Properties

Base field \(\Q(\sqrt{57}) \)
Label 2.2.57.1-121.1-a1
Conductor \((11)\)
Conductor norm \( 121 \)
CM no
base-change yes: 3971.b1,99.d1
Q-curve yes
Torsion order \( 1 \)
Rank not available

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Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{57}) \)

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

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

Weierstrass equation

\( y^2 + y = x^{3} + \left(a - 1\right) x^{2} + \left(-94469627 a - 309380202\right) x + 970977133866 a + 3179869733625 \)
sage: E = EllipticCurve(K, [0, a - 1, 1, -94469627*a - 309380202, 970977133866*a + 3179869733625])
 
gp: E = ellinit([0, a - 1, 1, -94469627*a - 309380202, 970977133866*a + 3179869733625],K)
 
magma: E := ChangeRing(EllipticCurve([0, a - 1, 1, -94469627*a - 309380202, 970977133866*a + 3179869733625]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((11)\) = \( \left(11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 121 \) = \( 121 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((11)\) = \( \left(11\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 121 \) = \( 121 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{52893159101157376}{11} \)
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: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(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\((j)_{-}\)
\( \left(11\right) \) \(121\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.4.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 121.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base-change of elliptic curves 3971.b1, 99.d1, defined over \(\Q\), so it is also a \(\Q\)-curve.