Properties

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

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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]);
 

\(y^2+y=x^3+x^2-14829659x-21985816061\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

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

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

Modular form   2093.2.a.h

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);
 

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

For more coefficients, see the Downloads section to the right.

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ordinary ordinary split ordinary split ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 2,7 1 1 4 1 2 1 1 1 1 1 1 1 1 1,1
$\mu$-invariant(s) 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) 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.