Base field 5.5.179024.1
Generator \(a\), with minimal polynomial \( x^{5} - 8 x^{3} + 6 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 6, 0, -8, 0, 1]))
gp: K = nfinit(Polrev([-2, 6, 0, -8, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 6, 0, -8, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([16,-15,-24,2,3]),K([14,-16,-24,2,3]),K([5,-7,-8,1,1]),K([73,-117,-125,16,16]),K([-1754,2050,3815,-266,-497])])
gp: E = ellinit([Polrev([16,-15,-24,2,3]),Polrev([14,-16,-24,2,3]),Polrev([5,-7,-8,1,1]),Polrev([73,-117,-125,16,16]),Polrev([-1754,2050,3815,-266,-497])], K);
magma: E := EllipticCurve([K![16,-15,-24,2,3],K![14,-16,-24,2,3],K![5,-7,-8,1,1],K![73,-117,-125,16,16],K![-1754,2050,3815,-266,-497]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1)\) | = | \((1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1 \) | = | 1 |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 1414198780324 a^{4} + 3815036657888 a^{3} - 1021893446812 a^{2} - 2756727621312 a + 1048460789992 \) | ||
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 | $\left(5 a^{4} + 2 a^{3} - 38 a^{2} - 16 a + 18 : -23 a^{4} - 12 a^{3} + 177 a^{2} + 91 a - 79 : 1\right)$ |
| Height | \(0.068291357706951455053523047916884818075\) |
| 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: | $\left(\frac{5}{2} a^{4} + \frac{3}{2} a^{3} - \frac{73}{4} a^{2} - 11 a + 3 : \frac{31}{4} a^{4} + \frac{37}{8} a^{3} - \frac{245}{4} a^{2} - \frac{141}{4} a + \frac{81}{2} : 1\right)$ |
| 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: | \( 0.068291357706951455053523047916884818075 \) | ||
| Period: | \( 4289.8026220562702952033230747474306985 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.865480808 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
No primes of bad reduction.
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
1.1-a
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.