Base field \(\Q(\zeta_{16})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 4 x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))
gp: K = nfinit(Polrev([2, 0, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,-3,0,1]),K([2,3,-1,-1]),K([1,0,0,0]),K([-455,-420,112,112]),K([-4038,-4334,1100,1225])])
gp: E = ellinit([Polrev([0,-3,0,1]),Polrev([2,3,-1,-1]),Polrev([1,0,0,0]),Polrev([-455,-420,112,112]),Polrev([-4038,-4334,1100,1225])], K);
magma: E := EllipticCurve([K![0,-3,0,1],K![2,3,-1,-1],K![1,0,0,0],K![-455,-420,112,112],K![-4038,-4334,1100,1225]]);
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: | \( -4374624644505560827923496904 a^{3} + 8083252342956303105729121856 a^{2} + 2562595786459777953350768848 a - 4735059594419705758581752312 \) | ||
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: | $\left(-2 a^{3} - \frac{19}{4} a^{2} + 11 a + 23 : -\frac{73}{8} a^{3} - \frac{7}{2} a^{2} + \frac{119}{4} a + \frac{17}{2} : 1\right)$ |
| 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: | \( 1.6302687553063589652205890031213109992 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.225151189850328 \) | ||
| Analytic order of Ш: | \( 25 \) (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 |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 5, 8, 10, 20 and 40.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 40.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.