Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([3,-7,-10,9,3,-2]),K([1,-5,-3,5,1,-1]),K([2,-1,-4,0,1,0]),K([-11,28,5,-57,-4,13]),K([-40,130,128,-114,-35,23])])
gp: E = ellinit([Polrev([3,-7,-10,9,3,-2]),Polrev([1,-5,-3,5,1,-1]),Polrev([2,-1,-4,0,1,0]),Polrev([-11,28,5,-57,-4,13]),Polrev([-40,130,128,-114,-35,23])], K);
magma: E := EllipticCurve([K![3,-7,-10,9,3,-2],K![1,-5,-3,5,1,-1],K![2,-1,-4,0,1,0],K![-11,28,5,-57,-4,13],K![-40,130,128,-114,-35,23]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) | = | \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(49\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((207a^5-225a^4-894a^3+1170a^2+393a-1387)\) | = | \((-2a^5+3a^4+10a^3-12a^2-11a+6)^{10}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -79792266297612001 \) | = | \(-49^{10}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{10036119836964947}{282475249} a^{5} - \frac{29399419342483055}{282475249} a^{4} - \frac{12822844127834407}{282475249} a^{3} + \frac{92208680747113198}{282475249} a^{2} - \frac{65621823291227116}{282475249} a + \frac{10801128327256298}{282475249} \) | ||
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: | $\left(-2 a^{5} + \frac{5}{2} a^{4} + \frac{37}{4} a^{3} - \frac{17}{2} a^{2} - \frac{35}{4} a + 2 : -\frac{23}{8} a^{5} + 4 a^{4} + \frac{111}{8} a^{3} - \frac{55}{4} a^{2} - \frac{109}{8} a + \frac{27}{8} : 1\right)$ |
| 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: | \( 2310.7349364208047835024738824872300473 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.65880 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) | \(49\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
49.1-c
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.