Properties

Label 20691.i2
Conductor $20691$
Discriminant $-8.858\times 10^{12}$
j-invariant \( -\frac{89915392}{6859} \)
CM no
Rank $1$
Torsion structure trivial

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -10164, -419598])
 
gp: E = ellinit([0, 0, 1, -10164, -419598])
 
magma: E := EllipticCurve([0, 0, 1, -10164, -419598]);
 

\(y^2+y=x^3-10164x-419598\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(\frac{2561}{4}, \frac{127877}{8}\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $6.8375054544538124927823273361$

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 20691 \)  =  \(3^{2} \cdot 11^{2} \cdot 19\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-8858178799371 \)  =  \(-1 \cdot 3^{6} \cdot 11^{6} \cdot 19^{3} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{89915392}{6859} \)  =  \(-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.2323867933905030755510666726\dots\)
Stable Faltings height: \(-0.51586698734273704217752773484\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(6.8375054544538124927823273361\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.23670378706858955357947015185\dots\)
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: \( 2 \)  = \( 2\cdot1\cdot1 \)
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)

Modular invariants

Modular form 20691.2.a.i

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 32400
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L'(E,1) \) ≈ \( 3.2369268703427097606953324582899095670 \)

Local data

This elliptic curve is not 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\((j)_{-}\)
\(3\) \(2\) \(I_0^{*}\) Additive -1 2 6 0
\(11\) \(1\) \(I_0^{*}\) Additive -1 2 6 0
\(19\) \(1\) \(I_{3}\) Non-split multiplicative 1 1 3 3

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(3\) Cs

$p$-adic data

$p$-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 add ordinary ordinary add ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2,5 - 1 1 - 3 1 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

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 \(d=\) 3.
Its isogeny class 20691.i consists of 2 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
$2$ \(\Q(\sqrt{33}) \) \(\Z/3\Z\) 2.2.33.1-361.1-d2
$2$ \(\Q(\sqrt{-11}) \) \(\Z/3\Z\) 2.0.11.1-29241.3-a2
$3$ 3.1.76.1 \(\Z/2\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-11})\) \(\Z/3\Z \times \Z/3\Z\) Not in database
$6$ 6.0.109744.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.2.207572112.3 \(\Z/6\Z\) Not in database
$6$ 6.0.7687856.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.6.102723230842848997856324612433.1 \(\Z/9\Z\) Not in database
$18$ 18.0.13265694774932146967552411030509446579.2 \(\Z/9\Z\) 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.