Properties

Label 2205i2
Conductor $2205$
Discriminant $-1.970\times 10^{14}$
j-invariant \( -\frac{19539165184}{46875} \)
CM no
Rank $1$
Torsion structure \(\Z/{3}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -90552, 10509777])
 
gp: E = ellinit([0, 0, 1, -90552, 10509777])
 
magma: E := EllipticCurve([0, 0, 1, -90552, 10509777]);
 

\(y^2+y=x^3-90552x+10509777\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{3}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(197, 562\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.2160420176800910763637668475$

Torsion generators

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

\( \left(147, 612\right) \)  Toggle raw display

Integral points

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

\( \left(-343, 1102\right) \), \( \left(-343, -1103\right) \), \( \left(135, 863\right) \), \( \left(135, -864\right) \), \( \left(147, 612\right) \), \( \left(147, -613\right) \), \( \left(197, 562\right) \), \( \left(197, -563\right) \), \( \left(14847, 1808712\right) \), \( \left(14847, -1808713\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 2205 \)  =  $3^{2} \cdot 5 \cdot 7^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-196994059171875 $  =  $-1 \cdot 3^{7} \cdot 5^{6} \cdot 7^{8} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{19539165184}{46875} \)  =  $-1 \cdot 2^{21} \cdot 3^{-1} \cdot 5^{-6} \cdot 7 \cdot 11^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.6221775593704048156923013142\dots$
Stable Faltings height: $-0.22440201766719223340888979989\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.2160420176800910763637668475\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.56681566115941669508946213301\dots$
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: $ 36 $  = $ 2\cdot( 2 \cdot 3 )\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $3$
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) $ ≈ $ 2.7570866409958876381512455261432755354 $

Modular invariants

Modular form   2205.2.a.f

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);
 

\( q - 2q^{4} + q^{5} - q^{13} + 4q^{16} - 6q^{17} + 5q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8064
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$3$ $2$ $I_1^{*}$ Additive -1 2 7 1
$5$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$7$ $3$ $IV^{*}$ Additive 1 2 8 0

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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 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]
 

$p$-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 ss add split add ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2,3 - 2 - 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 2205i consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.588.1 \(\Z/6\Z\) Not in database
$6$ 6.0.1037232.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.425329947.3 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.26377102494421875.3 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.315164892649262158461997559808.2 \(\Z/6\Z \times \Z/6\Z\) 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.