Base field \(\Q(\zeta_{11})^+\)
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,-3,0,1]),K([2,-4,-4,1,1]),K([1,0,-3,0,1]),K([-1224,491,2238,-683,-959]),K([-29653,42697,92332,-34613,-40616])])
gp: E = ellinit([Polrev([1,1,-3,0,1]),Polrev([2,-4,-4,1,1]),Polrev([1,0,-3,0,1]),Polrev([-1224,491,2238,-683,-959]),Polrev([-29653,42697,92332,-34613,-40616])], K);
magma: E := EllipticCurve([K![1,1,-3,0,1],K![2,-4,-4,1,1],K![1,0,-3,0,1],K![-1224,491,2238,-683,-959],K![-29653,42697,92332,-34613,-40616]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-3a^2-1)\) | = | \((a^4-3a^2-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 23 \) | = | \(23\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^4+2a^3+11a^2-3a-5)\) | = | \((a^4-3a^2-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -529 \) | = | \(-23^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1812431798583025575975116650734948165}{529} a^{4} - \frac{1296559921989620389944481933388349849}{529} a^{3} - \frac{7618766625551254351602825393413091355}{529} a^{2} + \frac{3268768315434618465890517821484807321}{529} a + \frac{6367683864076909597378964320128797635}{529} \) | ||
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: | $\left(-\frac{95}{4} a^{4} + \frac{1}{4} a^{3} + \frac{303}{4} a^{2} + \frac{13}{4} a - \frac{99}{4} : \frac{137}{8} a^{4} + \frac{69}{4} a^{3} - \frac{351}{8} a^{2} - \frac{107}{4} a + \frac{157}{8} : 1\right)$ |
| 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: | \( 0.31780034143956915037034160225321732999 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.820765345 \) | ||
| Analytic order of Ш: | \( 625 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-3a^2-1)\) | \(23\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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, 5 and 10.
Its isogeny class
23.1-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.