Properties

 Label 1859.b2 Conductor $1859$ Discriminant $-777362416259$ j-invariant $$-\frac{122023936}{161051}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2+y=x^3-x^2-1746x-50295$$ y^2+y=x^3-x^2-1746x-50295 (homogenize, simplify) $$y^2z+yz^2=x^3-x^2z-1746xz^2-50295z^3$$ y^2z+yz^2=x^3-x^2z-1746xz^2-50295z^3 (dehomogenize, simplify) $$y^2=x^3-2263248x-2373709104$$ y^2=x^3-2263248x-2373709104 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 1, -1746, -50295])

gp: E = ellinit([0, -1, 1, -1746, -50295])

magma: E := EllipticCurve([0, -1, 1, -1746, -50295]);

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);

 $P$ = $$\left(\frac{7045}{36}, \frac{573985}{216}\right)$$ (7045/36, 573985/216) $\hat{h}(P)$ ≈ $6.4244302828039167698966192616$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1859$$ = $11 \cdot 13^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-777362416259$ = $-1 \cdot 11^{5} \cdot 13^{6}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{122023936}{161051}$$ = $-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.97446483761236530333772945932\dots$ Stable Faltings height: $-0.30800984111840306468901426146\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $6.4244302828039167698966192616\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.35201532506737740953181959170\dots$ 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: $2$  = $1\cdot2$ 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: $L'(E,1)$ ≈ $4.5229958287478482864024234053$

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2160 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$11$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$13$ $2$ $I_0^{*}$ Additive 1 2 6 0

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
$5$ 5Cs.4.1 5.60.0.1

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss ord ord ord nonsplit add ord ss ord ss ord ord ord ord ord 2,3 1 1 1 1 - 1 1,1 1 1,1 1 1 1 1 1 0,0 0 1 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 $d=$ 5.
Its isogeny class 1859.b consists of 3 curves linked by isogenies of degrees dividing 25.

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 $K$ $2$ $$\Q(\sqrt{13})$$ $$\Z/5\Z$$ 2.2.13.1-121.1-a2 $3$ 3.1.44.1 $$\Z/2\Z$$ Not in database $4$ 4.0.21125.1 $$\Z/5\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.2.4253392.1 $$\Z/10\Z$$ Not in database $8$ 8.2.914519421387.3 $$\Z/3\Z$$ Not in database $8$ 8.0.446265625.1 $$\Z/5\Z \oplus \Z/5\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/10\Z$$ Not in database $12$ 12.0.2189052564185344.1 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ Deg 16 $$\Z/15\Z$$ 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.