Properties

 Label 9702d2 Conductor $9702$ Discriminant $-3.221\times 10^{13}$ j-invariant $$-\frac{61279455929796531}{681472}$$ CM no Rank $0$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -2969556, -1968891184])

gp: E = ellinit([1, -1, 0, -2969556, -1968891184])

magma: E := EllipticCurve([1, -1, 0, -2969556, -1968891184]);

$$y^2+xy=x^3-x^2-2969556x-1968891184$$

trivial

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$9702$$ = $2 \cdot 3^{2} \cdot 7^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-32205605515776$ = $-1 \cdot 2^{9} \cdot 3^{9} \cdot 7^{4} \cdot 11^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{61279455929796531}{681472}$$ = $-1 \cdot 2^{-9} \cdot 3^{6} \cdot 7^{2} \cdot 11^{-3} \cdot 11971^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.1609799973290750999400522130\dots$ Stable Faltings height: $0.68838406447622172969183403749\dots$

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: $0.057503816818278176827702882549\dots$ 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: $18$  = $1\cdot2\cdot3\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L(E,1)$ ≈ $1.0350687027290071828986518858816817931$

Modular invariants

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);

$$q - q^{2} + q^{4} - 3 q^{5} - q^{8} + 3 q^{10} + q^{11} + 2 q^{13} + q^{16} + 3 q^{17} + 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 155520 $\Gamma_0(N)$-optimal: no Manin constant: 1

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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$7$ $3$ $IV$ Additive 1 2 4 0
$11$ $3$ $I_{3}$ 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.1.2 9.24.0.4

$p$-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

$p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 7 11 nonsplit add add split 5 - - 1 0 - - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $d=$ 3.
Its isogeny class 9702d 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.12936.1 $$\Z/2\Z$$ Not in database $3$ 3.1.1323.1 $$\Z/3\Z$$ Not in database $6$ 6.0.44177785344.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.5250987.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.502020288.1 $$\Z/6\Z$$ Not in database $9$ 9.1.4734224010819072.4 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.8549394874383196572862347.3 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.67238630953847462271924956823552.3 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.45821244313380337809373244176459628544.2 $$\Z/2\Z \times \Z/6\Z$$ 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.