Base field \(\Q(\sqrt{5}, \sqrt{21})\)
Generator \(a\), with minimal polynomial \( x^{4} - 13 x^{2} + 16 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([16, 0, -13, 0, 1]))
gp: K = nfinit(Polrev([16, 0, -13, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 0, -13, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1/2,17/8,0,-1/8]),K([-3,1/2,1/2,0]),K([-3/2,13/8,1/2,-1/8]),K([-2303/2,-7751/8,199/2,667/8]),K([39273/2,133733/8,-3379/2,-11505/8])])
gp: E = ellinit([Polrev([1/2,17/8,0,-1/8]),Polrev([-3,1/2,1/2,0]),Polrev([-3/2,13/8,1/2,-1/8]),Polrev([-2303/2,-7751/8,199/2,667/8]),Polrev([39273/2,133733/8,-3379/2,-11505/8])], K);
magma: E := EllipticCurve([K![1/2,17/8,0,-1/8],K![-3,1/2,1/2,0],K![-3/2,13/8,1/2,-1/8],K![-2303/2,-7751/8,199/2,667/8],K![39273/2,133733/8,-3379/2,-11505/8]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-1/8a^3+1/2a^2+5/8a-7/2)\) | = | \((1/8a^3+1/2a^2+3/8a-1/2)\cdot(1/4a^3+1/2a^2-11/4a-6)\) |
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: | \((-31/4a^3+4a^2+403/4a-68)\) | = | \((1/8a^3+1/2a^2+3/8a-1/2)^{2}\cdot(1/4a^3+1/2a^2-11/4a-6)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6250000 \) | = | \(4^{2}\cdot5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{27629167345039}{2500} a^{3} + \frac{188393625220043}{5000} a^{2} - \frac{76063618383649}{5000} a - \frac{64831851375249}{1250} \) | ||
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 | $\left(-a^{3} - \frac{1}{2} a^{2} + \frac{25}{2} a + 17 : \frac{3}{2} a^{3} + \frac{3}{2} a^{2} - 8 a - 6 : 1\right)$ | |
| Height | \(2.0225803106623964293130006071963131149\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{3} + \frac{5}{2} a^{2} - \frac{47}{2} a - 31 : -3 a^{3} - \frac{7}{2} a^{2} + \frac{71}{2} a + 40 : 1\right)$ | $\left(-\frac{31}{32} a^{3} - \frac{3}{2} a^{2} + \frac{351}{32} a + \frac{125}{8} : \frac{53}{32} a^{3} + \frac{3}{2} a^{2} - \frac{597}{32} a - 18 : 1\right)$ |
| 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: | \( 2.0225803106623964293130006071963131149 \) | ||
| Period: | \( 169.37377029198122079936708055746809081 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.26259098033540 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((1/8a^3+1/2a^2+3/8a-1/2)\) | \(4\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((1/4a^3+1/2a^2-11/4a-6)\) | \(5\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
20.2-b
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.