# Properties

 Label 5610.q5 Conductor $5610$ Discriminant $5.862\times 10^{18}$ j-invariant $$\frac{476646772170172569823801}{5862293314453125000}$$ CM no Rank $0$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -1627388, 790397138])

gp: E = ellinit([1, 0, 1, -1627388, 790397138])

magma: E := EllipticCurve([1, 0, 1, -1627388, 790397138]);

$$y^2+xy+y=x^3-1627388x+790397138$$

## Mordell-Weil group structure

$\Z/{6}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2604, 117910\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(954, 9835\right)$$, $$\left(954, -10790\right)$$, $$\left(2604, 117910\right)$$, $$\left(2604, -120515\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5610$$ = $2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $5862293314453125000$ = $2^{3} \cdot 3^{3} \cdot 5^{12} \cdot 11^{3} \cdot 17^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{476646772170172569823801}{5862293314453125000}$$ = $2^{-3} \cdot 3^{-3} \cdot 5^{-12} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 107^{3} \cdot 31741^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.4114813045780072331167963565\dots$ Stable Faltings height: $2.4114813045780072331167963565\dots$

## 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: E.omega[1]  magma: RealPeriod(E); Real period: $0.24050500889864283326946327256\dots$ 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: $216$  = $1\cdot3\cdot( 2^{2} \cdot 3 )\cdot3\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $6$ 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: $L(E,1)$ ≈ $1.4430300533918569996167796353867365261$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 221184 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$2$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$5$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

## 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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1
$3$ 3B.1.1 3.8.0.1

## $p$-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 5 11 17 nonsplit split split split nonsplit 5 3 1 3 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 5610.q consists of 6 curves linked by isogenies of degrees dividing 12.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{66})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/12\Z$$ Not in database $2$ $$\Q(\sqrt{11})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{11})$$ $$\Z/2\Z \times \Z/12\Z$$ Not in database $6$ 6.0.2255067.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ 8.0.5416809268248576.19 $$\Z/2\Z \times \Z/12\Z$$ Not in database $8$ Deg 8 $$\Z/24\Z$$ Not in database $8$ Deg 8 $$\Z/24\Z$$ Not in database $9$ 9.3.33173450251082263546875.3 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/24\Z$$ Not in database $18$ 18.6.5308025359933309041398141791540791330720743424000000000000.1 $$\Z/2\Z \times \Z/18\Z$$ Not in database $18$ 18.6.3987998016478819715550820279144095665455104000000000000.1 $$\Z/36\Z$$ Not in database $18$ 18.6.383971741893323860054842432837152150659776000000000000.1 $$\Z/36\Z$$ 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.