Properties

Label 6.6.1229312.1-392.1-k5
Base field 6.6.1229312.1
Conductor norm \( 392 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 36 \)
Rank \( 0 \)

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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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, 0, 24, 0, -10, 0, 1]))
 
gp: K = nfinit(Polrev([-8, 0, 24, 0, -10, 0, 1]));
 
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([1,0,0,0,0,0]),K([0,0,0,0,0,0]),K([1,0,0,0,0,0]),K([-11,0,0,0,0,0]),K([12,0,0,0,0,0])])
 
gp: E = ellinit([Polrev([1,0,0,0,0,0]),Polrev([0,0,0,0,0,0]),Polrev([1,0,0,0,0,0]),Polrev([-11,0,0,0,0,0]),Polrev([12,0,0,0,0,0])], K);
 
magma: E := EllipticCurve([K![1,0,0,0,0,0],K![0,0,0,0,0,0],K![1,0,0,0,0,0],K![-11,0,0,0,0,0],K![12,0,0,0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/4a^5-5/2a^3+4a)\) = \((-1/4a^5+2a^3-1/2a^2-3a+1)\cdot(-1/2a^3+3a+1)\cdot(1/4a^5-2a^3+3a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 392 \) = \(7\cdot7\cdot8\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((98)\) = \((-1/4a^5+2a^3-1/2a^2-3a+1)^{6}\cdot(-1/2a^3+3a+1)^{6}\cdot(1/4a^5-2a^3+3a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 885842380864 \) = \(7^{6}\cdot7^{6}\cdot8^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{128787625}{98} \)
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: \(\Z/2\Z\oplus\Z/18\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{1}{2} a^{5} - 4 a^{3} + 6 a - 1 : -\frac{1}{4} a^{5} + 2 a^{3} - 3 a : 1\right)$ $\left(\frac{1}{2} a^{4} - \frac{7}{2} a^{2} + 3 : -\frac{3}{4} a^{4} + 5 a^{2} - 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 44104.626108278665262787570673624561189 \)
Tamagawa product: \( 72 \)  =  \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2\)
Torsion order: \(36\)
Leading coefficient: \( 2.20992 \)
Analytic order of Ш: \( 1 \) (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\((j)_{-}\)
\((-1/4a^5+2a^3-1/2a^2-3a+1)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((-1/2a^3+3a+1)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((1/4a^5-2a^3+3a)\) \(8\) \(2\) \(I_{2}\) 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 elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:

Base field Curve
\(\Q\) 14.a4
\(\Q\) 448.g4
\(\Q(\sqrt{2}) \) 2.2.8.1-98.1-a8
\(\Q(\zeta_{7})^+\) 3.3.49.1-56.1-a4
\(\Q(\zeta_{7})^+\) a curve with conductor norm 1835008 (not in the database)