Properties

Base field \(\Q(\sqrt{11}) \)
Label 2.2.44.1-361.2-b2
Conductor \((7 a + 30)\)
Conductor norm \( 361 \)
CM yes (\(-11\))
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

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Base field \(\Q(\sqrt{11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 11)
 
gp: K = nfinit(a^2 - 11);
 

Weierstrass equation

\( y^2 + y = x^{3} + \left(a + 1\right) x^{2} + \left(-106 a - 350\right) x + 968 a + 3210 \)
magma: E := ChangeRing(EllipticCurve([0, a + 1, 1, -106*a - 350, 968*a + 3210]),K);
 
sage: E = EllipticCurve(K, [0, a + 1, 1, -106*a - 350, 968*a + 3210])
 
gp: E = ellinit([0, a + 1, 1, -106*a - 350, 968*a + 3210],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((7 a + 30)\) = \( \left(2 a - 5\right)^{2} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 361 \) = \( 19^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((200 a + 6891)\) = \( \left(2 a - 5\right)^{6} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 47045881 \) = \( 19^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -32768 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: E.j
 
\( \text{End} (E) \) = \(\Z[(1+\sqrt{-11})/2]\)   ( 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()
 

Regulator: not available

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

Torsion subgroup

Structure: Trivial
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 

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\((j)_{-}\)
\( \left(2 a - 5\right) \) \(19\) \(2\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(11\) 11B.10.3

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 11.
Its isogeny class 361.2-b consists of curves linked by isogenies of degree 11.

Base change

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