# Properties

 Label 16830cd5 Conductor $16830$ Discriminant $1.050\times 10^{21}$ j-invariant $$\frac{1696892787277117093383481}{1440538624914939000}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -22364168, 40683483731])

gp: E = ellinit([1, -1, 1, -22364168, 40683483731])

magma: E := EllipticCurve([1, -1, 1, -22364168, 40683483731]);

$$y^2+xy+y=x^3-x^2-22364168x+40683483731$$

## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{11163}{4}, -\frac{11167}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$16830$$ = $2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1050152657562990531000$ = $2^{3} \cdot 3^{9} \cdot 5^{3} \cdot 11^{12} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1696892787277117093383481}{1440538624914939000}$$ = $2^{-3} \cdot 3^{-3} \cdot 5^{-3} \cdot 11^{-12} \cdot 17^{-1} \cdot 181^{3} \cdot 227^{3} \cdot 2903^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.9607874489120620788144189750\dots$ Stable Faltings height: $2.4114813045780072331167963565\dots$

## 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: $0.15446112241278367685600282863\dots$ 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: $12$  = $3\cdot2\cdot1\cdot2\cdot1$ 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: $4$ = $2^2$ (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: $L(E,1)$ ≈ $1.8535334689534041222720339435$

## Modular invariants

Modular form 16830.2.a.bk

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);

$$q + q^{2} + q^{4} - q^{5} - 4 q^{7} + q^{8} - q^{10} - q^{11} + 2 q^{13} - 4 q^{14} + q^{16} + q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1769472 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$3$ $2$ $I_{3}^{*}$ Additive -1 2 9 3
$5$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$11$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$17$ $1$ $I_{1}$ 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
$2$ 2B 8.12.0.8
$3$ 3B.1.2 3.8.0.2

## $p$-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) $\mu$-invariant(s) 2 3 5 11 17 split add nonsplit nonsplit split 6 - 0 0 1 0 - 0 0 0

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

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, 3, 4, 6 and 12.
Its isogeny class 16830cd consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $K$ $2$ $$\Q(\sqrt{510})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{255})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-3})$$ $$\Z/6\Z$$ Not in database $3$ 3.1.867.1 $$\Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{255})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-170})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-85})$$ $$\Z/12\Z$$ Not in database $6$ 6.0.2255067.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $6$ 6.2.2453512896000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.2.384864768.2 $$\Z/12\Z$$ Not in database $6$ 6.2.306689112000.2 $$\Z/12\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.0.277102632960000.179 $$\Z/2\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.0.29712900642147807147732528143704650293701171875.2 $$\Z/18\Z$$ 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.