Properties

Label 2093.g1
Conductor $2093$
Discriminant $-121324931$
j-invariant \( -\frac{9221261135586623488}{121324931} \)
CM no
Rank $1$
Torsion structure \(\Z/{3}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -43687, 3500082])
 
gp: E = ellinit([0, 1, 1, -43687, 3500082])
 
magma: E := EllipticCurve([0, 1, 1, -43687, 3500082]);
 

\(y^2+y=x^3+x^2-43687x+3500082\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{3}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(42, 1319\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.5043163845577101490304459331$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(120, 6\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(42, 1319\right) \), \( \left(42, -1320\right) \), \( \left(120, 6\right) \), \( \left(120, -7\right) \), \( \left(126, 108\right) \), \( \left(126, -109\right) \)  Toggle raw display

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: $-121324931 $  =  $-1 \cdot 7^{4} \cdot 13^{3} \cdot 23 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{9221261135586623488}{121324931} \)  =  $-1 \cdot 2^{15} \cdot 7^{-4} \cdot 13^{-3} \cdot 19^{3} \cdot 23^{-1} \cdot 3449^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.1095471340144167218391539356\dots$
Stable Faltings height: $1.1095471340144167218391539356\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.5043163845577101490304459331\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.3167669746701996108908448581\dots$
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 $  = $ 2^{2}\cdot3\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $3$
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) $ ≈ $ 2.6411121795211581025269456076754217951 $

Modular invariants

Modular form   2093.2.a.g

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3456
$ \Gamma_0(N) $-optimal: yes
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
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$23$ $1$ $I_{1}$ 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
$3$ 3B.1.1 3.8.0.1

$p$-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

$p$ 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 ordinary
$\lambda$-invariant(s) 2,7 1 1 2 1 2 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0,0 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.
Its isogeny class 2093.g consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.1196.1 \(\Z/6\Z\) Not in database
$6$ 6.0.427694384.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.18141252507.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.63834419075731858827.4 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.62442105498327731433149344118155881185783808.1 \(\Z/3\Z \times \Z/6\Z\) 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.