# Properties

 Label 136710du3 Conductor $136710$ Discriminant $1.090\times 10^{20}$ j-invariant $$\frac{5601911201812801}{1271193750000}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -1631709, 625898965]) # or

sage: E = EllipticCurve("136710du3")

gp: E = ellinit([1, -1, 0, -1631709, 625898965]) \\ or

gp: E = ellinit("136710du3")

magma: E := EllipticCurve([1, -1, 0, -1631709, 625898965]); // or

magma: E := EllipticCurve("136710du3");

$$y^2 + x y = x^{3} - x^{2} - 1631709 x + 625898965$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(86, 22007\right)$$ $$\hat{h}(P)$$ ≈ $0.9983449860205831$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(\frac{4019}{4}, -\frac{4019}{8}\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-514, 36707\right)$$, $$\left(-514, -36193\right)$$, $$\left(86, 22007\right)$$, $$\left(86, -22093\right)$$, $$\left(1161, 16607\right)$$, $$\left(1161, -17768\right)$$, $$\left(9935, 977360\right)$$, $$\left(9935, -987295\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$136710$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 31$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$109025356976943750000$$ = $$2^{4} \cdot 3^{14} \cdot 5^{8} \cdot 7^{6} \cdot 31$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{5601911201812801}{1271193750000}$$ = $$2^{-4} \cdot 3^{-8} \cdot 5^{-8} \cdot 31^{-1} \cdot 177601^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.998344986021$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.176962624163$$ 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: $$128$$  = $$2\cdot2^{2}\cdot2^{3}\cdot2\cdot1$$ 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)

## Modular invariants

Modular form 136710.2.a.dd

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);

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + 4q^{11} - 6q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6291456 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$5.65343195348$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$4$$ $$I_8^{*}$$ Additive -1 2 14 8
$$5$$ $$8$$ $$I_{8}$$ Split multiplicative -1 1 8 8
$$7$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$31$$ $$1$$ $$I_{1}$$ 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 X102.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 24.

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
$$2$$ 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.

## 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 nonsplit add split add ordinary ordinary ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ss 13 - 2 - 1 1 1 1 3 1 2 1 1 1 1,1 0 - 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 $$d=$$ 2, 4 and 8.
Its isogeny class 136710du 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
$2$ $$\Q(\sqrt{21})$$ $$\Z/8\Z$$ Not in database
$2$ $$\Q(\sqrt{31})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$2$ $$\Q(\sqrt{651})$$ $$\Z/4\Z$$ Not in database
$4$ $$\Q(\sqrt{21}, \sqrt{31})$$ $$\Z/2\Z \times \Z/8\Z$$ 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.