Properties

Label 1122m1
Conductor $1122$
Discriminant $1.115\times 10^{16}$
j-invariant \( \frac{959024269496848362625}{11151660319506432} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -205448, -35497920])
 
gp: E = ellinit([1, 0, 0, -205448, -35497920])
 
magma: E := EllipticCurve([1, 0, 0, -205448, -35497920]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3-205448x-35497920\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-205448xz^2-35497920z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-266260635x-1655392173642\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z/{6}\Z\)

Torsion generators

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

\( \left(-272, 664\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-272, 664\right) \), \( \left(-272, -392\right) \), \( \left(-240, 120\right) \), \( \left(784, 16504\right) \), \( \left(784, -17288\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1122 \)  =  $2 \cdot 3 \cdot 11 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $11151660319506432 $  =  $2^{30} \cdot 3^{3} \cdot 11^{3} \cdot 17^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{959024269496848362625}{11151660319506432} \)  =  $2^{-30} \cdot 3^{-3} \cdot 5^{3} \cdot 11^{-3} \cdot 17^{-2} \cdot 97^{3} \cdot 20333^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.8916387403250194511122064916\dots$
Stable Faltings height: $1.8916387403250194511122064916\dots$

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: $0.22440334251119608159877893810\dots$
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: $ 540 $  = $ ( 2 \cdot 3 \cdot 5 )\cdot3\cdot3\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 3.3660501376679412239816840715 $

Modular invariants

Modular form   1122.2.a.j

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);
 

\( q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{11} + q^{12} - 4 q^{13} + 2 q^{14} + q^{16} - q^{17} + q^{18} + 8 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 11520
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $30$ $I_{30}$ Split multiplicative -1 1 30 30
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{2}$ Non-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 8.6.0.4
$3$ 3B.1.1 3.8.0.1

$p$-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 11 17
Reduction type split split ss split nonsplit
$\lambda$-invariant(s) 6 3 2,2 1 0
$\mu$-invariant(s) 0 0 0,0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 1122m consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{33}) \) \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$4$ 4.0.2112.1 \(\Z/12\Z\) Not in database
$6$ 6.0.2255067.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$8$ 8.0.4857532416.8 \(\Z/2\Z \oplus \Z/12\Z\) Not in database
$9$ 9.3.135878452228432951488.2 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.6.663503169991663630174767723942598916340092928.1 \(\Z/2\Z \oplus \Z/18\Z\) 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.