# Properties

 Label 159936ip2 Conductor $159936$ Discriminant $2.264\times 10^{18}$ j-invariant $$\frac{3590714269297}{73410624}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -1000449, -377961471])

gp: E = ellinit([0, -1, 0, -1000449, -377961471])

magma: E := EllipticCurve([0, -1, 0, -1000449, -377961471]);

$$y^2=x^3-x^2-1000449x-377961471$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-541, 2156\right)$$ $$\left(1409, 31744\right)$$ $$\hat{h}(P)$$ ≈ $3.4964489805527382674915911094$ $4.4736706357541220129041771559$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-513, 0\right)$$, $$\left(1153, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-639, 0\right)$$, $$(-541,\pm 2156)$$, $$\left(-513, 0\right)$$, $$\left(1153, 0\right)$$, $$(1409,\pm 31744)$$, $$(1665,\pm 50688)$$, $$(2403,\pm 105300)$$, $$(6984,\pm 577269)$$, $$(12815,\pm 1446088)$$, $$(100611,\pm 31911300)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$159936$$ = $$2^{6} \cdot 3 \cdot 7^{2} \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$2264055546636140544$$ = $$2^{24} \cdot 3^{4} \cdot 7^{8} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3590714269297}{73410624}$$ = $$2^{-6} \cdot 3^{-4} \cdot 7^{-2} \cdot 17^{-2} \cdot 15313^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.3126521255602640254969806539\dots$$ Stable Faltings height: $$0.29997628019268940881845609999\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$14.865059288891014789676627196\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.15114387399843510701875637455\dots$$ 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: $$64$$  = $$2^{2}\cdot2\cdot2^{2}\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$4$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

Modular form 159936.2.a.bh

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);

$$q - q^{3} - 2q^{5} + q^{9} - 6q^{13} + 2q^{15} - q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3538944 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$8.9870505925576436486242589382779762481$$

## Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_{14}^{*}$$ Additive 1 6 24 6
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$7$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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 X38.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 6 & 7 \end{array}\right)$ and has index 12.

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

prime Image of Galois representation
$$2$$ Cs

## $p$-adic data

### $p$-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.

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 159936ip consists of 2 curves linked by isogenies of degrees dividing 4.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $4$ $$\Q(\sqrt{14}, \sqrt{-34})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{7}, \sqrt{34})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.172717185506805128768782336.1 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ 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.