Properties

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

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-x^2-1746x-50295\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-x^2z-1746xz^2-50295z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2263248x-2373709104\) Copy content Toggle raw display (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);
 

$P$ =  \(\left(\frac{7045}{36}, \frac{573985}{216}\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $6.4244302828039167698966192616$

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

Modular form   1859.2.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);
 

\( q + 2 q^{2} - q^{3} + 2 q^{4} - q^{5} - 2 q^{6} + 2 q^{7} - 2 q^{9} - 2 q^{10} - q^{11} - 2 q^{12} + 4 q^{14} + q^{15} - 4 q^{16} - 2 q^{17} - 4 q^{18} + O(q^{20}) \) Copy content Toggle raw display

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

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ord ord ord nonsplit add ord ss ord ss ord ord ord ord ord
$\lambda$-invariant(s) 2,3 1 1 1 1 - 1 1,1 1 1,1 1 1 1 1 1
$\mu$-invariant(s) 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{13}) \) \(\Z/5\Z\) 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.