Properties

Base field 3.3.733.1
Label 3.3.733.1-7.1-b2
Conductor \((7,-a^{2} - 2 a + 3)\)
Conductor norm \( 7 \)
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 \(1\).

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

\( y^2 + \left(a^{2} - 5\right) x y + y = x^{3} + \left(a^{2} - a - 5\right) x^{2} + \left(28 a^{2} + 141 a - 533\right) x - 38 a^{2} + 1741 a - 4144 \)
magma: E := ChangeRing(EllipticCurve([a^2 - 5, a^2 - a - 5, 1, 28*a^2 + 141*a - 533, -38*a^2 + 1741*a - 4144]),K);
 
sage: E = EllipticCurve(K, [a^2 - 5, a^2 - a - 5, 1, 28*a^2 + 141*a - 533, -38*a^2 + 1741*a - 4144])
 
gp (2.8): E = ellinit([a^2 - 5, a^2 - a - 5, 1, 28*a^2 + 141*a - 533, -38*a^2 + 1741*a - 4144],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()
 
\(\mathfrak{D}\) = \((7,a + 3,a^{2} - 2)\) = \( \left(a^{2} + 2 a - 3\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 7 \) = \( 7 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{84096891433574753384471}{7} a^{2} - \frac{14985568201611275023622}{7} a + \frac{571022332294137577763767}{7} \)
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: \(\Z/2\Z\)
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: $\left(\frac{7}{2} a^{2} - \frac{19}{4} a - \frac{37}{4} : \frac{7}{4} a^{2} + \frac{13}{2} a - \frac{229}{8} : 1\right)$
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\((j)_{-}\)
\( \left(a^{2} + 2 a - 3\right) \) \(7\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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.