Properties

 Label 159120de1 Conductor $159120$ Discriminant $3.101\times 10^{18}$ j-invariant $$\frac{2396726313900986596}{4154072495625}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2=x^3-2529147x+1545816314$$ y^2=x^3-2529147x+1545816314 (homogenize, simplify) $$y^2z=x^3-2529147xz^2+1545816314z^3$$ y^2z=x^3-2529147xz^2+1545816314z^3 (dehomogenize, simplify) $$y^2=x^3-2529147x+1545816314$$ y^2=x^3-2529147x+1545816314 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -2529147, 1545816314])

gp: E = ellinit([0, 0, 0, -2529147, 1545816314])

magma: E := EllipticCurve([0, 0, 0, -2529147, 1545816314]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(889, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(889, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$159120$$ = $2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $3100998501694080000$ = $2^{10} \cdot 3^{10} \cdot 5^{4} \cdot 13^{6} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2396726313900986596}{4154072495625}$$ = $2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 44371^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.4420701463122659625591847138\dots$ Stable Faltings height: $1.3151413515115900256805353275\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.25275721606843424614429939462\dots$ 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: $64$  = $2^{2}\cdot2\cdot2^{2}\cdot2\cdot1$ 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: $L(E,1)$ ≈ $4.0441154570949479383087903140$

Modular invariants

Modular form 159120.2.a.dr

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);

$$q + q^{5} + 2 q^{11} - q^{13} + q^{17} + 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2949120 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_{2}^{*}$ Additive 1 4 10 0
$3$ $2$ $I_{4}^{*}$ Additive -1 2 10 4
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$17$ $1$ $I_{1}$ 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

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $p$-adic regulators are identically $1$ since the rank is $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 159120de 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 $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.6619392.4 $$\Z/4\Z$$ Not in database $8$ 8.0.12662925279952896.234 $$\Z/2\Z \oplus \Z/4\Z$$ 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.