Properties

Base field 6.6.1241125.1
Label 6.6.1241125.1-25.2-b4
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

\( y^2 + \left(-3 a^{5} + 2 a^{4} + 20 a^{3} - 5 a^{2} - 30 a - 10\right) x y + \left(-3 a^{5} + a^{4} + 20 a^{3} - 29 a - 12\right) y = x^{3} + \left(-6 a^{5} + 2 a^{4} + 41 a^{3} - a^{2} - 65 a - 21\right) x^{2} + \left(-2 a^{5} - 3 a^{4} + 18 a^{3} + 16 a^{2} - 36 a - 16\right) x - 2 a^{5} - 2 a^{4} + 17 a^{3} + 16 a^{2} - 40 a - 20 \)
sage: E = EllipticCurve(K, [-3*a^5 + 2*a^4 + 20*a^3 - 5*a^2 - 30*a - 10, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 21, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -2*a^5 - 3*a^4 + 18*a^3 + 16*a^2 - 36*a - 16, -2*a^5 - 2*a^4 + 17*a^3 + 16*a^2 - 40*a - 20])
 
gp: E = ellinit([-3*a^5 + 2*a^4 + 20*a^3 - 5*a^2 - 30*a - 10, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 21, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -2*a^5 - 3*a^4 + 18*a^3 + 16*a^2 - 36*a - 16, -2*a^5 - 2*a^4 + 17*a^3 + 16*a^2 - 40*a - 20],K)
 
magma: E := ChangeRing(EllipticCurve([-3*a^5 + 2*a^4 + 20*a^3 - 5*a^2 - 30*a - 10, -6*a^5 + 2*a^4 + 41*a^3 - a^2 - 65*a - 21, -3*a^5 + a^4 + 20*a^3 - 29*a - 12, -2*a^5 - 3*a^4 + 18*a^3 + 16*a^2 - 36*a - 16, -2*a^5 - 2*a^4 + 17*a^3 + 16*a^2 - 40*a - 20]),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));
 
\(\mathfrak{D}\) = \((3125,625 a^{5} - 4375 a^{3} - 625 a^{2} + 6875 a + 3750,491 a^{5} + a^{4} - 3438 a^{3} - 495 a^{2} + 5404 a + 3393,315 a^{5} + a^{4} - 2205 a^{3} - 319 a^{2} + 3464 a + 2442,203 a^{5} - 1421 a^{3} - 203 a^{2} + 2234 a + 1886,247 a^{5} - 1729 a^{3} - 246 a^{2} + 2717 a + 1928)\) = \( \left(5, a - 2\right)^{9} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1953125 \) = \( 5^{9} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{3288568389824}{25} a^{5} - \frac{1515811489972}{25} a^{4} - \frac{22321293042632}{25} a^{3} + \frac{3711500587103}{25} a^{2} + \frac{6892701907756}{5} a + \frac{7134572445643}{25} \)
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: $\left(2 a^{5} - 15 a^{3} - 3 a^{2} + 26 a + 11 : 9 a^{5} - a^{4} - 63 a^{3} - 10 a^{2} + 103 a + 44 : 1\right)$
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\((j)_{-}\)
\( \left(5, a - 2\right) \) \(5\) \(2\) \(I_{3}^*\) Additive \(1\) \(2\) \(9\) \(3\)

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.