Properties

Label 6.6.1241125.1-25.2-c1
Base field 6.6.1241125.1
Conductor \((-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\)
Conductor norm \( 25 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1241125.1

Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 7*x^4 - 2*x^3 + 11*x^2 + 7*x + 1)
 
gp: K = nfinit(a^6 - 7*a^4 - 2*a^3 + 11*a^2 + 7*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\(y^2+\left(-2a^{5}+a^{4}+13a^{3}-2a^{2}-18a-6\right)xy+\left(-a^{5}+7a^{3}+2a^{2}-11a-5\right)y=x^{3}+\left(-a^{2}+a+2\right)x^{2}+\left(26a^{5}-12a^{4}-176a^{3}+29a^{2}+270a+59\right)x-51a^{5}+24a^{4}+346a^{3}-61a^{2}-534a-106\)
sage: E = EllipticCurve(K, [-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106])
 
gp: E = ellinit([-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106],K)
 
magma: E := ChangeRing(EllipticCurve([-2*a^5 + a^4 + 13*a^3 - 2*a^2 - 18*a - 6, -a^2 + a + 2, -a^5 + 7*a^3 + 2*a^2 - 11*a - 5, 26*a^5 - 12*a^4 - 176*a^3 + 29*a^2 + 270*a + 59, -51*a^5 + 24*a^4 + 346*a^3 - 61*a^2 - 534*a - 106]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\) = \( \left(5, a - 2\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \( 5^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((-14 a^{5} + 5 a^{4} + 162 a^{3} - 23 a^{2} - 282 a + 33)\) = \( \left(5, a - 2\right)^{19} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19073486328125 \) = \( 5^{19} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1376405109}{78125} a^{5} - \frac{1325860482}{78125} a^{4} - \frac{7597745332}{78125} a^{3} + \frac{2698464798}{78125} a^{2} + \frac{2247897599}{15625} a + \frac{2321410818}{78125} \)
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: 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: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: not available
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\( \left(5, a - 2\right) \) \(5\) \(2\) \(I_{13}^*\) Additive \(1\) \(2\) \(19\) \(13\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 25.2-c consists of this curve only.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.