Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 5, -3, -2, 1]))
gp: K = nfinit(Polrev([-1, 1, 5, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0,0,0]),K([-1,3,1,-1,0]),K([3,7,-7,-3,2]),K([-5334,-4556,11325,2553,-4706]),K([-18618,-211344,-305029,106339,186623])])
gp: E = ellinit([Polrev([1,0,0,0,0]),Polrev([-1,3,1,-1,0]),Polrev([3,7,-7,-3,2]),Polrev([-5334,-4556,11325,2553,-4706]),Polrev([-18618,-211344,-305029,106339,186623])], K);
magma: E := EllipticCurve([K![1,0,0,0,0],K![-1,3,1,-1,0],K![3,7,-7,-3,2],K![-5334,-4556,11325,2553,-4706],K![-18618,-211344,-305029,106339,186623]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^4+3a^3+7a^2-6a-2)\) | = | \((a^2-1)\cdot(a^3-3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 39 \) | = | \(3\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^4+22a^3+15a^2-57a+47)\) | = | \((a^2-1)\cdot(a^3-3a)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 188245551 \) | = | \(3\cdot13^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{167642752752633447006093524748799040238434}{188245551} a^{4} + \frac{140053631818257799009656597663946084463786}{62748517} a^{3} + \frac{96735383245269016352815198554224769104727}{62748517} a^{2} - \frac{985141440142117916445879153670063426054138}{188245551} a + \frac{331121807700898395019402938118329735513617}{188245551} \) | ||
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: | $\left(4 a^{4} - 21 a^{3} - 67 a^{2} + 71 a + \frac{267}{4} : -3 a^{4} + 12 a^{3} + 37 a^{2} - 39 a - \frac{279}{8} : 1\right)$ |
| 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: | \( 26.028153154633316474744048596697764773 \) | ||
| Tamagawa product: | \( 7 \) = \(1\cdot7\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.953702256 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-1)\) | \(3\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((a^3-3a)\) | \(13\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
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 |
| \(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 7, 14 and 28.
Its isogeny class
39.1-a
consists of curves linked by isogenies of
degrees dividing 28.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.