# Properties

 Label 116688.w1 Conductor $116688$ Discriminant $1.247\times 10^{15}$ j-invariant $$\frac{3047678972871625}{304559880768}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -48328, 3703412])

gp: E = ellinit([0, 1, 0, -48328, 3703412])

magma: E := EllipticCurve([0, 1, 0, -48328, 3703412]);

$$y^2=x^3+x^2-48328x+3703412$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(68, 858\right)$$ $$\left(158, 192\right)$$ $\hat{h}(P)$ ≈ $0.49298143076197516217913643292$ $1.8349635734936355405554469906$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(94, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-244,\pm 1014)$$, $$(-34,\pm 2304)$$, $$(-4,\pm 1974)$$, $$(44,\pm 1290)$$, $$(68,\pm 858)$$, $$\left(94, 0\right)$$, $$(158,\pm 192)$$, $$(211,\pm 1716)$$, $$(263,\pm 3042)$$, $$(926,\pm 27456)$$, $$(4583,\pm 309942)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$116688$$ = $2^{4} \cdot 3 \cdot 11 \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $1247477271625728$ = $2^{18} \cdot 3^{4} \cdot 11^{2} \cdot 13^{4} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3047678972871625}{304559880768}$$ = $2^{-6} \cdot 3^{-4} \cdot 5^{3} \cdot 11^{-2} \cdot 13^{-4} \cdot 17^{-1} \cdot 107^{3} \cdot 271^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.6330277172820514195460261863\dots$ Stable Faltings height: $0.93988053672210611012879406484\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.90415095273169990780288316152\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.47097152223074995409211512937\dots$ 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: $128$  = $2^{2}\cdot2^{2}\cdot2\cdot2^{2}\cdot1$ 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: $1$ (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: $L^{(2)}(E,1)/2!$ ≈ $13.626539217101809729931694995713262878$

## Modular invariants

Modular form 116688.2.a.w

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);

$$q + q^{3} - 2 q^{7} + q^{9} - q^{11} + q^{13} - q^{17} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 516096 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_{10}^{*}$ Additive -1 4 18 6
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 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 8.6.0.4

## $p$-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 split ss ordinary nonsplit split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - 5 2,2 2 2 3 2 2 2 2 2 2 2 4 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 116688.w 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 $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.0.4352.1 $$\Z/4\Z$$ Not in database $8$ 8.4.90469925818624.5 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.5473632256.5 $$\Z/2\Z \times \Z/4\Z$$ 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.