Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-448.4-a1
Conductor \((-16 a + 8)\)
Conductor norm \( 448 \)
CM no
base-change yes: 56.a4,392.d4
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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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: K = nfinit(a^2 - a + 2);
 

Weierstrass equation

\( y^2 = x^{3} + x + 2 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 0, 1, 2]),K);
 
sage: E = EllipticCurve(K, [0, 0, 0, 1, 2])
 
gp: E = ellinit([0, 0, 0, 1, 2],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-16 a + 8)\) = \( \left(a\right)^{3} \cdot \left(-a + 1\right)^{3} \cdot \left(-2 a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 448 \) = \( 2^{6} \cdot 7 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1792)\) = \( \left(a\right)^{8} \cdot \left(-a + 1\right)^{8} \cdot \left(-2 a + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 3211264 \) = \( 2^{16} \cdot 7^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{432}{7} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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: \( 1 \)

magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(-2 a + 5 : 6 a - 10 : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 

Height: 0.4729897507253302

magma: [Height(P):P in gens];
 
sage: [P.height() for P in gens]
 

Regulator: 0.472989750725

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(-3 : -4 a + 2 : 1\right)$,$\left(-a + 1 : 0 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(-a + 1\right) \) \(2\) \(4\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(-2 a + 1\right) \) \(7\) \(2\) \(I_{2}\) Non-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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 448.4-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 56.a4, 392.d4, defined over \(\Q\), so it is also a \(\Q\)-curve.