Properties

Base field 6.6.1229312.1
Label 6.6.1229312.1-392.1-k5
Conductor \((14,-\frac{1}{2} a^{3} + 4 a)\)
Conductor norm \( 392 \)
CM no
base-change yes: 14.a4,448.g4
Q-curve yes
Torsion order \( 36 \)
Rank not available

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Base field 6.6.1229312.1

Generator \(a\), with minimal polynomial \( x^{6} - 10 x^{4} + 24 x^{2} - 8 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 10*x^4 + 24*x^2 - 8)
 
gp: K = nfinit(a^6 - 10*a^4 + 24*a^2 - 8);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} - 11 x + 12 \)
sage: E = EllipticCurve(K, [1, 0, 1, -11, 12])
 
gp: E = ellinit([1, 0, 1, -11, 12],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -11, 12]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((14,-\frac{1}{2} a^{3} + 4 a)\) = \( \left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right) \cdot \left(7, a + 1\right) \cdot \left(7, a - 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 392 \) = \( 7^{2} \cdot 8 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((98,49 a^{2} - 196,\frac{49}{2} a^{4} - 147 a^{2} + 98,\frac{49}{2} a^{5} - 196 a^{3} + 294 a,98 a,49 a^{3} - 294 a)\) = \( \left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right)^{2} \cdot \left(7, a + 1\right)^{6} \cdot \left(7, a - 1\right)^{6} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 885842380864 \) = \( 7^{12} \cdot 8^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{128787625}{98} \)
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: \(\Z/2\Z\times\Z/18\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(\frac{1}{2} a^{4} - \frac{7}{2} a^{2} + 3 : -\frac{3}{4} a^{4} + 5 a^{2} - 3 : 1\right)$,$\left(\frac{1}{2} a^{5} - 4 a^{3} + 6 a - 1 : -\frac{1}{4} a^{5} + 2 a^{3} - 3 a : 1\right)$
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\((j)_{-}\)
\( \left(7, a + 1\right) \) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(7, a - 1\right) \) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(2, -\frac{1}{4} a^{5} + 2 a^{3} - 3 a\right) \) \(8\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 392.1-k consists of curves linked by isogenies of degrees dividing 36.

Base change

This curve is the base-change of elliptic curves 14.a4, 448.g4, defined over \(\Q\), so it is also a \(\Q\)-curve.