Properties

Base field \(\Q(\sqrt{33}) \)
Label 2.2.33.1-12.1-a5
Conductor \((-4 a + 14)\)
Conductor norm \( 12 \)
CM no
base-change no
Q-curve yes
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

\( y^2 + x y + y = x^{3} + \left(-a - 1\right) x^{2} + \left(-1110 a - 2709\right) x - 26351 a - 62809 \)
magma: E := ChangeRing(EllipticCurve([1, -a - 1, 1, -1110*a - 2709, -26351*a - 62809]),K);
 
sage: E = EllipticCurve(K, [1, -a - 1, 1, -1110*a - 2709, -26351*a - 62809])
 
gp: E = ellinit([1, -a - 1, 1, -1110*a - 2709, -26351*a - 62809],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-4 a + 14)\) = \( \left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 12 \) = \( 2^{2} \cdot 3 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((90 a + 342)\) = \( \left(-a - 2\right) \cdot \left(-a + 3\right)^{9} \cdot \left(-2 a + 7\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 82944 \) = \( 2^{10} \cdot 3^{4} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( -\frac{2879604455941411323125}{4608} a + \frac{404618180215561464625}{192} \)
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: $\left(17 a + \frac{139}{4} : -\frac{17}{2} a - \frac{143}{8} : 1\right)$
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\((j)_{-}\)
\( \left(-a - 2\right) \) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(-a + 3\right) \) \(2\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\( \left(-2 a + 7\right) \) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class 12.1-a consists of curves linked by isogenies of degrees dividing 36.

Base change

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