Properties

 Label 60984.g1 Conductor $60984$ Discriminant $2.640\times 10^{13}$ j-invariant $$\frac{598885164}{539}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -64251, 6263686])

gp: E = ellinit([0, 0, 0, -64251, 6263686])

magma: E := EllipticCurve([0, 0, 0, -64251, 6263686]);

$$y^2=x^3-64251x+6263686$$

Mordell-Weil group structure

$\Z^2 \times \Z/{2}\Z$

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(110, 726\right)$$ $$\left(1199, 40656\right)$$ $\hat{h}(P)$ ≈ $1.1017402619666572675563392508$ $1.5058462633918083819435208111$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(143, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-241,\pm 2784)$$, $$(110,\pm 726)$$, $$(135,\pm 224)$$, $$\left(143, 0\right)$$, $$(170,\pm 504)$$, $$(231,\pm 1936)$$, $$(506,\pm 10164)$$, $$(1199,\pm 40656)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$60984$$ = $2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $26400283886592$ = $2^{10} \cdot 3^{3} \cdot 7^{2} \cdot 11^{7}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{598885164}{539}$$ = $2^{2} \cdot 3^{6} \cdot 7^{-2} \cdot 11^{-1} \cdot 59^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.4997755032122017712917241397\dots$ Stable Faltings height: $-0.55144785582063201476908572639\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.6227073215291293984217611393\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.66438297316062495298298440639\dots$ 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: $32$  = $2\cdot2\cdot2\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $8.6247929187762974644072971445343914929$

Modular invariants

Modular form 60984.2.a.g

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);

$$q - 2q^{5} - q^{7} - 4q^{13} - 6q^{17} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 215040 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive 1 3 10 0
$3$ $2$ $III$ Additive 1 2 3 0
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$11$ $4$ $I_1^{*}$ Additive -1 2 7 1

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 2.3.0.1

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ordinary nonsplit add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - - 2 2 - 2 2 2 2 2 2 2 2 2 2 - - 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 $d=$ 2.
Its isogeny class 60984.g consists of 2 curves linked by isogenies of degree 2.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{33})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.19008.2 $$\Z/4\Z$$ Not in database $8$ 8.4.3175234143814656.26 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.43717791744.1 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.