Properties

 Label 127050.gx1 Conductor $127050$ Discriminant $2.943\times 10^{28}$ j-invariant $$\frac{260744057755293612689909}{8504954620259328}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -20130703513, -1099328827258969])

gp: E = ellinit([1, 1, 1, -20130703513, -1099328827258969])

magma: E := EllipticCurve([1, 1, 1, -20130703513, -1099328827258969]);

$$y^2+xy+y=x^3+x^2-20130703513x-1099328827258969$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-81671, 168088\right)$$ $\hat{h}(P)$ ≈ $4.1942428982882384845530228923$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-\frac{329085}{4}, \frac{329081}{8}\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-81671, 168088\right)$$, $$\left(-81671, -86418\right)$$, $$\left(208885, 61620682\right)$$, $$\left(208885, -61829568\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$127050$$ = $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $29427824046916475334000000000$ = $2^{10} \cdot 3^{5} \cdot 5^{9} \cdot 7^{10} \cdot 11^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{260744057755293612689909}{8504954620259328}$$ = $2^{-10} \cdot 3^{-5} \cdot 7^{-10} \cdot 11^{-2} \cdot 29^{3} \cdot 2202961^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $4.5566687174669092569465459211\dots$ Stable Faltings height: $2.1506426467421487039650046322\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $4.1942428982882384845530228923\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.012674640982546326800032905446\dots$ 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: $800$  = $( 2 \cdot 5 )\cdot1\cdot2\cdot( 2 \cdot 5 )\cdot2^{2}$ 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: $L'(E,1)$ ≈ $10.632104585879598489217574028798480463$

Modular invariants

Modular form 127050.2.a.gx

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);

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + 4 q^{13} + q^{14} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 230400000 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$3$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$5$ $2$ $III^{*}$ Additive -1 2 9 0
$7$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$11$ $4$ $I_{2}^{*}$ Additive -1 2 8 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 2.3.0.1
$5$ 5B.4.1 5.12.0.1

$p$-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 split nonsplit add split add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary 3 3 - 2 - 1 1 1,1 3 1,1 1 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, 5 and 10.
Its isogeny class 127050.gx consists of 4 curves linked by isogenies of degrees dividing 10.

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 $K$ $2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-55})$$ $$\Z/10\Z$$ Not in database $4$ 4.0.8893500.4 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{15}, \sqrt{-33})$$ $$\Z/2\Z \times \Z/10\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.11389585284000000.205 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $8$ 8.0.79094342250000.9 $$\Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/30\Z$$ Not in database $20$ 20.4.169675210983039290802001953125.1 $$\Z/10\Z$$ 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.