Properties

Label 6.6.1241125.1-41.1-a1
Base field 6.6.1241125.1
Conductor norm \( 41 \)
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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
 
gp: K = nfinit(Polrev([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

\({y}^2+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-52a-19\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-4178a^{5}+5149a^{4}+22801a^{3}-19626a^{2}-21217a-3693\right){x}-269508a^{5}+337696a^{4}+1461510a^{3}-1289635a^{2}-1338554a-220717\)
sage: E = EllipticCurve([K([-19,-52,-2,34,2,-5]),K([1,-3,-1,1,0,0]),K([1,1,0,0,0,0]),K([-3693,-21217,-19626,22801,5149,-4178]),K([-220717,-1338554,-1289635,1461510,337696,-269508])])
 
gp: E = ellinit([Polrev([-19,-52,-2,34,2,-5]),Polrev([1,-3,-1,1,0,0]),Polrev([1,1,0,0,0,0]),Polrev([-3693,-21217,-19626,22801,5149,-4178]),Polrev([-220717,-1338554,-1289635,1461510,337696,-269508])], K);
 
magma: E := EllipticCurve([K![-19,-52,-2,34,2,-5],K![1,-3,-1,1,0,0],K![1,1,0,0,0,0],K![-3693,-21217,-19626,22801,5149,-4178],K![-220717,-1338554,-1289635,1461510,337696,-269508]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4+a^3-4a^2-3a+1)\) = \((a^4+a^3-4a^2-3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-26a^5+42a^4+176a^3-166a^2-301a+21)\) = \((a^4+a^3-4a^2-3a+1)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 194754273881 \) = \(41^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{300039257776916723031380}{194754273881} a^{5} - \frac{135201991016220514678646}{194754273881} a^{4} - \frac{1435403563400771691241019}{194754273881} a^{3} + \frac{343098620572538429827465}{194754273881} a^{2} + \frac{360645134275870330036563}{194754273881} a - \frac{34885214721449234564804}{194754273881} \)
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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.55357546064756299660352397232241976964 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.19306 \)
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\((j)_{-}\)
\((a^4+a^3-4a^2-3a+1)\) \(41\) \(1\) \(I_{7}\) Non-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 41.1-a consists of curves linked by isogenies of degree 7.

Base change

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

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