Properties

Base field \(\Q(\sqrt{-1}) \)
Label 2.0.4.1-5929.1-c2
Conductor \((77)\)
Conductor norm \( 5929 \)
CM no
base-change yes: 1232.a1,77.c1
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)
 
gp: K = nfinit(i^2 + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\( y^2 + x y = x^{3} + x^{2} - 51 x + 110 \)
sage: E = EllipticCurve(K, [1, 1, 0, -51, 110])
 
gp: E = ellinit([1, 1, 0, -51, 110],K)
 
magma: E := ChangeRing(EllipticCurve([1, 1, 0, -51, 110]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((77)\) = \( \left(7\right) \cdot \left(11\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 5929 \) = \( 49 \cdot 121 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1294139)\) = \( \left(7\right)^{6} \cdot \left(11\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 1674795751321 \) = \( 49^{6} \cdot 121 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{15124197817}{1294139} \)
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: \( 1 \)

sage: E.rank()
 
magma: Rank(E);
 

Generator: $\left(-\frac{19}{2} : -\frac{49}{4} i + \frac{19}{4} : 1\right)$

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 

Height: 0.6755381582936599

sage: [P.height() for P in gens]
 
magma: [Height(P):P in gens];
 

Regulator: 0.675538158294

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: $\left(\frac{11}{4} : -\frac{11}{8} : 1\right)$
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\((j)_{-}\)
\( \left(7\right) \) \(49\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(11\right) \) \(121\) \(1\) \(I_{1}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 5929.1-c consists of curves linked by isogenies of degree 2.

Base change

This curve is the base-change of elliptic curves 1232.a1, 77.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.