# Properties

 Base field $$\Q(\sqrt{13})$$ Label 2.2.13.1-441.1-a3 Conductor $$(21)$$ Conductor norm $$441$$ CM no base-change yes: 21.a5,3549.c5 Q-curve yes Torsion order $$8$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{13})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 3)

gp (2.8): K = nfinit(a^2 - a - 3);

## Weierstrass equation

$$y^2 + x y = x^{3} - 4 x - 1$$
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -4, -1]),K);

sage: E = EllipticCurve(K, [1, 0, 0, -4, -1])

gp (2.8): E = ellinit([1, 0, 0, -4, -1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(21)$$ = $$\left(-a\right) \cdot \left(-a + 1\right) \cdot \left(7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$441$$ = $$3^{2} \cdot 49$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(3969)$$ = $$\left(-a\right)^{4} \cdot \left(-a + 1\right)^{4} \cdot \left(7\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$15752961$$ = $$3^{8} \cdot 49^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{7189057}{3969}$$ 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: $$\Z/2\Z\times\Z/4\Z$$ 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] $\left(-1 : -1 : 1\right)$,$\left(-\frac{1}{4} : \frac{1}{8} : 1\right)$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(-a + 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(7\right)$$ $$49$$ $$2$$ $$I_{2}$$ 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 and 4.
Its isogeny class 441.1-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base-change of elliptic curves 21.a5, 3549.c5, defined over $$\Q$$, so it is also a $$\Q$$-curve.