Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-28672.7-o1
Conductor \((-128 a + 64)\)
Conductor norm \( 28672 \)
CM no
base-change yes: 448.g2,3136.e2
Q-curve yes
Torsion order \( 4 \)
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

\( y^2 = x^{3} - x^{2} - 10913 x - 436447 \)
magma: E := ChangeRing(EllipticCurve([0, -1, 0, -10913, -436447]),K);
 
sage: E = EllipticCurve(K, [0, -1, 0, -10913, -436447])
 
gp (2.8): E = ellinit([0, -1, 0, -10913, -436447],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-128 a + 64)\) = \( \left(a\right)^{6} \cdot \left(-a + 1\right)^{6} \cdot \left(-2 a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 28672 \) = \( 2^{12} \cdot 7 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((481036337152)\) = \( \left(a\right)^{36} \cdot \left(-a + 1\right)^{36} \cdot \left(-2 a + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 231395957660612615471104 \) = \( 2^{72} \cdot 7^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{548347731625}{1835008} \)
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: \(\Z/2\Z\times\Z/2\Z\)
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]
 
Generators: $\left(-2 a - 59 : 0 : 1\right)$,$\left(2 a - 61 : 0 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{26}^*\) Additive \(1\) \(6\) \(36\) \(18\)
\( \left(-a + 1\right) \) \(2\) \(4\) \(I_{26}^*\) Additive \(1\) \(6\) \(36\) \(18\)
\( \left(-2 a + 1\right) \) \(7\) \(2\) \(I_{2}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 28672.7-o consists of curves linked by isogenies of degrees dividing 36.

Base change

This curve is the base-change of elliptic curves 448.g2, 3136.e2, defined over \(\Q\), so it is also a \(\Q\)-curve.