Properties

Base field \(\Q(\sqrt{3}) \)
Label 2.2.12.1-75.1-b7
Conductor \((5 a)\)
Conductor norm \( 75 \)
CM no
base-change yes: 15.a4,720.c4
Q-curve yes
Torsion order \( 8 \)
Rank not available

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Base field \(\Q(\sqrt{3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)
 
gp (2.8): K = nfinit(a^2 - 3);
 

Weierstrass equation

\( y^2 + a x y = x^{3} + \left(a - 1\right) x^{2} + \left(4481 a - 7761\right) x - 217866 a + 377355 \)
magma: E := ChangeRing(EllipticCurve([a, a - 1, 0, 4481*a - 7761, -217866*a + 377355]),K);
 
sage: E = EllipticCurve(K, [a, a - 1, 0, 4481*a - 7761, -217866*a + 377355])
 
gp (2.8): E = ellinit([a, a - 1, 0, 4481*a - 7761, -217866*a + 377355],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((5 a)\) = \( \left(a\right) \cdot \left(5\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 75 \) = \( 3 \cdot 25 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((15)\) = \( \left(a\right)^{2} \cdot \left(5\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 225 \) = \( 3^{2} \cdot 25 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{56667352321}{15} \)
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: \(\Z/8\Z\)
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]
 
Generator: $\left(-21 a + 36 : -17 a + 30 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(5\right) \) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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, 8 and 16.
Its isogeny class 75.1-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 15.a4, 720.c4, defined over \(\Q\), so it is also a \(\Q\)-curve.