Properties

Base field 3.3.733.1
Label 3.3.733.1-7.1-a1
Conductor \((7,-a^{2} - 2 a + 3)\)
Conductor norm \( 7 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
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

\( y^2 + \left(a^{2} + a - 4\right) x y + \left(a + 1\right) y = x^{3} + \left(a^{2} + a - 6\right) x^{2} + \left(77 a^{2} + 9 a - 494\right) x + 499 a^{2} + 93 a - 3377 \)
magma: E := ChangeRing(EllipticCurve([a^2 + a - 4, a^2 + a - 6, a + 1, 77*a^2 + 9*a - 494, 499*a^2 + 93*a - 3377]),K);
sage: E = EllipticCurve(K, [a^2 + a - 4, a^2 + a - 6, a + 1, 77*a^2 + 9*a - 494, 499*a^2 + 93*a - 3377])
gp (2.8): E = ellinit([a^2 + a - 4, a^2 + a - 6, a + 1, 77*a^2 + 9*a - 494, 499*a^2 + 93*a - 3377],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()
\(\mathfrak{D}\) = \((40353607,a + 31688086,a^{2} + 29019583)\) = \( \left(a^{2} + 2 a - 3\right)^{9} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 40353607 \) = \( 7^{9} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{16146410084579303}{40353607} a^{2} - \frac{24513786018922996}{40353607} a + \frac{51294295430574624}{40353607} \)
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: Trivial
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]

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\((j)_{-}\)
\( \left(a^{2} + 2 a - 3\right) \) \(7\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 7.1-a consists of curves linked by isogenies of degree3.

Base change

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