Properties

Label 22386bc2
Conductor 22386
Discriminant 21429653098766238144
j-invariant \( \frac{120986373702456846135875233}{21429653098766238144} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

Related objects

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -10303962, -12729674844]); // or
 
magma: E := EllipticCurve("22386bc2");
 
sage: E = EllipticCurve([1, 0, 0, -10303962, -12729674844]) # or
 
sage: E = EllipticCurve("22386bc2")
 
gp: E = ellinit([1, 0, 0, -10303962, -12729674844]) \\ or
 
gp: E = ellinit("22386bc2")
 

\( y^2 + x y = x^{3} - 10303962 x - 12729674844 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(-1866, 1962\right) \)
\(\hat{h}(P)\) ≈  0.435839725398

Integral points

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

\( \left(-1866, 1962\right) \), \( \left(-1866, -96\right) \), \( \left(3744, 32442\right) \), \( \left(3744, -36186\right) \), \( \left(8424, 701682\right) \), \( \left(8424, -710106\right) \)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 22386 \)  =  \(2 \cdot 3 \cdot 7 \cdot 13 \cdot 41\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(21429653098766238144 \)  =  \(2^{6} \cdot 3^{3} \cdot 7^{12} \cdot 13 \cdot 41^{3} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{120986373702456846135875233}{21429653098766238144} \)  =  \(2^{-6} \cdot 3^{-3} \cdot 7^{-12} \cdot 13^{-1} \cdot 41^{-3} \cdot 89^{3} \cdot 919^{3} \cdot 6047^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.435839725398\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.0842661415116\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 216 \)  = \( ( 2 \cdot 3 )\cdot3\cdot( 2^{2} \cdot 3 )\cdot1\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 22386.2.a.u

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - 3q^{10} - 3q^{11} + q^{12} + q^{13} + q^{14} - 3q^{15} + q^{16} + 3q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 1306368
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 7.93293090699 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(3\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(7\) \(12\) \( I_{12} \) Split multiplicative -1 1 12 12
\(13\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(41\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3

Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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

prime Image of Galois representation
\(3\) B.1.2

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 split split ordinary split ordinary split ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary
$\lambda$-invariant(s) 4 2 1 4 1 2 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0 1 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 22386bc consists of 2 curves linked by isogenies of degree 3.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 3.3.6396.1 \(\Z/2\Z\) Not in database
3.1.4563.1 \(\Z/3\Z\) Not in database
6 6.0.122726448.1 \(\Z/6\Z\) Not in database
6.0.62462907.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
6.6.261652787136.1 \(\Z/2\Z \times \Z/2\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.