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([2,0,-4,0,1]),K([3,-4,-4,1,1]),K([3,-3,-4,1,1]),K([-7,-3,17,1,-5]),K([-11,-7,10,1,-3])])
gp: E = ellinit([Polrev([2,0,-4,0,1]),Polrev([3,-4,-4,1,1]),Polrev([3,-3,-4,1,1]),Polrev([-7,-3,17,1,-5]),Polrev([-11,-7,10,1,-3])], K);
magma: E := EllipticCurve([K![2,0,-4,0,1],K![3,-4,-4,1,1],K![3,-3,-4,1,1],K![-7,-3,17,1,-5],K![-11,-7,10,1,-3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4+a^3-4a^2-2a+3)\) | = | \((a^4+a^3-4a^2-2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-24a^4+16a^3+67a^2-53a+32)\) | = | \((a^4+a^3-4a^2-2a+3)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -147008443 \) | = | \(-43^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{3026241451169}{147008443} a^{4} + \frac{3418994898940}{147008443} a^{3} + \frac{8931388378881}{147008443} a^{2} - \frac{5169364782818}{147008443} a - \frac{7100903835142}{147008443} \) | ||
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: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(3 a^{4} - 2 a^{3} - 12 a^{2} + 6 a + 10 : 6 a^{4} - 7 a^{3} - 27 a^{2} + 18 a + 25 : 1\right)$ |
| 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: | \( 537.84618223957766282372374176009879432 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 0.889001954 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4+a^3-4a^2-2a+3)\) | \(43\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
43.5-b
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.