# Properties

 Label 585.c1 Conductor $585$ Discriminant $-19990546875$ j-invariant $$-\frac{32278933504}{27421875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+y=x^3-597x+8820$$ y^2+y=x^3-597x+8820 (homogenize, simplify) $$y^2z+yz^2=x^3-597xz^2+8820z^3$$ y^2z+yz^2=x^3-597xz^2+8820z^3 (dehomogenize, simplify) $$y^2=x^3-9552x+564496$$ y^2=x^3-9552x+564496 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 1, -597, 8820])

gp: E = ellinit([0, 0, 1, -597, 8820])

magma: E := EllipticCurve([0, 0, 1, -597, 8820]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(8, 67\right)$$ (8, 67) $\hat{h}(P)$ ≈ $0.037671158137087150705923009058$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-27, 72\right)$$, $$\left(-27, -73\right)$$, $$\left(-7, 112\right)$$, $$\left(-7, -113\right)$$, $$\left(8, 67\right)$$, $$\left(8, -68\right)$$, $$\left(18, 62\right)$$, $$\left(18, -63\right)$$, $$\left(35, 175\right)$$, $$\left(35, -176\right)$$, $$\left(38, 202\right)$$, $$\left(38, -203\right)$$, $$\left(143, 1687\right)$$, $$\left(143, -1688\right)$$, $$\left(15803, 1986592\right)$$, $$\left(15803, -1986593\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$585$$ = $3^{2} \cdot 5 \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-19990546875$ = $-1 \cdot 3^{9} \cdot 5^{7} \cdot 13$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{32278933504}{27421875}$$ = $-1 \cdot 2^{12} \cdot 3^{-3} \cdot 5^{-7} \cdot 13^{-1} \cdot 199^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.67415848514371860324113093389\dots$ Stable Faltings height: $0.12485234080966375754350831543\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.037671158137087150705923009058\dots$ 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: $1.1139515047269911012369397194\dots$ 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: $28$  = $2^{2}\cdot7\cdot1$ 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: $1$ (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: $L'(E,1)$ ≈ $1.1749876121652666605129056615$

## 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);

$$q - 2 q^{2} + 2 q^{4} + q^{5} - q^{7} - 2 q^{10} - 5 q^{11} - q^{13} + 2 q^{14} - 4 q^{16} + 7 q^{17} - 6 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 672 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$3$ $4$ $I_{3}^{*}$ Additive -1 2 9 3
$5$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$13$ $1$ $I_{1}$ 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 $\ell$.

The image of the adelic Galois representation has level $390$, index $2$, and genus $0$.

## $p$-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 add split ord ord nonsplit ord ord ord ord ord ord ord ord ord 1,2 - 2 1 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

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has no rational isogenies. Its isogeny class 585.c 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 $K$ $3$ 3.1.780.1 $$\Z/2\Z$$ Not in database $6$ 6.0.118638000.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $8$ 8.2.39039316875.2 $$\Z/3\Z$$ 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.