Properties

Base field 3.3.733.1
Label 3.3.733.1-7.1-b6
Conductor \((7,-a^{2} - 2 a + 3)\)
Conductor norm \( 7 \)
CM no
base-change no
Q-curve no
Torsion order \( 4 \)
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 \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 7*x + 8)
 
gp (2.8): K = nfinit(a^3 - a^2 - 7*a + 8);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a^{2} - 4\right) y = x^{3} + \left(-a^{2} + 4\right) x^{2} + \left(454041244 a^{2} + 80907460 a - 3082964015\right) x + 1380930983897 a^{2} + 246073726249 a - 9376594279770 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -a^2 + 4, a^2 - 4, 454041244*a^2 + 80907460*a - 3082964015, 1380930983897*a^2 + 246073726249*a - 9376594279770]),K);
 
sage: E = EllipticCurve(K, [a + 1, -a^2 + 4, a^2 - 4, 454041244*a^2 + 80907460*a - 3082964015, 1380930983897*a^2 + 246073726249*a - 9376594279770])
 
gp (2.8): E = ellinit([a + 1, -a^2 + 4, a^2 - 4, 454041244*a^2 + 80907460*a - 3082964015, 1380930983897*a^2 + 246073726249*a - 9376594279770],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((7,-a^{2} - 2 a + 3)\) = \( \left(a^{2} + 2 a - 3\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 7 \) = \( 7 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((49,a + 31,a^{2} + 19)\) = \( \left(a^{2} + 2 a - 3\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 49 \) = \( 7^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{24849000}{49} a^{2} + \frac{37748161}{49} a - \frac{78881609}{49} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): 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()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: not available

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/4\Z\)
magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[2]
 
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp (2.8): elltors(E)[1]
 
Generator: $\left(-23446 a^{2} - 4179 a + 159202 : 7509500 a^{2} + 1338146 a - 50989896 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(a^{2} + 2 a - 3\right) \) \(7\) \(2\) \(I_{2}\) 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, 4 and 8.
Its isogeny class 7.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

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