Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-47432.2-b2
Conductor \((-77 a - 154)\)
Conductor norm \( 47432 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{-7}) \)

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

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

Weierstrass equation

\( y^2 + a x y + a y = x^{3} - x^{2} + \left(-1001 a + 3369\right) x + 41643 a + 14291 \)
magma: E := ChangeRing(EllipticCurve([a, -1, a, -1001*a + 3369, 41643*a + 14291]),K);
sage: E = EllipticCurve(K, [a, -1, a, -1001*a + 3369, 41643*a + 14291])
gp (2.8): E = ellinit([a, -1, a, -1001*a + 3369, 41643*a + 14291],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-77 a - 154)\) = \( \left(a\right)^{3} \cdot \left(-2 a + 1\right)^{2} \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 47432 \) = \( 2^{3} \cdot 7^{2} \cdot 11^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((183188888113 a + 62681067834)\) = \( \left(a\right)^{10} \cdot \left(-2 a + 1\right)^{10} \cdot \left(-2 a + 3\right)^{3} \cdot \left(2 a + 1\right)^{8} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 82527728843210964110336 \) = \( 2^{10} \cdot 7^{10} \cdot 11^{11} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{71704185617021}{10503585169} a + \frac{49079299276110}{10503585169} \)
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: $\left(-\frac{113}{4} a + \frac{7}{2} : \frac{95}{8} a - \frac{113}{4} : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(III^*\) Additive \(-1\) \(3\) \(10\) \(0\)
\( \left(-2 a + 1\right) \) \(7\) \(4\) \(I_{4}^*\) Additive \(-1\) \(2\) \(10\) \(4\)
\( \left(-2 a + 3\right) \) \(11\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(2 a + 1\right) \) \(11\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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 and 4.
Its isogeny class 47432.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

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