10, -11, -1, 1]))
gp: K = nfinit(Polrev([20, 10, -11, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 10, -11, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-7,-5/2,3/2,1/2]),K([4,9/2,-1/2,-1/2]),K([-7,-5/2,3/2,1/2]),K([125,113,-29,-17]),K([1462,2563/2,-653/2,-393/2])])
gp: E = ellinit([Polrev([-7,-5/2,3/2,1/2]),Polrev([4,9/2,-1/2,-1/2]),Polrev([-7,-5/2,3/2,1/2]),Polrev([125,113,-29,-17]),Polrev([1462,2563/2,-653/2,-393/2])], K);
magma: E := EllipticCurve([K![-7,-5/2,3/2,1/2],K![4,9/2,-1/2,-1/2],K![-7,-5/2,3/2,1/2],K![125,113,-29,-17],K![1462,2563/2,-653/2,-393/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3-2a^2+7a+10)\) | = | \((-2a^3-3a^2+13a+14)\cdot(-a^3-a^2+8a+5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a)\) | = | \((-2a^3-3a^2+13a+14)\cdot(-a^3-a^2+8a+5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 20 \) | = | \(4\cdot5\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{108228121969295379}{20} a^{3} + \frac{179433515162009521}{20} a^{2} - \frac{713589500604580069}{20} a - \frac{81438223753542063}{2} \) | ||
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: | \(0\) |
| 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: | $\left(-\frac{9}{8} a^{3} - \frac{15}{8} a^{2} + \frac{49}{8} a + 7 : 5 a^{3} + \frac{33}{4} a^{2} - 32 a - \frac{281}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 120.91066369813436053421903100352375035 \) | ||
| Tamagawa product: | \( 1 \) = \(1\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.20759808163784 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^3-3a^2+13a+14)\) | \(4\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((-a^3-a^2+8a+5)\) | \(5\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
20.1-d
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.