# Properties

 Label 2093.h1 Conductor $2093$ Discriminant $-5.622\times 10^{16}$ j-invariant $$-\frac{360675992659311050823073792}{56219378022244619}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -14829659, -21985816061])

gp: E = ellinit([0, 1, 1, -14829659, -21985816061])

magma: E := EllipticCurve([0, 1, 1, -14829659, -21985816061]);

$$y^2+y=x^3+x^2-14829659x-21985816061$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{14674995661}{1512900}, \frac{1608233748263209}{1860867000}\right)$$ $\hat{h}(P)$ ≈ $22.156379505909238641840451294$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2093$$ = $7 \cdot 13 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-56219378022244619$ = $-1 \cdot 7^{4} \cdot 13 \cdot 23^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{360675992659311050823073792}{56219378022244619}$$ = $-1 \cdot 2^{15} \cdot 7^{-4} \cdot 13^{-1} \cdot 23^{-9} \cdot 47^{3} \cdot 473287^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.6183204437568972490579051056\dots$ Stable Faltings height: $2.6183204437568972490579051056\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $22.156379505909238641840451294\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.038466862166583668996954842824\dots$ 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: $4$  = $2^{2}\cdot1\cdot1$ 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: $L'(E,1)$ ≈ $3.4091455862573194285761702474879761253$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 69984 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$23$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9

## 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 9.24.0.3

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

## 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 ss ordinary ordinary split ordinary split ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss 2,7 1 1 4 1 2 1 1 1 1 1 1 1 1 1,1 0,0 2 0 0 0 0 0 0 0 0 0 0 0 0 0,0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 2093.h 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 $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.1196.1 $$\Z/2\Z$$ Not in database $3$ 3.1.24843.1 $$\Z/3\Z$$ Not in database $6$ 6.0.427694384.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.1851523947.3 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.47258883.1 $$\Z/9\Z$$ Not in database $6$ 6.0.27546141987.4 $$\Z/9\Z$$ Not in database $6$ 6.0.38621232.1 $$\Z/6\Z$$ Not in database $9$ 9.1.155209733218789075008.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.2459068265791141604143011767237631147.19 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.650431654717887275719121056913118755401728.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.308913889405770026658073732789422918807552.1 $$\Z/18\Z$$ Not in database $18$ 18.0.2141884331629533998639849947891416830851166208.1 $$\Z/18\Z$$ Not in database $18$ 18.0.3810349083643812899547003581222403950394322944.2 $$\Z/2\Z \times \Z/6\Z$$ 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.