# Properties

 Label 3264bc6 Conductor $3264$ Discriminant $-3.292\times 10^{16}$ j-invariant $$-\frac{491411892194497}{125563633938}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-105217x+15737087$$ y^2=x^3+x^2-105217x+15737087 (homogenize, simplify) $$y^2z=x^3+x^2z-105217xz^2+15737087z^3$$ y^2z=x^3+x^2z-105217xz^2+15737087z^3 (dehomogenize, simplify) $$y^2=x^3-8522604x+11497904208$$ y^2=x^3-8522604x+11497904208 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -105217, 15737087])

gp: E = ellinit([0, 1, 0, -105217, 15737087])

magma: E := EllipticCurve([0, 1, 0, -105217, 15737087]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-383, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-383, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3264$$ = $2^{6} \cdot 3 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-32915753255043072$ = $-1 \cdot 2^{19} \cdot 3^{2} \cdot 17^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{491411892194497}{125563633938}$$ = $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 47^{3} \cdot 73^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.8867990564813902183636752389\dots$ Stable Faltings height: $0.84707828564147225423782705671\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $0.35124887956770803887453982351\dots$ 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: $32$  = $2\cdot2\cdot2^{3}$ 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) 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: $L(E,1)$ ≈ $2.8099910365416643109963185881$

## 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);

$$q + q^{3} + 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + q^{17} + 4 q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 24576 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $2$ $I_{9}^{*}$ Additive -1 6 19 1
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$17$ $8$ $I_{8}$ Split multiplicative -1 1 8 8

## 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 8.96.0.169
sage: gens = [[241, 16, 24, 129], [188, 203, 249, 126], [257, 16, 256, 17], [1, 16, 0, 1], [162, 203, 23, 126], [1, 0, 16, 1], [5, 4, 268, 269], [15, 2, 174, 259]]

sage: GL(2,Integers(272)).subgroup(gens)

magma: Gens := [[241, 16, 24, 129], [188, 203, 249, 126], [257, 16, 256, 17], [1, 16, 0, 1], [162, 203, 23, 126], [1, 0, 16, 1], [5, 4, 268, 269], [15, 2, 174, 259]];

magma: sub<GL(2,Integers(272))|Gens>;

The image of the adelic Galois representation has level $272$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 241 & 16 \\ 24 & 129 \end{array}\right),\left(\begin{array}{rr} 188 & 203 \\ 249 & 126 \end{array}\right),\left(\begin{array}{rr} 257 & 16 \\ 256 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 162 & 203 \\ 23 & 126 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 268 & 269 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 174 & 259 \end{array}\right)$

## $p$-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) 2 3 17 add split split - 1 1 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ 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 $d=$ 2, 4 and 8.
Its isogeny class 3264bc consists of 6 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 $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.8.1-5202.5-i1 $2$ $$\Q(\sqrt{2})$$ $$\Z/8\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ Not in database $4$ 4.0.18432.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.1358954496.9 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.1565515579392.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.28375309615104.33 $$\Z/16\Z$$ Not in database $8$ deg 8 $$\Z/6\Z$$ Not in database $16$ 16.0.9803356117276277820358656.198 $$\Z/8\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ 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.