Base field 4.4.16448.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -6, -2, 1]))
gp: K = nfinit(Polrev([2, 0, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,-3,-3,1]),K([3,2,-1,0]),K([2,-7,-5,2]),K([-26,21,11,-4]),K([-63,92,51,-20])])
gp: E = ellinit([Polrev([2,-3,-3,1]),Polrev([3,2,-1,0]),Polrev([2,-7,-5,2]),Polrev([-26,21,11,-4]),Polrev([-63,92,51,-20])], K);
magma: E := EllipticCurve([K![2,-3,-3,1],K![3,2,-1,0],K![2,-7,-5,2],K![-26,21,11,-4],K![-63,92,51,-20]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((a)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2)\) | = | \((a)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 6918173802044 a^{3} + 11184587375248 a^{2} - 1057784853504 a - 3825686656612 \) | ||
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 | $\left(-\frac{306}{25} a^{3} + \frac{4179}{100} a^{2} + \frac{367}{25} a - \frac{921}{50} : \frac{88491}{1000} a^{3} - \frac{38242}{125} a^{2} - \frac{10739}{125} a + \frac{61289}{500} : 1\right)$ |
| Height | \(3.6438702911844038521845942785890012136\) |
| 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: | $\left(\frac{3}{4} a^{2} - a - \frac{11}{2} : \frac{7}{8} a^{3} - \frac{11}{4} a^{2} - 3 a + \frac{11}{4} : 1\right)$ |
| 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: | \( 3.6438702911844038521845942785890012136 \) | ||
| Period: | \( 127.09572790998066739791362414628822842 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.61108169170893 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(II\) | Additive | \(-1\) | \(4\) | \(4\) | \(0\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
16.1-c
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.