Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-567.1-a8
Conductor \((-18 a + 9)\)
Conductor norm \( 567 \)
CM no
base-change yes: 441.f1,63.a1
Q-curve yes
Torsion order \( 4 \)
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

\( y^2 + x y = x^{3} - x^{2} - 7056 x + 229905 \)
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -7056, 229905]),K);
 
sage: E = EllipticCurve(K, [1, -1, 0, -7056, 229905])
 
gp: E = ellinit([1, -1, 0, -7056, 229905],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-18 a + 9)\) = \( \left(3\right)^{2} \cdot \left(-2 a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 567 \) = \( 7 \cdot 9^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((107163)\) = \( \left(3\right)^{7} \cdot \left(-2 a + 1\right)^{4} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 11483908569 \) = \( 7^{4} \cdot 9^{7} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{53297461115137}{147} \)
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: $\left(-\frac{33457}{100} : -\frac{1117141}{250} a + \frac{2401567}{1000} : 1\right)$

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

Height: 10.554035292536632

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

Regulator: 10.5540352925

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

Torsion subgroup

Structure: \(\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generator: $\left(48 : -33 : 1\right)$
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\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(3\right) \) \(9\) \(4\) \(I_{1}^*\) Additive \(1\) \(2\) \(7\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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, 8 and 16.
Its isogeny class 567.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 441.f1, 63.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.