Properties

Label 2.2.120.1-15.1-a5
Base field \(\Q(\sqrt{30}) \)
Conductor \((15,a)\)
Conductor norm \( 15 \)
CM no
Base change yes: 75.b6,2880.bc6
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{30}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 30 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-30, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-30, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-30, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-224a-1188\right){x}+3752a+20593\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,0]),K([-1188,-224]),K([20593,3752])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-1188,-224])),Pol(Vecrev([20593,3752]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,0],K![-1188,-224],K![20593,3752]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((15,a)\) = \((3,a)\cdot(a+5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 15 \) = \(3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((225)\) = \((3,a)^{4}\cdot(a+5)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 50625 \) = \(3^{4}\cdot5^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{13997521}{225} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-5 a - 31 : 18 a + 90 : 1\right)$ $\left(\frac{5}{2} a + \frac{41}{4} : -\frac{51}{8} a - \frac{345}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 31.3870221150614 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 2.86523000422211 \)
Analytic order of Ш: \( 1 \) (rounded)

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\((j)_{-}\)
\((3,a)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a+5)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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, 4 and 8.
Its isogeny class 15.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of 75.b6, 2880.bc6, defined over \(\Q\), so it is also a \(\Q\)-curve.