Properties

Label 22218bj1
Conductor $22218$
Discriminant $-1.034\times 10^{22}$
j-invariant \( \frac{83228502970940543}{69854999176704} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, 4811773, -2725907391])
 
gp: E = ellinit([1, 0, 0, 4811773, -2725907391])
 
magma: E := EllipticCurve([1, 0, 0, 4811773, -2725907391]);
 

\(y^2+xy=x^3+4811773x-2725907391\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(1332, 77097\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.27172435486009336452578041264$

Integral points

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

\( \left(1332, 77097\right) \), \( \left(1332, -78429\right) \), \( \left(467910, 319838553\right) \), \( \left(467910, -320306463\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 22218 \)  =  \(2 \cdot 3 \cdot 7 \cdot 23^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-10341046904217644729856 \)  =  \(-1 \cdot 2^{9} \cdot 3 \cdot 7^{11} \cdot 23^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{83228502970940543}{69854999176704} \)  =  \(2^{-9} \cdot 3^{-1} \cdot 7^{-11} \cdot 23^{-1} \cdot 436607^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(2.9118776753720351988886427607\dots\)
Stable Faltings height: \(1.3441305674074603534852663448\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.27172435486009336452578041264\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.071021640073201287920131756454\dots\)
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: \( 396 \)  = \( 3^{2}\cdot1\cdot11\cdot2^{2} \)
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 22218.2.a.bd

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2090880
\( \Gamma_0(N) \)-optimal: yes
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);
 

\( L'(E,1) \) ≈ \( 7.6421304946785641185587397497905459910 \)

Local data

This elliptic curve is not semistable. There are 4 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\((j)_{-}\)
\(2\) \(9\) \(I_{9}\) Split multiplicative -1 1 9 9
\(3\) \(1\) \(I_{1}\) Split multiplicative -1 1 1 1
\(7\) \(11\) \(I_{11}\) Split multiplicative -1 1 11 11
\(23\) \(4\) \(I_1^{*}\) Additive -1 2 7 1

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 \( p \) .

$p$-adic data

$p$-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 split split ordinary split ordinary ordinary ordinary ss add ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 3 2 3 2 1 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 no rational isogenies. Its isogeny class 22218bj consists of this curve only.

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
$3$ 3.1.3864.1 \(\Z/2\Z\) Not in database
$6$ 6.0.57691436544.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ Deg 8 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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.