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Properties

Label 9660.d2
Conductor $9660$
Discriminant $-5.257\times 10^{14}$
j-invariant \( \frac{2477112820760576}{2053567248075} \)
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, 1, 0, 17899, -600201])
 
gp: E = ellinit([0, 1, 0, 17899, -600201])
 
magma: E := EllipticCurve([0, 1, 0, 17899, -600201]);
 

\(y^2=x^3+x^2+17899x-600201\) Copy content 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);
 

$P$ =  \(\left(49, 630\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.31806963648412334511625555494$

Torsion generators

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

\( \left(85, 1242\right) \) Copy content Toggle raw display

Integral points

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

\((49,\pm 630)\), \((85,\pm 1242)\), \((154,\pm 2415)\), \((274,\pm 4995)\), \((301,\pm 5670)\), \((637,\pm 16422)\), \((729,\pm 20010)\), \((2569,\pm 130410)\), \((155449,\pm 61289130)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 9660 \)  =  $2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-525713215507200 $  =  $-1 \cdot 2^{8} \cdot 3^{9} \cdot 5^{2} \cdot 7^{3} \cdot 23^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2477112820760576}{2053567248075} \)  =  $2^{22} \cdot 3^{-9} \cdot 5^{-2} \cdot 7^{-3} \cdot 23^{-3} \cdot 839^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.5123802024419454973786965010\dots$
Stable Faltings height: $1.0502820820686486244338750867\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.31806963648412334511625555494\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.28823619902609830158796933013\dots$
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: $ 486 $  = $ 3\cdot3^{2}\cdot2\cdot3\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) $ ≈ $ 4.9506758844730117636903094600 $

Modular invariants

Modular form   9660.2.a.d

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);
 

\( q + q^{3} - q^{5} + q^{7} + q^{9} - 3 q^{11} - 4 q^{13} - q^{15} + 6 q^{17} - q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 38880
$ \Gamma_0(N) $-optimal: yes
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$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$3$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$23$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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
$3$ 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]
 

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add split nonsplit split ord ord ord ord split ord ord ord ord ord ord
$\lambda$-invariant(s) - 4 1 2 1 1 1 1 4 1 1 1 1 1 1
$\mu$-invariant(s) - 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 9660.d 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
$3$ 3.1.1932.1 \(\Z/6\Z\) Not in database
$6$ 6.0.1802857392.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.270000.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
$9$ 9.3.116551616127480000.11 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$18$ 18.0.342804527144547363000000000000.1 \(\Z/3\Z \times \Z/6\Z\) 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.