# Properties

 Label 1050i1 Conductor $1050$ Discriminant $3402000$ j-invariant $$\frac{4386781853}{27216}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -171, 838])

gp: E = ellinit([1, 0, 1, -171, 838])

magma: E := EllipticCurve([1, 0, 1, -171, 838]);

## Simplified equation

 $$y^2+xy+y=x^3-171x+838$$ y^2+xy+y=x^3-171x+838 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-171xz^2+838z^3$$ y^2z+xyz+yz^2=x^3-171xz^2+838z^3 (dehomogenize, simplify) $$y^2=x^3-220995x+39772350$$ y^2=x^3-220995x+39772350 (homogenize, minimize)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(2, 21\right)$$ (2, 21) $\hat{h}(P)$ ≈ $0.16875893648028187333446620325$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(7, -4\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-13, 36\right)$$, $$\left(-13, -24\right)$$, $$\left(2, 21\right)$$, $$\left(2, -24\right)$$, $$\left(7, -4\right)$$, $$\left(8, -3\right)$$, $$\left(8, -6\right)$$, $$\left(11, 12\right)$$, $$\left(11, -24\right)$$, $$\left(23, 84\right)$$, $$\left(23, -108\right)$$, $$\left(56, 381\right)$$, $$\left(56, -438\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1050$$ = $2 \cdot 3 \cdot 5^{2} \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $3402000$ = $2^{4} \cdot 3^{5} \cdot 5^{3} \cdot 7$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4386781853}{27216}$$ = $2^{-4} \cdot 3^{-5} \cdot 7^{-1} \cdot 1637^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.091346603799118917943577558753\dots$ Stable Faltings height: $-0.31101287430940617570661227455\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.16875893648028187333446620325\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.5211567220374541509662547583\dots$ 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: $20$  = $2\cdot5\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $2.1273386355557719404788954866$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 320 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $1$ $I_{1}$ Non-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
$2$ 2B 2.3.0.1

## $p$-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 nonsplit split add nonsplit ord ord ord ord ss ord ord ord ord ord ord 1 8 - 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 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 $d=$ 2.
Its isogeny class 1050i consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{105})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.42000.2 $$\Z/4\Z$$ Not in database $8$ 8.4.21441530250000.10 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.777924000000.11 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.106332486750000.8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.