# Properties

 Label 9702.f1 Conductor $9702$ Discriminant $-5.731\times 10^{19}$ j-invariant $$-\frac{34068278205171}{10307264}$$ CM no Rank $0$ Torsion structure trivial

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -11964633, 15936473069])

gp: E = ellinit([1, -1, 0, -11964633, 15936473069])

magma: E := EllipticCurve([1, -1, 0, -11964633, 15936473069]);

## Simplified equation

 $$y^2+xy=x^3-x^2-11964633x+15936473069$$ y^2+xy=x^3-x^2-11964633x+15936473069 (homogenize, simplify) $$y^2z+xyz=x^3-x^2z-11964633xz^2+15936473069z^3$$ y^2z+xyz=x^3-x^2z-11964633xz^2+15936473069z^3 (dehomogenize, simplify) $$y^2=x^3-191434131x+1019742842286$$ y^2=x^3-191434131x+1019742842286 (homogenize, minimize)

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: $-57307978910298650688$ = $-1 \cdot 2^{6} \cdot 3^{9} \cdot 7^{10} \cdot 11^{5}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{34068278205171}{10307264}$$ = $-1 \cdot 2^{-6} \cdot 3^{3} \cdot 7^{2} \cdot 11^{-5} \cdot 2953^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.7688462006097401859222322511\dots$ Stable Faltings height: $0.32329519322923016312133770387\dots$

## 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: $0.19393398146725376875331045033\dots$ 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: $4$  = $2\cdot2\cdot1\cdot1$ 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: $L(E,1)$ ≈ $0.77573592586901507501324180132$

## 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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 423360 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$7$ $1$ $II^{*}$ Additive -1 2 10 0
$11$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5

## 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 $\ell$.

## $p$-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 5 7 11 13 17 19 23 29 31 37 41 43 47 nonsplit add ord add nonsplit ord ord ord ord ord ord ord ord ord ord 3 - 0 - 0 0 2 0 2 2 0 0 0 2 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 no rational isogenies. Its isogeny class 9702.f consists of this curve only.

## 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 $K$ $3$ 3.1.6468.1 $$\Z/2\Z$$ Not in database $6$ 6.0.5522223168.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.3767105332683.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.