Properties

 Label 16650.cb1 Conductor $16650$ Discriminant $2.845\times 10^{17}$ j-invariant $$\frac{4385367890843575421521}{24975000000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -76726130, -258660935503]) # or

sage: E = EllipticCurve("16650.cb1")

gp: E = ellinit([1, -1, 1, -76726130, -258660935503]) \\ or

gp: E = ellinit("16650.cb1")

magma: E := EllipticCurve([1, -1, 1, -76726130, -258660935503]); // or

magma: E := EllipticCurve("16650.cb1");

$$y^2+xy+y=x^3-x^2-76726130x-258660935503$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(92599, 28000575\right)$$ $$\hat{h}(P)$$ ≈ $6.0832695845577521817387242395$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{20229}{4}, \frac{20225}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(92599, 28000575\right)$$, $$\left(92599, -28093175\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$16650$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 37$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$284480859375000000$$ = $$2^{6} \cdot 3^{9} \cdot 5^{14} \cdot 37$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{4385367890843575421521}{24975000000}$$ = $$2^{-6} \cdot 3^{-3} \cdot 5^{-8} \cdot 37^{-1} \cdot 311^{3} \cdot 52631^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$6.0832695845577521817387242395$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.051010987618151991358682114805$$ 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: $$96$$  = $$( 2 \cdot 3 )\cdot2^{2}\cdot2^{2}\cdot1$$ 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: $$1$$ (exact)

Modular invariants

Modular form 16650.2.a.cb

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1327104 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

Special L-value

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);

$$L'(E,1)$$ ≈ $$7.4475261469381465189913535534351907122$$

Local data

This elliptic curve is 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$$(j)_{-}$$
$$2$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$3$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3
$$5$$ $$4$$ $$I_8^{*}$$ Additive 1 2 14 8
$$37$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X36b.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 8 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 24.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 split add add ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary ss 3 - - 1,1 1 3 1 3 1 1 1 1 1 1 1,1 1 - - 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 $$d=$$ 2, 4 and 8.
Its isogeny class 16650.cb consists of 4 curves linked by isogenies of degrees dividing 8.

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 $K$ $2$ $$\Q(\sqrt{111})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-185})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-15})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-15}, \sqrt{111})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{-10})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{-15}, \sqrt{74})$$ $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.0.6218016399360000.98 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/16\Z$$ Not in database $8$ 8.2.64043595421875.8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.