Properties

Label 6.6.703493.1-13.1-b1
Base field 6.6.703493.1
Conductor norm \( 13 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+9a-3\right){x}{y}+\left(a^{5}-6a^{3}+8a+2\right){y}={x}^{3}+\left(-a^{5}+7a^{3}-a^{2}-9a+2\right){x}^{2}+\left(-a^{5}-a^{4}+8a^{3}+5a^{2}-20a+6\right){x}+2a^{5}-4a^{4}-8a^{3}+16a^{2}-a-2\)
sage: E = EllipticCurve([K([-3,9,10,-11,-2,2]),K([2,-9,-1,7,0,-1]),K([2,8,0,-6,0,1]),K([6,-20,5,8,-1,-1]),K([-2,-1,16,-8,-4,2])])
 
gp: E = ellinit([Polrev([-3,9,10,-11,-2,2]),Polrev([2,-9,-1,7,0,-1]),Polrev([2,8,0,-6,0,1]),Polrev([6,-20,5,8,-1,-1]),Polrev([-2,-1,16,-8,-4,2])], K);
 
magma: E := EllipticCurve([K![-3,9,10,-11,-2,2],K![2,-9,-1,7,0,-1],K![2,8,0,-6,0,1],K![6,-20,5,8,-1,-1],K![-2,-1,16,-8,-4,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3a^5-2a^4-17a^3+10a^2+16a-3)\) = \((3a^5-2a^4-17a^3+10a^2+16a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 13 \) = \(13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((6a^4+2a^3-21a^2-7a-10)\) = \((3a^5-2a^4-17a^3+10a^2+16a-3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -62748517 \) = \(-13^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{33889457777}{62748517} a^{5} - \frac{83113788886}{62748517} a^{4} - \frac{102300181186}{62748517} a^{3} + \frac{371472408155}{62748517} a^{2} - \frac{227617587843}{62748517} a + \frac{15310089083}{62748517} \)
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 $\left(12 a^{5} - 8 a^{4} - 72 a^{3} + 39 a^{2} + 78 a - 10 : 40 a^{5} - 28 a^{4} - 236 a^{3} + 134 a^{2} + 247 a - 32 : 1\right)$
Height \(0.0023491183090318897384186167089784011836\)
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: \( 0.0023491183090318897384186167089784011836 \)
Period: \( 17461.840809504202347106487193485287286 \)
Tamagawa product: \( 7 \)
Torsion order: \(1\)
Leading coefficient: \( 2.05407 \)
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\((j)_{-}\)
\((3a^5-2a^4-17a^3+10a^2+16a-3)\) \(13\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 13.1-b consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.