# Properties

 Label 462.a3 Conductor $462$ Discriminant $213444$ j-invariant $$\frac{498677257}{213444}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -16, -20])

gp: E = ellinit([1, 1, 0, -16, -20])

magma: E := EllipticCurve([1, 1, 0, -16, -20]);

$$y^2+xy=x^3+x^2-16x-20$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-3, 5\right)$$ $\hat{h}(P)$ ≈ $0.50340411585919074963442577700$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-4, 2\right)$$, $$\left(4, -2\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-4, 2\right)$$, $$\left(-3, 5\right)$$, $$\left(-3, -2\right)$$, $$\left(-2, 4\right)$$, $$\left(-2, -2\right)$$, $$\left(4, -2\right)$$, $$\left(5, 5\right)$$, $$\left(5, -10\right)$$, $$\left(7, 13\right)$$, $$\left(7, -20\right)$$, $$\left(18, 68\right)$$, $$\left(18, -86\right)$$, $$\left(46, 292\right)$$, $$\left(46, -338\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$462$$ = $2 \cdot 3 \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $213444$ = $2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{498677257}{213444}$$ = $2^{-2} \cdot 3^{-2} \cdot 7^{-2} \cdot 11^{-2} \cdot 13^{3} \cdot 61^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.28091299485118297012048166399\dots$ Stable Faltings height: $-0.28091299485118297012048166399\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.50340411585919074963442577700\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.4613024380517452452137946611\dots$ 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: $16$  = $2\cdot2\cdot2\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L'(E,1)$ ≈ $1.2390297776895094261990270968597971796$

## 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);

$$q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + q^{11} - q^{12} + 2 q^{13} - q^{14} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 64 $\Gamma_0(N)$-optimal: no Manin constant: 1

## 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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $2$ $I_{2}$ 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.12.0.1

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 nonsplit nonsplit ordinary split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary 2 1 1 2 2 1 1 1 1 1 1 1 1 1,1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0,0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 462.a consists of 2 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{2}, \sqrt{21})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{-11})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{11}, \sqrt{-21})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.99636092064432.4 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.34822159495883806555961819136.14 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ 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.