Properties

Label 22386bc1
Conductor 22386
Discriminant 1115930860975816704
j-invariant \( \frac{3215014175651328584353}{1115930860975816704} \)
CM no
Rank 1
Torsion Structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -307482, 41491044]); // or
 
magma: E := EllipticCurve("22386bc1");
 
sage: E = EllipticCurve([1, 0, 0, -307482, 41491044]) # or
 
sage: E = EllipticCurve("22386bc1")
 
gp: E = ellinit([1, 0, 0, -307482, 41491044]) \\ or
 
gp: E = ellinit("22386bc1")
 

\( y^2 + x y = x^{3} - 307482 x + 41491044 \)

Mordell-Weil group structure

\(\Z\times \Z/{3}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(-588, 4662\right) \)
\(\hat{h}(P)\) ≈  0.145279908466

Torsion generators

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

\( \left(660, 10902\right) \)

Integral points

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

\( \left(-588, 4662\right) \), \( \left(-588, -4074\right) \), \( \left(-462, 9450\right) \), \( \left(-462, -8988\right) \), \( \left(-420, 10038\right) \), \( \left(-420, -9618\right) \), \( \left(-42, 7392\right) \), \( \left(-42, -7350\right) \), \( \left(84, 3990\right) \), \( \left(84, -4074\right) \), \( \left(140, 1022\right) \), \( \left(140, -1162\right) \), \( \left(468, 150\right) \), \( \left(468, -618\right) \), \( \left(504, 3570\right) \), \( \left(504, -4074\right) \), \( \left(660, 10902\right) \), \( \left(660, -11562\right) \), \( \left(1092, 31206\right) \), \( \left(1092, -32298\right) \), \( \left(2064, 89526\right) \), \( \left(2064, -91590\right) \), \( \left(2532, 123222\right) \), \( \left(2532, -125754\right) \), \( \left(8148, 729750\right) \), \( \left(8148, -737898\right) \), \( \left(206544, 93764730\right) \), \( \left(206544, -93971274\right) \)

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: \(1115930860975816704 \)  =  \(2^{18} \cdot 3^{9} \cdot 7^{4} \cdot 13^{3} \cdot 41 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{3215014175651328584353}{1115930860975816704} \)  =  \(2^{-18} \cdot 3^{-9} \cdot 7^{-4} \cdot 13^{-3} \cdot 41^{-1} \cdot 89^{3} \cdot 165833^{3}\)
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: \(0.145279908466\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.252798424535\)
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: \( 1944 \)  = \( ( 2 \cdot 3^{2} )\cdot3^{2}\cdot2^{2}\cdot3\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(3\)
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: 435456
\( \Gamma_0(N) \)-optimal: yes
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\((j)_{-}\)
\(2\) \(18\) \( I_{18} \) Split multiplicative -1 1 18 18
\(3\) \(9\) \( I_{9} \) Split multiplicative -1 1 9 9
\(7\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4
\(13\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(41\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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.1.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.6396.1 \(\Z/6\Z\) Not in database
6 6.6.261652787136.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
6.0.183185608347.1 \(\Z/3\Z \times \Z/3\Z\) 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.