Properties

Base field 6.6.1241125.1
Label 6.6.1241125.1-41.1-a1
Conductor \((41,a^{4} - a^{3} - 5 a^{2} + 3 a + 3)\)
Conductor norm \( 41 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((41,a^{4} - a^{3} - 5 a^{2} + 3 a + 3)\) = \( \left(41, a - 12\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 41 \) = \( 41 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((194754273881,a^{5} - 7 a^{3} - a^{2} + 11 a + 45379617960,-2 a^{5} + a^{4} + 13 a^{3} - 2 a^{2} - 19 a + 77762713635,-2 a^{5} + a^{4} + 14 a^{3} - 2 a^{2} - 23 a + 114653504175,a + 130493523986,-a^{5} + 7 a^{3} + 2 a^{2} - 11 a + 150298996103)\) = \( \left(41, a - 12\right)^{7} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 194754273881 \) = \( 41^{7} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{300039257776916723031380}{194754273881} a^{5} - \frac{135201991016220514678646}{194754273881} a^{4} - \frac{1435403563400771691241019}{194754273881} a^{3} + \frac{343098620572538429827465}{194754273881} a^{2} + \frac{360645134275870330036563}{194754273881} a - \frac{34885214721449234564804}{194754273881} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

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\((j)_{-}\)
\( \left(41, a - 12\right) \) \(41\) \(1\) \(I_{7}\) 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 curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.