Properties

Label 22050bo2
Conductor $22050$
Discriminant $-3.163\times 10^{20}$
j-invariant \( -\frac{16591834777}{98304} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -7844517, -8497850859]) # or
 
sage: E = EllipticCurve("22050bo2")
 
gp: E = ellinit([1, -1, 0, -7844517, -8497850859]) \\ or
 
gp: E = ellinit("22050bo2")
 
magma: E := EllipticCurve([1, -1, 0, -7844517, -8497850859]); // or
 
magma: E := EllipticCurve("22050bo2");
 

\( y^2 + x y = x^{3} - x^{2} - 7844517 x - 8497850859 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{272988195}{75076}, -\frac{2199227456091}{20570824}\right) \)
\(\hat{h}(P)\) ≈  $18.419879049559864$

Integral points

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

None

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: \(-316299965216256000000 \)  =  \(-1 \cdot 2^{15} \cdot 3^{7} \cdot 5^{6} \cdot 7^{10} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{16591834777}{98304} \)  =  \(-1 \cdot 2^{-15} \cdot 3^{-1} \cdot 7^{2} \cdot 17^{3} \cdot 41^{3}\)
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: \(18.4198790496\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.0450893726182\)
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: \( 4 \)  = \( 1\cdot2^{2}\cdot1\cdot1 \)
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 22050.2.a.k

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);
 

\( q - q^{2} + q^{4} - q^{8} - 3q^{11} - 4q^{13} + q^{16} + 4q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1088640
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 3.32216316019 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{15} \) Non-split multiplicative 1 1 15 15
\(3\) \(4\) \( I_1^{*} \) Additive -1 2 7 1
\(5\) \(1\) \( I_0^{*} \) Additive 1 2 6 0
\(7\) \(1\) \( II^{*} \) Additive -1 2 10 0

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
\(3\) B

$p$-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 nonsplit add add add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) 2 - - - 3 1 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 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 22050bo 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
$2$ \(\Q(\sqrt{-35}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.1176.1 \(\Z/2\Z\) Not in database
$6$ 6.0.33191424.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.1210104000.1 \(\Z/6\Z\) Not in database
$6$ 6.2.372163703625.2 \(\Z/3\Z\) 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.