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Properties

Label 585f2
Conductor $585$
Discriminant $20208044025$
j-invariant \( \frac{168288035761}{27720225} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy=x^3-x^2-1035x-10584\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3-x^2z-1035xz^2-10584z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-16563x-693938\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 0, -1035, -10584])
 
gp: E = ellinit([1, -1, 0, -1035, -10584])
 
magma: E := EllipticCurve([1, -1, 0, -1035, -10584]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(40, 84\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.7724276067073725060266561064$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-24, 12\right) \), \( \left(-12, 6\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-24, 12\right) \), \( \left(-12, 6\right) \), \( \left(40, 84\right) \), \( \left(40, -124\right) \), \( \left(948, 28686\right) \), \( \left(948, -29634\right) \) Copy content Toggle raw display

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: $20208044025 $  =  $3^{14} \cdot 5^{2} \cdot 13^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{168288035761}{27720225} \)  =  $3^{-8} \cdot 5^{-2} \cdot 13^{-2} \cdot 5521^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.69900244128877076854597130980\dots$
Stable Faltings height: $0.14969629695471592284834869134\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $2.7724276067073725060266561064\dots$
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: $0.85108145292620109686565595561\dots$
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^{2}\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) $ ≈ $ 2.3595617156492210218470390653 $

Modular invariants

Modular form   585.2.a.g

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);
 

\( q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 4 q^{11} + q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$3$ $4$ $I_{8}^{*}$ Additive -1 2 14 8
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $2$ $I_{2}$ 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 16.48.0.4
sage: gens = [[13, 8, 2330, 3077], [1, 16, 0, 1], [633, 16, 818, 345], [3111, 3112, 1084, 39], [15, 8, 448, 1409], [1, 16, 4, 65], [1, 0, 16, 1], [5, 16, 4, 2353], [3105, 16, 3104, 17]]
 
sage: GL(2,Integers(3120)).subgroup(gens)
 
magma: Gens := [[13, 8, 2330, 3077], [1, 16, 0, 1], [633, 16, 818, 345], [3111, 3112, 1084, 39], [15, 8, 448, 1409], [1, 16, 4, 65], [1, 0, 16, 1], [5, 16, 4, 2353], [3105, 16, 3104, 17]];
 
magma: sub<GL(2,Integers(3120))|Gens>;
 

The image of the adelic Galois representation has level $3120$, index $768$, genus $13$, and generators

$\left(\begin{array}{rr} 13 & 8 \\ 2330 & 3077 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 633 & 16 \\ 818 & 345 \end{array}\right),\left(\begin{array}{rr} 3111 & 3112 \\ 1084 & 39 \end{array}\right),\left(\begin{array}{rr} 15 & 8 \\ 448 & 1409 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 4 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 16 \\ 4 & 2353 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ord add nonsplit ss ord split ord ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) 2 - 1 1,1 1 2 1 1 1 1 1 1 3 3 1
$\mu$-invariant(s) 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, 4 and 8.
Its isogeny class 585f consists of 8 curves linked by isogenies of degrees dividing 16.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/2\Z \oplus \Z/4\Z\) 2.0.3.1-12675.2-b4
$4$ \(\Q(\sqrt{3}, \sqrt{65})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{3}, \sqrt{-65})\) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{13})\) \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.370150560000.10 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.12960000.1 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.0.390971529.1 \(\Z/2\Z \oplus \Z/16\Z\) Not in database
$8$ 8.2.3162184666875.6 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.137011437068313600000000.1 \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.3913183654108104729600000000.3 \(\Z/2\Z \oplus \Z/16\Z\) Not in database
$16$ deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.