Properties

 Label 135240.e1 Conductor $135240$ Discriminant $3.137\times 10^{18}$ j-invariant $$\frac{155324313723954725282}{13018359375}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -69685856, 223928874156])

gp: E = ellinit([0, -1, 0, -69685856, 223928874156])

magma: E := EllipticCurve([0, -1, 0, -69685856, 223928874156]);

$$y^2=x^3-x^2-69685856x+223928874156$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(4793, 3234\right)$$ (4793, 3234) $\hat{h}(P)$ ≈ $2.8477996606393477183113258178$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(4821, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(4793,\pm 3234)$$, $$\left(4821, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$135240$$ = $2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $3136710578400000000$ = $2^{11} \cdot 3^{2} \cdot 5^{8} \cdot 7^{7} \cdot 23^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{155324313723954725282}{13018359375}$$ = $2 \cdot 3^{-2} \cdot 5^{-8} \cdot 7^{-1} \cdot 23^{-2} \cdot 4266481^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.9916103966809121293800111604\dots$ Stable Faltings height: $1.3832704066399722765282053440\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.8477996606393477183113258178\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.19295979106366704940018696892\dots$ 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: $32$  = $1\cdot2\cdot2\cdot2^{2}\cdot2$ 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: $L'(E,1)$ ≈ $4.3960866200652037102195307161$

Modular invariants

Modular form 135240.2.a.e

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 11010048 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 3 11 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8
$7$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$23$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 8.24.0.98

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add nonsplit nonsplit add ord ord ord ord nonsplit ord ord ord ord ord ord - 3 1 - 1 1 1 1 1 1 1 1 1 1 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 $d=$ 2, 4 and 8.
Its isogeny class 135240.e consists of 6 curves linked by isogenies of degrees dividing 8.

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 $K$ $2$ $$\Q(\sqrt{14})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{2})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{7})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{7}, \sqrt{23})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{7}, \sqrt{46})$$ $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.39969909374976.52 $$\Z/8\Z$$ Not in database $8$ 8.8.44033523122176.3 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.