# Properties

 Label 32487.j1 Conductor $32487$ Discriminant $2.349\times 10^{14}$ j-invariant $$\frac{3885442650361}{1996623837}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-16049x+260579$$ y^2+xy+y=x^3-16049x+260579 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-16049xz^2+260579z^3$$ y^2z+xyz+yz^2=x^3-16049xz^2+260579z^3 (dehomogenize, simplify) $$y^2=x^3-20798883x+12219982110$$ y^2=x^3-20798883x+12219982110 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 1, -16049, 260579])

gp: E = ellinit([1, 0, 1, -16049, 260579])

magma: E := EllipticCurve([1, 0, 1, -16049, 260579]);

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);

$$\left(-\frac{537}{4}, \frac{533}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$32487$$ = $3 \cdot 7^{2} \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $234900797799213$ = $3^{12} \cdot 7^{6} \cdot 13 \cdot 17^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3885442650361}{1996623837}$$ = $3^{-12} \cdot 13^{-1} \cdot 17^{-2} \cdot 79^{3} \cdot 199^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.4496992410525925251632500466\dots$ Stable Faltings height: $0.47674416652493587261057367488\dots$

## 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: $0.49122486764827179158995759185\dots$ 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: $48$  = $( 2^{2} \cdot 3 )\cdot2\cdot1\cdot2$ 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: $L(E,1)$ ≈ $5.8946984117792614990794911022$

## Modular invariants

Modular form 32487.2.a.j

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);

$$q + q^{2} + q^{3} - q^{4} + 4 q^{5} + q^{6} - 3 q^{8} + q^{9} + 4 q^{10} + 6 q^{11} - q^{12} + q^{13} + 4 q^{15} - q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 207360 $\Gamma_0(N)$-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$(j)_{-}$
$3$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$7$ $2$ $I_0^{*}$ Additive -1 2 6 0
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$17$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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
sage: gens = [[1, 0, 4, 1], [3, 4, 8, 11], [1, 2, 2, 5], [1, 4, 0, 1], [881, 4, 880, 5], [105, 4, 210, 9], [342, 1, 271, 0], [665, 222, 220, 663]]

sage: GL(2,Integers(884)).subgroup(gens)

magma: Gens := [[1, 0, 4, 1], [3, 4, 8, 11], [1, 2, 2, 5], [1, 4, 0, 1], [881, 4, 880, 5], [105, 4, 210, 9], [342, 1, 271, 0], [665, 222, 220, 663]];

magma: sub<GL(2,Integers(884))|Gens>;

The image of the adelic Galois representation has level $884$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 881 & 4 \\ 880 & 5 \end{array}\right),\left(\begin{array}{rr} 105 & 4 \\ 210 & 9 \end{array}\right),\left(\begin{array}{rr} 342 & 1 \\ 271 & 0 \end{array}\right),\left(\begin{array}{rr} 665 & 222 \\ 220 & 663 \end{array}\right)$

## $p$-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) $\mu$-invariant(s) 2 3 7 13 17 ord split add split nonsplit 6 1 - 1 0 1 0 - 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 $d=$ 2.
Its isogeny class 32487.j 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 $K$ $2$ $$\Q(\sqrt{13})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.2945488.1 $$\Z/4\Z$$ Not in database $8$ 8.0.1466227025326336.72 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ deg 8 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.