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,-1,1,0,0]),K([2,1,-1,0,0]),K([-1,-1,1,0,0]),K([-20375,59680,-18068,-25455,10238]),K([-1771850,5265005,-1558277,-2245513,897162])])
gp: E = ellinit([Polrev([-1,-1,1,0,0]),Polrev([2,1,-1,0,0]),Polrev([-1,-1,1,0,0]),Polrev([-20375,59680,-18068,-25455,10238]),Polrev([-1771850,5265005,-1558277,-2245513,897162])], K);
magma: E := EllipticCurve([K![-1,-1,1,0,0],K![2,1,-1,0,0],K![-1,-1,1,0,0],K![-20375,59680,-18068,-25455,10238],K![-1771850,5265005,-1558277,-2245513,897162]]);
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: | \((416132a^4+1381052a^3-2778251a^2-7898059a+900721)\) | = | \((a^2-1)^{4}\cdot(a^3-3a)^{28}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1255737557015654093436832343547201 \) | = | \(3^{4}\cdot13^{28}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{719563077579420768826778357723754883977964219744702}{1255737557015654093436832343547201} a^{4} - \frac{381278393594661046595984821496687550631447920681206}{418579185671884697812277447849067} a^{3} - \frac{876030344833436911152144359572205314535143035049625}{418579185671884697812277447849067} a^{2} + \frac{2519311520172502928352178004640753313415969058351782}{1255737557015654093436832343547201} a + \frac{1753426510580954709523077468627597477978011837927249}{1255737557015654093436832343547201} \) | ||
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: | $\left(-\frac{241}{4} a^{4} + \frac{273}{2} a^{3} + \frac{533}{4} a^{2} - \frac{635}{2} a + \frac{351}{4} : -\frac{115}{8} a^{4} + \frac{229}{8} a^{3} + 24 a^{2} - \frac{369}{8} a + \frac{25}{4} : 1\right)$ |
| 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: | \( 0.059646735251789572901522092038168693268 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.49927138 \) | ||
| Analytic order of Ш: | \( 2401 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-1)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a^3-3a)\) | \(13\) | \(2\) | \(I_{28}\) | Non-split multiplicative | \(1\) | \(1\) | \(28\) | \(28\) |
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.1.3 |
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-b
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.