# Properties

 Label 32487.a1 Conductor $32487$ Discriminant $-4.706\times 10^{12}$ j-invariant $$-\frac{325660672}{40000779}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3-x^2-702x+104852$$ y^2+y=x^3-x^2-702x+104852 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z-702xz^2+104852z^3$$ y^2z+yz^2=x^3-x^2z-702xz^2+104852z^3 (dehomogenize, simplify) $$y^2=x^3-910224x+4881065616$$ y^2=x^3-910224x+4881065616 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 1, -702, 104852])

gp: E = ellinit([0, -1, 1, -702, 104852])

magma: E := EllipticCurve([0, -1, 1, -702, 104852]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(96, 955\right)$$ (96, 955) $$\left(47, 416\right)$$ (47, 416) $\hat{h}(P)$ ≈ $0.16363152227611104561417852688$ $0.62143516255980567073602138201$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-51, 73\right)$$, $$\left(-51, -74\right)$$, $$\left(-44, 220\right)$$, $$\left(-44, -221\right)$$, $$\left(-21, 331\right)$$, $$\left(-21, -332\right)$$, $$\left(-8, 331\right)$$, $$\left(-8, -332\right)$$, $$\left(5, 318\right)$$, $$\left(5, -319\right)$$, $$\left(30, 331\right)$$, $$\left(30, -332\right)$$, $$\left(44, 396\right)$$, $$\left(44, -397\right)$$, $$\left(47, 416\right)$$, $$\left(47, -417\right)$$, $$\left(96, 955\right)$$, $$\left(96, -956\right)$$, $$\left(161, 2034\right)$$, $$\left(161, -2035\right)$$, $$\left(537, 12421\right)$$, $$\left(537, -12422\right)$$, $$\left(642, 16243\right)$$, $$\left(642, -16244\right)$$, $$\left(1860, 80188\right)$$, $$\left(1860, -80189\right)$$, $$\left(2504, 125268\right)$$, $$\left(2504, -125269\right)$$, $$\left(2631, 134920\right)$$, $$\left(2631, -134921\right)$$, $$\left(73351, 19865800\right)$$, $$\left(73351, -19865801\right)$$

## 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: $-4706051648571$ = $-1 \cdot 3^{2} \cdot 7^{7} \cdot 13^{3} \cdot 17^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{325660672}{40000779}$$ = $-1 \cdot 2^{12} \cdot 3^{-2} \cdot 7^{-1} \cdot 13^{-3} \cdot 17^{-2} \cdot 43^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.1111144493925204616629758621\dots$ Stable Faltings height: $0.13815937486486380911029949038\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.099161187770099366668695682584\dots$ 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: $0.63291503084419577836096063563\dots$ 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: $48$  = $2\cdot2^{2}\cdot3\cdot2$ 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: $1$ (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: $L^{(2)}(E,1)/2!$ ≈ $3.0125090983708574328419968010$

## Modular invariants

Modular form 32487.2.a.a

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);

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

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 105984 $\Gamma_0(N)$-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$(j)_{-}$
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{2}$ 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 $\ell$.

sage: gens = [[181, 2, 180, 3], [1, 1, 181, 0], [1, 2, 0, 1], [1, 0, 2, 1], [15, 2, 15, 3], [157, 2, 157, 3]]

sage: GL(2,Integers(182)).subgroup(gens)

magma: Gens := [[181, 2, 180, 3], [1, 1, 181, 0], [1, 2, 0, 1], [1, 0, 2, 1], [15, 2, 15, 3], [157, 2, 157, 3]];

magma: sub<GL(2,Integers(182))|Gens>;

The image of the adelic Galois representation has level $182$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 181 & 2 \\ 180 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 181 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 15 & 3 \end{array}\right),\left(\begin{array}{rr} 157 & 2 \\ 157 & 3 \end{array}\right)$

## $p$-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 ss nonsplit ord add ord split split ord ord ord ord ord ord ord ord 23,8 2 2 - 2 3 3 2 2 2 2 2 2 2 2 0,0 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 32487.a 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 $K$ $3$ 3.1.364.1 $$\Z/2\Z$$ Not in database $6$ 6.0.12057136.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.1740675142665963.6 $$\Z/3\Z$$ 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.