Base field 5.5.38569.1
Generator \(a\), with minimal polynomial \( x^{5} - 5 x^{3} + 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,-3,-4,1,1]),K([1,0,0,0,0]),K([1,-3,-4,1,1]),K([10,-40,-55,5,10]),K([64,-161,-226,12,38])])
gp: E = ellinit([Polrev([1,-3,-4,1,1]),Polrev([1,0,0,0,0]),Polrev([1,-3,-4,1,1]),Polrev([10,-40,-55,5,10]),Polrev([64,-161,-226,12,38])], K);
magma: E := EllipticCurve([K![1,-3,-4,1,1],K![1,0,0,0,0],K![1,-3,-4,1,1],K![10,-40,-55,5,10],K![64,-161,-226,12,38]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a-1)\) | = | \((a^3-3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 17 \) | = | \(17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((22a^4+61a^3-130a^2-191a+150)\) | = | \((a^3-3a-1)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6975757441 \) | = | \(17^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{63035509038380061495748094}{6975757441} a^{4} + \frac{128497588344189495041715327}{6975757441} a^{3} - \frac{53235816441571130890411893}{6975757441} a^{2} - \frac{108520961136492735262181368}{6975757441} a + \frac{30922567893134525558722397}{6975757441} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a^{4} + 2 a^{3} - 8 a^{2} - 6 a + 3 : a^{4} - 7 a^{2} - 5 a + 2 : 1\right)$ | $\left(\frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{11}{4} a^{2} - 2 a - \frac{9}{4} : \frac{9}{4} a^{4} + \frac{19}{8} a^{3} - \frac{17}{2} a^{2} - \frac{55}{8} a + \frac{19}{8} : 1\right)$ |
| 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: | \( 119.39362332875188596118815369413857716 \) | ||
| Tamagawa product: | \( 8 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.21588343 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a-1)\) | \(17\) | \(8\) | \(I_{8}\) | 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 and 4.
Its isogeny class
17.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.