Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-3600.1-g8
Conductor \((60)\)
Conductor norm \( 3600 \)
CM no
base-change yes: 240.b1,960.p1
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

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

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

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

Weierstrass equation

\(y^2+axy=x^{3}-x^{2}-21334x+1202942\)
sage: E = EllipticCurve(K, [a, -1, 0, -21334, 1202942])
 
gp: E = ellinit([a, -1, 0, -21334, 1202942],K)
 
magma: E := ChangeRing(EllipticCurve([a, -1, 0, -21334, 1202942]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((60)\) = \( \left(a\right)^{4} \cdot \left(3\right) \cdot \left(5\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 3600 \) = \( 2^{4} \cdot 9 \cdot 25 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((5184000)\) = \( \left(a\right)^{18} \cdot \left(3\right)^{4} \cdot \left(5\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 26873856000000 \) = \( 2^{18} \cdot 9^{4} \cdot 25^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{16778985534208729}{81000} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 0 \)

sage: E.rank()
 
magma: Rank(E);
 

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(86 : -43 a - 24 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a\right) \) \(2\) \(4\) \(I_{10}^*\) Additive \(1\) \(4\) \(18\) \(6\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(5\right) \) \(25\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 3600.1-g consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 240.b1, 960.p1, defined over \(\Q\), so it is also a \(\Q\)-curve.