# Properties

 Label 19.a2 Conductor $19$ Discriminant $-6859$ j-invariant $-\frac{89915392}{6859}$ CM no Rank $0$ Torsion Structure $\Z/{3}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

This is a model for the modular curve $X_0(19)$.

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -9, -15]); // or
magma: E := EllipticCurve("19a1");
sage: E = EllipticCurve([0, 1, 1, -9, -15]) # or
sage: E = EllipticCurve("19a1")
gp: E = ellinit([0, 1, 1, -9, -15]) \\ or
gp: E = ellinit("19a1")

$y^2 + y = x^{3} + x^{2} - 9 x - 15$

## Mordell-Weil group structure

$\Z/{3}\Z$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$\left(5, 9\right)$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$\left(5, 9\right)$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $19$ = $19$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-6859$ = $-1 \cdot 19^{3}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $-\frac{89915392}{6859}$ = $-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$ $\text{End} (E)$ = $\Z$ (no Complex Multiplication) $\text{ST} (E)$ = $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $1.35975973349$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $3$  = $3$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $3$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form19.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1 : curve is $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

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])

$L(E,1)$ ≈ $0.453253244496$

## 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$(j)_{-}$
$19$ $3$ $I_{3}$ Split multiplicative -1 1 3 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
$3$ Cs.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 ordinary split 0,3 0 1 0,0 1 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 up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{-3})$ $\Z/3\Z \times \Z/3\Z$ 2.0.3.1-361.2-a4
3 3.1.76.1 $\Z/6\Z$ Not in database
6 6.0.155952.1 $\Z/3\Z \times \Z/6\Z$ Not in database
6.0.109744.2 $\Z/2\Z \times \Z/6\Z$ 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.