Properties

Base field \(\Q(\sqrt{33}) \)
Label 2.2.33.1-132.1-a6
Conductor \((-4 a + 2)\)
Conductor norm \( 132 \)
CM no
base-change yes: 66.a1,2178.b1
Q-curve yes
Torsion order \( 4 \)
Rank not available

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

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

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

Weierstrass equation

\( y^2 + x y + y = x^{3} - 81 x - 284 \)
sage: E = EllipticCurve(K, [1, 0, 1, -81, -284])
 
gp: E = ellinit([1, 0, 1, -81, -284],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -81, -284]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-4 a + 2)\) = \( \left(-a - 2\right) \cdot \left(-a + 3\right) \cdot \left(-2 a + 7\right) \cdot \left(-4 a - 9\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 132 \) = \( 2^{2} \cdot 3 \cdot 11 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((255552)\) = \( \left(-a - 2\right)^{6} \cdot \left(-a + 3\right)^{6} \cdot \left(-2 a + 7\right)^{2} \cdot \left(-4 a - 9\right)^{6} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 65306824704 \) = \( 2^{12} \cdot 3^{2} \cdot 11^{6} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{57736239625}{255552} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(-\frac{11}{4} a + \frac{15}{4} : \frac{11}{8} a - \frac{19}{8} : 1\right)$,$\left(-5 : 2 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a - 2\right) \) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(-a + 3\right) \) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(-2 a + 7\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(-4 a - 9\right) \) \(11\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 132.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 66.a1, 2178.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.