Properties

Label 6.6.1241125.1-29.1-a2
Base field 6.6.1241125.1
Conductor \((a^4-a^3-5a^2+3a+4)\)
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, 7, 11, -2, -7, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}+\left(-a^{5}+a^{4}+6a^{3}-3a^{2}-7a-2\right){y}={x}^{3}+\left(-2a^{5}+a^{4}+13a^{3}-2a^{2}-19a-5\right){x}^{2}+\left(1531a^{5}-743a^{4}-10369a^{3}+1820a^{2}+16027a+3286\right){x}+33441a^{5}-15713a^{4}-226728a^{3}+38407a^{2}+350175a+72382\)
sage: E = EllipticCurve([K([-3,0,1,0,0,0]),K([-5,-19,-2,13,1,-2]),K([-2,-7,-3,6,1,-1]),K([3286,16027,1820,-10369,-743,1531]),K([72382,350175,38407,-226728,-15713,33441])])
 
gp: E = ellinit([Pol(Vecrev([-3,0,1,0,0,0])),Pol(Vecrev([-5,-19,-2,13,1,-2])),Pol(Vecrev([-2,-7,-3,6,1,-1])),Pol(Vecrev([3286,16027,1820,-10369,-743,1531])),Pol(Vecrev([72382,350175,38407,-226728,-15713,33441]))], K);
 
magma: E := EllipticCurve([K![-3,0,1,0,0,0],K![-5,-19,-2,13,1,-2],K![-2,-7,-3,6,1,-1],K![3286,16027,1820,-10369,-743,1531],K![72382,350175,38407,-226728,-15713,33441]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-5a^2+3a+4)\) = \((a^4-a^3-5a^2+3a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((29a^5-12a^4-194a^3+25a^2+280a+18)\) = \((a^4-a^3-5a^2+3a+4)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 17249876309 \) = \(29^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{13815841607213083975511386207}{17249876309} a^{5} - \frac{28062038812038612265502450888}{17249876309} a^{4} - \frac{39710623624428834678366798314}{17249876309} a^{3} + \frac{53026981056198164144695142492}{17249876309} a^{2} + \frac{44263596385560929969398311274}{17249876309} a + \frac{6801758503148238763013009216}{17249876309} \)
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 $\left(-\frac{158}{5} a^{5} + \frac{78}{5} a^{4} + 211 a^{3} - \frac{196}{5} a^{2} - \frac{1627}{5} a - 61 : -\frac{2818}{25} a^{5} + \frac{1404}{25} a^{4} + \frac{18974}{25} a^{3} - \frac{3496}{25} a^{2} - \frac{5867}{5} a - \frac{5676}{25} : 1\right)$
Height \(1.54501074951103\)
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: \(1\)
Regulator: \( 1.54501074951103 \)
Period: not available
Tamagawa product: \( 7 \)
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\((j)_{-}\)
\((a^4-a^3-5a^2+3a+4)\) \(29\) \(7\) \(I_{7}\) 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
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 29.1-a consists of curves linked by isogenies of degree 7.

Base change

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