# Properties

 Label 1850k1 Conductor $1850$ Discriminant $-68450$ j-invariant $$-\frac{121945}{2738}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -3, -13])

gp: E = ellinit([1, 0, 0, -3, -13])

magma: E := EllipticCurve([1, 0, 0, -3, -13]);

## Simplified equation

 $$y^2+xy=x^3-3x-13$$ y^2+xy=x^3-3x-13 (homogenize, simplify) $$y^2z+xyz=x^3-3xz^2-13z^3$$ y^2z+xyz=x^3-3xz^2-13z^3 (dehomogenize, simplify) $$y^2=x^3-3915x-594810$$ y^2=x^3-3915x-594810 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{19}{4}, \frac{55}{8}\right)$$ (19/4, 55/8) $\hat{h}(P)$ ≈ $1.4493794213936244159169100521$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1850$$ = $2 \cdot 5^{2} \cdot 37$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-68450$ = $-1 \cdot 2 \cdot 5^{2} \cdot 37^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{121945}{2738}$$ = $-1 \cdot 2^{-1} \cdot 5 \cdot 29^{3} \cdot 37^{-2}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.39139985922221508672737444821\dots$ Stable Faltings height: $-0.65963951129456514916083433708\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.4493794213936244159169100521\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.5011184556692405062406279268\dots$ 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: $2$  = $1\cdot1\cdot2$ 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: $L'(E,1)$ ≈ $4.3513803974423496955593740406$

## 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);

$$q + q^{2} + q^{3} + q^{4} + q^{6} - 4 q^{7} + q^{8} - 2 q^{9} + 3 q^{11} + q^{12} - 6 q^{13} - 4 q^{14} + q^{16} - 3 q^{17} - 2 q^{18} - 3 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 192 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 primes 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$(j)_{-}$
$2$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $1$ $II$ Additive 1 2 2 0
$37$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## 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
$2$ 2G 8.2.0.1

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 split ord add ord ord ord ord ord ord ss ss split ord ord ord 3 1 - 1 1 1 1 1 1 3,1 1,1 2 1 1 1 0 0 - 0 0 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 no rational isogenies. Its isogeny class 1850k consists of this curve only.

## 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 $K$ $3$ 3.1.200.1 $$\Z/2\Z$$ Not in database $6$ 6.0.320000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.