Properties

Base field 4.4.18625.1
Label 4.4.18625.1-11.1-a2
Conductor \((11,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{11}{6})\)
Conductor norm \( 11 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 4.4.18625.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 14*x^2 + 9*x + 41)
 
gp: K = nfinit(a^4 - a^3 - 14*a^2 + 9*a + 41);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 9, -14, -1, 1]);
 

Weierstrass equation

\( y^2 + \left(\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{35}{6}\right) x y + \left(a + 1\right) y = x^{3} + \left(\frac{1}{6} a^{3} - \frac{4}{3} a + \frac{1}{6}\right) x^{2} + \left(\frac{47}{6} a^{3} + 17 a^{2} - \frac{155}{3} a - \frac{547}{6}\right) x + \frac{611}{3} a^{3} + 320 a^{2} - \frac{6034}{3} a - \frac{9808}{3} \)
sage: E = EllipticCurve(K, [1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3])
 
gp: E = ellinit([1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3],K)
 
magma: E := ChangeRing(EllipticCurve([1/6*a^3 + a^2 - 1/3*a - 35/6, 1/6*a^3 - 4/3*a + 1/6, a + 1, 47/6*a^3 + 17*a^2 - 155/3*a - 547/6, 611/3*a^3 + 320*a^2 - 6034/3*a - 9808/3]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((11,-\frac{1}{6} a^{3} + \frac{7}{3} a + \frac{11}{6})\) = \( \left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 11 \) = \( 11 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((14641,\frac{1}{6} a^{3} - \frac{4}{3} a + \frac{24175}{6},a + 12237,a^{2} + 3979)\) = \( \left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 14641 \) = \( 11^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{1060007393}{87846} a^{3} + \frac{746018190}{14641} a^{2} + \frac{411434192}{43923} a - \frac{13361296451}{87846} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(-\frac{29}{24} a^{3} - \frac{7}{4} a^{2} + \frac{67}{6} a + \frac{379}{24} : \frac{55}{24} a^{3} + \frac{41}{8} a^{2} - \frac{437}{24} a - \frac{439}{12} : 1\right)$
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\((j)_{-}\)
\( \left(\frac{1}{6} a^{3} - \frac{7}{3} a - \frac{11}{6}\right) \) \(11\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 11.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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