# Properties

 Label 19.a1 Conductor $19$ Discriminant $-19$ j-invariant $$-\frac{50357871050752}{19}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -769, -8470])

gp: E = ellinit([0, 1, 1, -769, -8470])

magma: E := EllipticCurve([0, 1, 1, -769, -8470]);

$$y^2+y=x^3+x^2-769x-8470$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$19$$ = $19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-19$ = $-1 \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{50357871050752}{19}$$ = $-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.033439156991317803520094883584\dots$ Stable Faltings height: $0.033439156991317803520094883584\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.45325324449610360357883918706\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.45325324449610360357883918707$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

$$q - 2 q^{3} - 2 q^{4} + 3 q^{5} - q^{7} + q^{9} + 3 q^{11} + 4 q^{12} - 4 q^{13} - 6 q^{15} + 4 q^{16} - 3 q^{17} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is semistable. There is only one prime of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$19$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

## Galois representations

sage: rho = E.galois_representation();

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

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

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B.1.2 27.72.0.2

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 19 ss ord split 0,3 0 1 0,0 2 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 19.a consists of 3 curves linked by isogenies of degrees dividing 9.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ 2.0.3.1-361.2-a1 $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $3$ 3.1.1083.1 $$\Z/3\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.3518667.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.2565108243.2 $$\Z/9\Z$$ Not in database $6$ 6.0.7105563.2 $$\Z/9\Z$$ Not in database $6$ 6.0.155952.1 $$\Z/6\Z$$ Not in database $9$ 9.1.1544610364992.1 $$\Z/6\Z$$ Not in database $12$ 12.2.937292452593664.2 $$\Z/4\Z$$ Not in database $12$ 12.0.8779890495744.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.16877848680315122776257224907.2 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.64417171850299425397321728.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.24956532218240038183849403152084992.1 $$\Z/18\Z$$ Not in database $18$ 18.0.191500465912938345959971172352.1 $$\Z/18\Z$$ Not in database $18$ 18.0.45330602413173669724041216.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive.