Properties

Base field 6.6.1241125.1
Label 6.6.1241125.1-25.2-b2
Conductor \((5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\)
Conductor norm \( 25 \)
CM no
base-change no
Q-curve no
Torsion order \( 5 \)
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

\( y^2 + \left(-a^{5} + 7 a^{3} + 2 a^{2} - 11 a - 6\right) x y + \left(-5 a^{5} + 2 a^{4} + 34 a^{3} - 2 a^{2} - 52 a - 19\right) y = x^{3} + \left(-3 a^{5} + a^{4} + 20 a^{3} - 30 a - 12\right) x^{2} + \left(-5 a^{5} + a^{4} + 36 a^{3} + 3 a^{2} - 62 a - 25\right) x - 6 a^{5} + a^{4} + 41 a^{3} + 4 a^{2} - 64 a - 27 \)
sage: E = EllipticCurve(K, [-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27])
 
gp: E = ellinit([-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27],K)
 
magma: E := ChangeRing(EllipticCurve([-a^5 + 7*a^3 + 2*a^2 - 11*a - 6, -3*a^5 + a^4 + 20*a^3 - 30*a - 12, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -5*a^5 + a^4 + 36*a^3 + 3*a^2 - 62*a - 25, -6*a^5 + a^4 + 41*a^3 + 4*a^2 - 64*a - 27]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((5,-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\) = \( \left(5, a - 2\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 25 \) = \( 5^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((625,125 a^{5} - 875 a^{3} - 125 a^{2} + 1375 a + 750,116 a^{5} + a^{4} - 813 a^{3} - 120 a^{2} + 1279 a + 518,65 a^{5} + a^{4} - 455 a^{3} - 69 a^{2} + 714 a + 317,78 a^{5} - 546 a^{3} - 78 a^{2} + 859 a + 511,122 a^{5} - 854 a^{3} - 121 a^{2} + 1342 a + 553)\) = \( \left(5, a - 2\right)^{7} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 78125 \) = \( 5^{7} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{1332544}{5} a^{5} + \frac{2260678}{5} a^{4} - \frac{5421687}{5} a^{3} - \frac{11771837}{5} a^{2} - 1128428 a - \frac{732232}{5} \)
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: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(0 : 3 a^{5} - a^{4} - 21 a^{3} + 34 a + 13 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(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\((j)_{-}\)
\( \left(5, a - 2\right) \) \(5\) \(2\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B
\(5\) 5B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 25.2-b consists of curves linked by isogenies of degrees dividing 15.

Base change

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