Properties

Base field \(\Q(\sqrt{77}) \)
Label 2.2.77.1-11.1-a1
Conductor \((a - 6)\)
Conductor norm \( 11 \)
CM no
base-change yes: 121.d1,539.a1
Q-curve yes
Torsion order \( 1 \)
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

\( y^2 + y = x^{3} + \left(a + 1\right) x^{2} + \left(-70382 a - 273705\right) x + 21156761 a + 82247005 \)
magma: E := ChangeRing(EllipticCurve([0, a + 1, 1, -70382*a - 273705, 21156761*a + 82247005]),K);
 
sage: E = EllipticCurve(K, [0, a + 1, 1, -70382*a - 273705, 21156761*a + 82247005])
 
gp (2.8): E = ellinit([0, a + 1, 1, -70382*a - 273705, 21156761*a + 82247005],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((a - 6)\) = \( \left(a - 6\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 11 \) = \( 11 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((11)\) = \( \left(a - 6\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 121 \) = \( 11^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{52893159101157376}{11} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

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: Trivial
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]
 

Local data at primes of bad reduction

magma: LocalInformation(E);
 
sage: E.local_data()
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a - 6\right) \) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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 11.1-a consists of curves linked by isogenies of degrees dividing 25.

Base change

This curve is the base-change of elliptic curves 121.d1, 539.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.