Base field 5.5.70601.1
Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 2 x^{2} + 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 2, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, 3, 2, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 2, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([5,2,-15,-2,3]),K([-4,4,2,-1,0]),K([3,0,-10,-1,2]),K([-3,2,-10,-6,4]),K([17,-29,-83,-11,19])])
gp: E = ellinit([Polrev([5,2,-15,-2,3]),Polrev([-4,4,2,-1,0]),Polrev([3,0,-10,-1,2]),Polrev([-3,2,-10,-6,4]),Polrev([17,-29,-83,-11,19])], K);
magma: E := EllipticCurve([K![5,2,-15,-2,3],K![-4,4,2,-1,0],K![3,0,-10,-1,2],K![-3,2,-10,-6,4],K![17,-29,-83,-11,19]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+6a^2-2)\) | = | \((-a^4+a^3+6a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 47 \) | = | \(47\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-a^2-4a-1)\) | = | \((-a^4+a^3+6a^2-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -47 \) | = | \(-47\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{42542580556}{47} a^{4} - \frac{21887546864}{47} a^{3} - \frac{232611726616}{47} a^{2} - \frac{13686260207}{47} a + \frac{157328368641}{47} \) | ||
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 | $\left(-11 a^{4} + a^{3} + 59 a^{2} + 25 a - 15 : -90 a^{4} + 20 a^{3} + 458 a^{2} + 191 a - 108 : 1\right)$ |
| Height | \(0.091928162302122458653375804289476531199\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.091928162302122458653375804289476531199 \) | ||
| Period: | \( 1617.8010240740069759601083287909978291 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.79858346 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+6a^2-2)\) | \(47\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
47.2-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.