# Properties

 Label 19600.bd1 Conductor 19600 Discriminant -807072140000000 j-invariant $$-\frac{225637236736}{1715}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -986533, -376825063]); // or

magma: E := EllipticCurve("19600cj2");

sage: E = EllipticCurve([0, -1, 0, -986533, -376825063]) # or

sage: E = EllipticCurve("19600cj2")

gp: E = ellinit([0, -1, 0, -986533, -376825063]) \\ or

gp: E = ellinit("19600cj2")

$$y^2 = x^{3} - x^{2} - 986533 x - 376825063$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(1237, 17150\right)$$ $$\hat{h}(P)$$ ≈ 1.3908594829

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(1237,\pm 17150)$$, $$(8552,\pm 785225)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$19600$$ = $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-807072140000000$$ = $$-1 \cdot 2^{8} \cdot 5^{7} \cdot 7^{9}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{225637236736}{1715}$$ = $$-1 \cdot 2^{16} \cdot 5^{-1} \cdot 7^{-3} \cdot 151^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.3908594829$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0757428112036$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$32$$  = $$2\cdot2^{2}\cdot2^{2}$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 19600.2.a.bd

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{3} - 2q^{9} - 3q^{11} - q^{13} - 3q^{17} + 2q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 165888 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$3.37112343117$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_0^{*}$$ Additive -1 4 8 0
$$5$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$7$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add ordinary add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary - 1 - - 1 1 1 1 1 1 1 1 1,1 1 1 - 1 - - 0 0 0 0 0 0 0 0 0,0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 19600.bd consists of 2 curves linked by isogenies of degree 3.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-105})$$ $$\Z/3\Z$$ Not in database
3 3.1.140.1 $$\Z/2\Z$$ Not in database
6 6.0.296352000.1 $$\Z/6\Z$$ Not in database
6.2.200037600000.3 $$\Z/3\Z$$ Not in database
6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.