Properties

Label 22050bl1
Conductor $22050$
Discriminant $-2.802\times 10^{17}$
j-invariant \( \frac{46969655}{130691232} \)
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([1, -1, 0, 9693, -25468619]) # or
 
sage: E = EllipticCurve("22050bl1")
 
gp: E = ellinit([1, -1, 0, 9693, -25468619]) \\ or
 
gp: E = ellinit("22050bl1")
 
magma: E := EllipticCurve([1, -1, 0, 9693, -25468619]); // or
 
magma: E := EllipticCurve("22050bl1");
 

\( y^2 + x y = x^{3} - x^{2} + 9693 x - 25468619 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(6365, 504629\right) \)
\(\hat{h}(P)\) ≈  $3.309769278720521$

Integral points

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

\( \left(6365, 504629\right) \), \( \left(6365, -510994\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 22050 \)  =  \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-280222000433776800 \)  =  \(-1 \cdot 2^{5} \cdot 3^{11} \cdot 5^{2} \cdot 7^{11} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{46969655}{130691232} \)  =  \(2^{-5} \cdot 3^{-5} \cdot 5 \cdot 7^{-5} \cdot 211^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(3.30976927872\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.143214659489\)
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: \( 8 \)  = \( 1\cdot2\cdot1\cdot2^{2} \)
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 22050.2.a.w

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);
 

\( q - q^{2} + q^{4} - q^{8} - 2q^{11} + q^{13} + q^{16} + 3q^{17} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 230400
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 3.79205984191 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(1\) \( I_{5} \) Non-split multiplicative 1 1 5 5
\(3\) \(2\) \( I_5^{*} \) Additive -1 2 11 5
\(5\) \(1\) \( II \) Additive 1 2 2 0
\(7\) \(4\) \( I_5^{*} \) Additive -1 2 11 5

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
\(5\) B.4.1

$p$-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 nonsplit add add add ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4 - - - 1 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

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 \(d=\) 5.
Its isogeny class 22050bl consists of 2 curves linked by isogenies of degree 5.

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
$2$ \(\Q(\sqrt{21}) \) \(\Z/5\Z\) Not in database
$3$ 3.1.4200.1 \(\Z/2\Z\) Not in database
$6$ 6.2.370440000.1 \(\Z/10\Z\) Not in database
$6$ 6.0.2963520000.1 \(\Z/2\Z \times \Z/2\Z\) 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.