Properties

Base field \(\Q(\sqrt{3}, \sqrt{5})\)
Label 4.4.3600.1-225.1-e7
Conductor \((15,a^{2} - a - 4)\)
Conductor norm \( 225 \)
CM no
base-change yes: 15.a4,75.b4,720.c4,3600.u4
Q-curve yes
Torsion order \( 16 \)
Rank not available

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{3}, \sqrt{5})\)

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 8, -7, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 7*x^2 + 8*x + 1)
gp (2.8): K = nfinit(a^4 - 2*a^3 - 7*a^2 + 8*a + 1);

Weierstrass equation

\( y^2 + x y + y = x^{3} + x^{2} - 80 x + 242 \)
magma: E := ChangeRing(EllipticCurve([1, 1, 1, -80, 242]),K);
sage: E = EllipticCurve(K, [1, 1, 1, -80, 242])
gp (2.8): E = ellinit([1, 1, 1, -80, 242],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((15,a^{2} - a - 4)\) = \( \left(-\frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{19}{7} a - \frac{10}{7}\right) \cdot \left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{13}{7}\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 225 \) = \( 9 \cdot 25 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((15,-\frac{30}{7} a^{3} + \frac{45}{7} a^{2} + \frac{180}{7} a - \frac{150}{7},-\frac{30}{7} a^{3} + \frac{45}{7} a^{2} + \frac{285}{7} a - \frac{150}{7},\frac{15}{7} a^{3} + \frac{30}{7} a^{2} - \frac{195}{7} a - \frac{135}{7})\) = \( \left(-\frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{19}{7} a - \frac{10}{7}\right)^{2} \cdot \left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{13}{7}\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 50625 \) = \( 9^{2} \cdot 25^{2} \)
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: \(\Z/2\Z\times\Z/8\Z\)
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]
Generators: $\left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{62}{7} : \frac{16}{7} a^{3} - \frac{24}{7} a^{2} - \frac{96}{7} a + \frac{94}{7} : 1\right)$,$\left(2 a^{2} - 2 a - 11 : -a^{2} + a + 5 : 1\right)$
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\((j)_{-}\)
\( \left(-\frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{19}{7} a - \frac{10}{7}\right) \) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{13}{7}\right) \) \(25\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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, 8, 16 and 32.
Its isogeny class 225.1-e consists of curves linked by isogenies of degrees dividing 64.

Base change

This curve is the base-change of elliptic curves 15.a4, 75.b4, 720.c4, 3600.u4, defined over \(\Q\), so it is also a \(\Q\)-curve.