Properties

Base field \(\Q(\sqrt{33}) \)
Label 2.2.33.1-16.2-a2
Conductor \((-2 a + 8)\)
Conductor norm \( 16 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{33}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 8 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 8)
 
gp: K = nfinit(a^2 - a - 8);
 

Weierstrass equation

\( y^2 + a x y + a y = x^{3} + \left(a - 1\right) x^{2} + a x + 130 a - 436 \)
magma: E := ChangeRing(EllipticCurve([a, a - 1, a, a, 130*a - 436]),K);
 
sage: E = EllipticCurve(K, [a, a - 1, a, a, 130*a - 436])
 
gp: E = ellinit([a, a - 1, a, a, 130*a - 436],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-2 a + 8)\) = \( \left(-a - 2\right)^{3} \cdot \left(-a + 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 16 \) = \( 2^{4} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((-36 a - 32)\) = \( \left(-a - 2\right)^{11} \cdot \left(-a + 3\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 8192 \) = \( 2^{13} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{1241}{4} a + 736 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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()
 

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: \(\Z/2\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(-2 a + 5 : -2 a + 8 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[3]
 

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\((j)_{-}\)
\( \left(-a - 2\right) \) \(2\) \(1\) \(II^*\) Additive \(1\) \(3\) \(11\) \(0\)
\( \left(-a + 3\right) \) \(2\) \(2\) \(I_{2}\) Non-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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 16.2-a consists of curves linked by isogenies of degree 2.

Base change

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