Properties

Base field 3.3.733.1
Label 3.3.733.1-8.1-c4
Conductor \((2,2)\)
Conductor norm \( 8 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field 3.3.733.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 7*x + 8)
 
gp: K = nfinit(a^3 - a^2 - 7*a + 8);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 

Weierstrass equation

\( y^2 + x y + a y = x^{3} + \left(-a^{2} + a + 4\right) x^{2} + \left(-174 a^{2} - 328 a + 377\right) x - 3028 a^{2} - 5091 a + 8285 \)
sage: E = EllipticCurve(K, [1, -a^2 + a + 4, a, -174*a^2 - 328*a + 377, -3028*a^2 - 5091*a + 8285])
 
gp: E = ellinit([1, -a^2 + a + 4, a, -174*a^2 - 328*a + 377, -3028*a^2 - 5091*a + 8285],K)
 
magma: E := ChangeRing(EllipticCurve([1, -a^2 + a + 4, a, -174*a^2 - 328*a + 377, -3028*a^2 - 5091*a + 8285]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((2,2)\) = \( \left(a^{2} + a - 3\right) \cdot \left(a^{2} - 6\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 8 \) = \( 2 \cdot 4 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1024,1024 a,4 a^{2} + 988 a + 260)\) = \( \left(a^{2} + a - 3\right)^{10} \cdot \left(a^{2} - 6\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 4194304 \) = \( 2^{2} \cdot 4^{10} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{33212059667827837}{128} a^{2} + \frac{403380055280934631}{1024} a - \frac{105510530322967865}{128} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(-\frac{25}{4} a^{2} - \frac{27}{4} a + \frac{135}{4} : \frac{25}{8} a^{2} + \frac{23}{8} a - \frac{135}{8} : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} - 6\right) \) \(2\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(a^{2} + a - 3\right) \) \(4\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 8.1-c consists of curves linked by isogenies of degrees dividing 10.

Base change

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