# 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

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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: 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: 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: E.disc $$N(\mathfrak{D})$$ = $$50625$$ = $$9^{2} \cdot 25^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{56667352321}{15}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/8\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(\frac{4}{7} a^{3} - \frac{6}{7} a^{2} - \frac{24}{7} a + \frac{62}{7} : \frac{16}{7} a^{3} - \frac{24}{7} a^{2} - \frac{96}{7} a + \frac{94}{7} : 1\right)$,$\left(2 a^{2} - 2 a - 11 : -a^{2} + a + 5 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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.