Properties

Base field \(\Q(\sqrt{77}) \)
Label 2.2.77.1-1.1-a4
Conductor \((1)\)
Conductor norm \( 1 \)
CM yes (\(-28\))
base-change no
Q-curve yes
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{77}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 19 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 19)
 
gp (2.8): K = nfinit(a^2 - a - 19);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} + x^{2} + \left(7 a - 21\right) x - 6 a + 52 \)
magma: E := ChangeRing(EllipticCurve([a + 1, 1, a + 1, 7*a - 21, -6*a + 52]),K);
 
sage: E = EllipticCurve(K, [a + 1, 1, a + 1, 7*a - 21, -6*a + 52])
 
gp (2.8): E = ellinit([a + 1, 1, a + 1, 7*a - 21, -6*a + 52],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((1)\) = \((1)\)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1 \) = 1
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1)\) = \((1)\)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1 \) = 1
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( 16581375 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z[\sqrt{-7}]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $N(\mathrm{U}(1))$

Mordell-Weil group

Rank not available.
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(-\frac{3}{4} a : \frac{1}{4} a + \frac{53}{8} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[3]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
No primes of bad reduction.

Galois Representations

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

prime Image of Galois Representation
\(7\) 7B.6.2
\(11\) 11Ns.3.1

For all other primes \(p\), the image is a Borel subgroup if \(p=2\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 14.

Base change

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