Properties

Label 11550.cu1
Conductor $11550$
Discriminant $1.063\times 10^{18}$
j-invariant \( \frac{260744057755293612689909}{8504954620259328} \)
CM no
Rank $0$
Torsion structure \(\Z/{10}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -6654778, 6606931172])
 
gp: E = ellinit([1, 0, 0, -6654778, 6606931172])
 
magma: E := EllipticCurve([1, 0, 0, -6654778, 6606931172]);
 

\(y^2+xy=x^3-6654778x+6606931172\)  Toggle raw display

Mordell-Weil group structure

$\Z/{10}\Z$

Torsion generators

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

\( \left(1592, 6134\right) \)  Toggle raw display

Integral points

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

\( \left(1508, 422\right) \), \( \left(1508, -1930\right) \), \( \left(1592, 6134\right) \), \( \left(1592, -7726\right) \), \( \left(2096, 42170\right) \), \( \left(2096, -44266\right) \), \( \left(6212, 449654\right) \), \( \left(6212, -455866\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 11550 \)  =  $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1063119327532416000 $  =  $2^{10} \cdot 3^{5} \cdot 5^{3} \cdot 7^{10} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{260744057755293612689909}{8504954620259328} \)  =  $2^{-10} \cdot 3^{-5} \cdot 7^{-10} \cdot 11^{-2} \cdot 29^{3} \cdot 2202961^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.5530021248506737976151944655\dots$
Stable Faltings height: $2.1506426467421487039650046322\dots$

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: $0.25788941075355022381234201319\dots$
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: $ 2000 $  = $ ( 2 \cdot 5 )\cdot5\cdot2\cdot( 2 \cdot 5 )\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $10$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L(E,1) $ ≈ $ 5.1577882150710044762468402637956388177 $

Modular invariants

Modular form 11550.2.a.cu

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384000
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $5$ $I_{5}$ Split multiplicative -1 1 5 5
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$11$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1
$5$ 5B.1.1 5.24.0.1

$p$-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 7 11
Reduction type split split add split split
$\lambda$-invariant(s) 4 1 - 1 1
$\mu$-invariant(s) 0 0 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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, 5 and 10.
Its isogeny class 11550.cu consists of 4 curves linked by isogenies of degrees dividing 10.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{15}) \) \(\Z/2\Z \times \Z/10\Z\) Not in database
$4$ 4.4.8893500.3 \(\Z/20\Z\) Not in database
$8$ 8.8.11389585284000000.4 \(\Z/2\Z \times \Z/20\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/20\Z\) Not in database
$8$ Deg 8 \(\Z/30\Z\) Not in database
$16$ Deg 16 \(\Z/40\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/30\Z\) Not in database
$20$ 20.0.1402274470934209014892578125.2 \(\Z/5\Z \times \Z/10\Z\) 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.