Properties

Base field \(\Q(\sqrt{15}) \)
Label 2.2.60.1-30.1-c2
Conductor \((30,a + 15)\)
Conductor norm \( 30 \)
CM no
base-change yes: 150.b8,720.j8
Q-curve yes
Torsion order \( 4 \)
Rank not available

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Base field \(\Q(\sqrt{15}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 15 \); class number \(2\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 15)
 
gp: K = nfinit(a^2 - 15);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
 

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} + x^{2} + \left(48 a + 185\right) x + 1182 a + 4577 \)
sage: E = EllipticCurve(K, [a + 1, 1, a + 1, 48*a + 185, 1182*a + 4577])
 
gp: E = ellinit([a + 1, 1, a + 1, 48*a + 185, 1182*a + 4577],K)
 
magma: E := ChangeRing(EllipticCurve([a + 1, 1, a + 1, 48*a + 185, 1182*a + 4577]),K);
 

This is not a global minimal model: it is minimal at all primes except \((2,a + 1)\). No global minimal model exists.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((30,a + 15)\) = \( \left(2, a + 1\right) \cdot \left(3, a\right) \cdot \left(5, a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 30 \) = \( 2 \cdot 3 \cdot 5 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\((\Delta)\) = \((138240)\) = \( \left(2, a + 1\right)^{20} \cdot \left(3, a\right)^{6} \cdot \left(5, a\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\Delta)\) = \( 19110297600 \) = \( 2^{20} \cdot 3^{6} \cdot 5^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(\mathfrak{D}\) = \((2160)\) = \( \left(2, a + 1\right)^{8} \cdot \left(3, a\right)^{6} \cdot \left(5, a\right)^{2} \)
\(N(\mathfrak{D})\) = \( 4665600 \) = \( 2^{8} \cdot 3^{6} \cdot 5^{2} \)
\(j\) = \( \frac{357911}{2160} \)
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 not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

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: $\left(2 a + 7 : -23 a - 89 : 1\right)$
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);
 
Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(2, a + 1\right) \) \(2\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(3, a\right) \) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(5, a\right) \) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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 30.1-c consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 150.b8, 720.j8, defined over \(\Q\), so it is also a \(\Q\)-curve.