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([1,-7/2,0,1/2]),K([7/2,-7/2,-1/2,1/2]),K([1,-5/2,0,1/2]),K([1009,421,-280,-106]),K([-41841/2,-6690,10915/2,1829])])
gp: E = ellinit([Polrev([1,-7/2,0,1/2]),Polrev([7/2,-7/2,-1/2,1/2]),Polrev([1,-5/2,0,1/2]),Polrev([1009,421,-280,-106]),Polrev([-41841/2,-6690,10915/2,1829])], K);
magma: E := EllipticCurve([K![1,-7/2,0,1/2],K![7/2,-7/2,-1/2,1/2],K![1,-5/2,0,1/2],K![1009,421,-280,-106],K![-41841/2,-6690,10915/2,1829]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/2a^3+1/2a^2+7/2a-7/2)\) | = | \((-a+3)\cdot(-1/2a^3-3/2a^2+3/2a+11/2)\) |
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: | \((100a^3-176a^2+12a-816)\) | = | \((-a+3)^{6}\cdot(-1/2a^3-3/2a^2+3/2a+11/2)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1000000000000 \) | = | \(4^{6}\cdot5^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2028817931053}{250000} a^{3} + \frac{3032952547803}{125000} a^{2} - \frac{7629088540237}{250000} a - \frac{1142292115229}{12500} \) | ||
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 | $\left(-\frac{7}{2} a^{3} - 9 a^{2} + \frac{67}{2} a + 89 : -66 a^{3} - \frac{215}{2} a^{2} + 536 a + \frac{1731}{2} : 1\right)$ |
| Height | \(0.50398168699254402461364858193911517982\) |
| 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: | $\left(\frac{3}{2} a^{3} + 2 a^{2} - \frac{15}{2} a - 2 : -3 a^{3} - \frac{5}{2} a^{2} + 17 a + \frac{1}{2} : 1\right)$ |
| 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: | \( 0.50398168699254402461364858193911517982 \) | ||
| Period: | \( 22.841198040814572663103017098452791424 \) | ||
| Tamagawa product: | \( 24 \) = \(2\cdot( 2^{2} \cdot 3 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.48104153222950 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+3)\) | \(4\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((-1/2a^3-3/2a^2+3/2a+11/2)\) | \(5\) | \(12\) | \(I_{12}\) | Split multiplicative | \(-1\) | \(1\) | \(12\) | \(12\) |
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
20.2-d
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.