# Properties

 Label 570l1 Conductor $570$ Discriminant $-2.875\times 10^{14}$ j-invariant $$\frac{89962967236397039}{287450726400000}$$ CM no Rank $0$ Torsion structure $$\Z/{10}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, 9335, -737383])

gp: E = ellinit([1, 0, 0, 9335, -737383])

magma: E := EllipticCurve([1, 0, 0, 9335, -737383]);

$$y^2+xy=x^3+9335x-737383$$

## Mordell-Weil group structure

$$\Z/{10}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(134, 1643\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(58, -29\right)$$, $$\left(74, 563\right)$$, $$\left(74, -637\right)$$, $$\left(134, 1643\right)$$, $$\left(134, -1777\right)$$, $$\left(314, 5603\right)$$, $$\left(314, -5917\right)$$, $$\left(1274, 44963\right)$$, $$\left(1274, -46237\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$570$$ = $$2 \cdot 3 \cdot 5 \cdot 19$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-287450726400000$$ = $$-1 \cdot 2^{20} \cdot 3^{5} \cdot 5^{5} \cdot 19^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{89962967236397039}{287450726400000}$$ = $$2^{-20} \cdot 3^{-5} \cdot 5^{-5} \cdot 19^{-2} \cdot 29^{3} \cdot 15451^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.4575365745150151398012579683\dots$$ Stable Faltings height: $$1.4575365745150151398012579683\dots$$

## 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: $$0.28000312519189812375146695111\dots$$ 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: $$1000$$  = $$( 2^{2} \cdot 5 )\cdot5\cdot5\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$10$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2400 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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);

$$L(E,1)$$ ≈ $$2.8000312519189812375146695110478135866$$

## 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$$(j)_{-}$$
$$2$$ $$20$$ $$I_{20}$$ Split multiplicative -1 1 20 20
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$19$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B
$$5$$ B.1.1

## $p$-adic data

### $p$-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 19 split split split nonsplit 1 3 3 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 570l consists of 4 curves linked by isogenies of degrees dividing 10.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/10\Z$$ Not in database $4$ 4.2.86640.2 $$\Z/20\Z$$ Not in database $8$ 8.0.1688960160000.18 $$\Z/2\Z \times \Z/20\Z$$ Not in database $8$ 8.0.65792250000.7 $$\Z/2\Z \times \Z/20\Z$$ Not in database $8$ 8.2.230859741870000.11 $$\Z/30\Z$$ Not in database $16$ Deg 16 $$\Z/40\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/30\Z$$ Not in database $20$ 20.0.8802533373035313955108642578125.1 $$\Z/5\Z \times \Z/10\Z$$ 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.