# Properties

 Label 18735.b2 Conductor $18735$ Discriminant $281025$ j-invariant $$\frac{3301293169}{281025}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2+xy=x^3-31x-64$$ y^2+xy=x^3-31x-64 (homogenize, simplify) $$y^2z+xyz=x^3-31xz^2-64z^3$$ y^2z+xyz=x^3-31xz^2-64z^3 (dehomogenize, simplify) $$y^2=x^3-40203x-2865402$$ y^2=x^3-40203x-2865402 (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 0, -31, -64])

gp: E = ellinit([1, 0, 0, -31, -64])

magma: E := EllipticCurve([1, 0, 0, -31, -64]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(16, 52\right)$$ (16, 52) $\hat{h}(P)$ ≈ $2.1528198061358049011182391136$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

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

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-4, 2\right)$$, $$\left(16, 52\right)$$, $$\left(16, -68\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$18735$$ = $3 \cdot 5 \cdot 1249$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $281025$ = $3^{2} \cdot 5^{2} \cdot 1249$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3301293169}{281025}$$ = $3^{-2} \cdot 5^{-2} \cdot 1249^{-1} \cdot 1489^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.21311966694598403247621748082\dots$ Stable Faltings height: $-0.21311966694598403247621748082\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.1528198061358049011182391136\dots$ 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: $2.0339235080878289788595240171\dots$ 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: $4$  = $2\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $4.3786708123766961941274682223$

## Modular invariants

Modular form 18735.2.a.b

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2816 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is 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$(j)_{-}$
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$1249$ $1$ $I_{1}$ 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

The image of the adelic Galois representation has level $74940$, index $12$, 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 1249 ord split nonsplit ord ord ord ord ord ord ord ord ord ss ord ss split 8 2 1 1 1 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 0,0 ?

An entry ? indicates that the invariants have not yet been computed.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 18735.b consists of 2 curves linked by isogenies of degree 2.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{1249})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.4496400.1 $$\Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.