Properties

 Label 159936fx1 Conductor 159936 Discriminant 10871815349993472 j-invariant $$\frac{4354703137}{352512}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -106689, -12475233]); // or

magma: E := EllipticCurve("159936fx1");

sage: E = EllipticCurve([0, 1, 0, -106689, -12475233]) # or

sage: E = EllipticCurve("159936fx1")

gp: E = ellinit([0, 1, 0, -106689, -12475233]) \\ or

gp: E = ellinit("159936fx1")

$$y^2 = x^{3} + x^{2} - 106689 x - 12475233$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-\frac{867}{4}, \frac{5733}{8}\right)$$ $$\hat{h}(P)$$ ≈ 3.7429872968

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-229, 0\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-229, 0\right)$$, $$(3867,\pm 239616)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$159936$$ = $$2^{6} \cdot 3 \cdot 7^{2} \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$10871815349993472$$ = $$2^{26} \cdot 3^{4} \cdot 7^{6} \cdot 17$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{4354703137}{352512}$$ = $$2^{-8} \cdot 3^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 71^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$3.7429872968$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.265519195322$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$32$$  = $$2^{2}\cdot2^{2}\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 159936.2.a.gx

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q + q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - q^{17} + 4q^{19} + O(q^{20})$$

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1179648 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$7.95067980116$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_16^{*}$$ Additive 1 6 26 8
$$3$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$7$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X223.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 159936fx consists of 6 curves linked by isogenies of degrees dividing 8.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-14})$$ $$\Z/8\Z$$ Not in database
$$\Q(\sqrt{-238})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$\Q(\sqrt{-14}, \sqrt{17})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.