Properties

Label 22080.bv3
Conductor $22080$
Discriminant $1.032\times 10^{18}$
j-invariant \( \frac{39248884582600321}{3935264062500} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -453121, -106897921])
 
gp: E = ellinit([0, 1, 0, -453121, -106897921])
 
magma: E := EllipticCurve([0, 1, 0, -453121, -106897921]);
 

\(y^2=x^3+x^2-453121x-106897921\)  Toggle raw display

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{2}\Z\)

Torsion generators

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

\( \left(-289, 0\right) \), \( \left(769, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-481, 0\right) \), \( \left(-289, 0\right) \), \( \left(769, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 22080 \)  =  \(2^{6} \cdot 3 \cdot 5 \cdot 23\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(1031605862400000000 \)  =  \(2^{20} \cdot 3^{2} \cdot 5^{8} \cdot 23^{4} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{39248884582600321}{3935264062500} \)  =  \(2^{-2} \cdot 3^{-2} \cdot 5^{-8} \cdot 23^{-4} \cdot 339841^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(2.1927378411924389994939741313\dots\)
Stable Faltings height: \(1.1530170703525210353681259491\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.18520054825748351985707753203\dots\)
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: \( 64 \)  = \( 2^{2}\cdot2\cdot2\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(4\) = $2^2$ (exact)

Modular invariants

Modular form 22080.2.a.bv

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);
 

\( q + q^{3} - q^{5} + q^{9} + 4q^{11} + 2q^{13} - q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 294912
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 2.9632087721197363177132405124652647544 \)

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\((j)_{-}\)
\(2\) \(4\) \(I_{10}^{*}\) Additive -1 6 20 2
\(3\) \(2\) \(I_{2}\) Split multiplicative -1 1 2 2
\(5\) \(2\) \(I_{8}\) Non-split multiplicative 1 1 8 8
\(23\) \(4\) \(I_{4}\) Split multiplicative -1 1 4 4

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X101e.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 4 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 4 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 0 & 1 \end{array}\right)$ and has index 48.

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
\(2\) Cs

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 23
Reduction type add split nonsplit split
$\lambda$-invariant(s) - 1 0 1
$\mu$-invariant(s) - 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 \(d=\) 2 and 4.
Its isogeny class 22080.bv consists of 3 curves linked by isogenies of degrees dividing 8.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-23})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ \(\Q(\zeta_{24})\) \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.0.3057647616.9 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.8.1485512441856.2 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ 16.0.149587343098087735296.14 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.2206747214908975780724736.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) 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.