Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-22050.2-c4
Conductor \((105 a)\)
Conductor norm \( 22050 \)
CM no
base-change yes: 6720.k4,210.c4
Q-curve yes
Torsion order \( 4 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{-2}) \)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)
 
gp: K = nfinit(a^2 + 2);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 370 x + 2435 \)
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -370, 2435]),K);
 
sage: E = EllipticCurve(K, [1, 1, 1, -370, 2435])
 
gp: E = ellinit([1, 1, 1, -370, 2435],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((105 a)\) = \( \left(a\right) \cdot \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right) \cdot \left(7\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 22050 \) = \( 2 \cdot 3^{2} \cdot 25 \cdot 49 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((345888060)\) = \( \left(a\right)^{4} \cdot \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right) \cdot \left(7\right)^{8} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 119638550050563600 \) = \( 2^{4} \cdot 3^{2} \cdot 25 \cdot 49^{8} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{5602762882081}{345888060} \)
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 not available.

magma: Rank(E);
 
sage: E.rank()
 

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 
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: $\left(25 : -111 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(-a - 1\right) \) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(a - 1\right) \) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(5\right) \) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(7\right) \) \(49\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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 and 8.
Its isogeny class 22050.2-c consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base-change of elliptic curves 6720.k4, 210.c4, defined over \(\Q\), so it is also a \(\Q\)-curve.