Properties

 Label 58443.k1 Conductor $58443$ Discriminant $3.385\times 10^{12}$ j-invariant $$\frac{564661380021747712}{27978783021}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -85169, 9538139]) # or

sage: E = EllipticCurve("58443.k1")

gp: E = ellinit([0, 1, 1, -85169, 9538139]) \\ or

gp: E = ellinit("58443.k1")

magma: E := EllipticCurve([0, 1, 1, -85169, 9538139]); // or

magma: E := EllipticCurve("58443.k1");

$$y^2+y=x^3+x^2-85169x+9538139$$

Mordell-Weil group structure

$$\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{4039}{25}, \frac{18188}{125}\right)$$ $$\hat{h}(P)$$ ≈ $1.6126374703338473131278090356$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$58443$$ = $$3 \cdot 7 \cdot 11^{2} \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$3385432745541$$ = $$3^{3} \cdot 7 \cdot 11^{2} \cdot 23^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{564661380021747712}{27978783021}$$ = $$2^{18} \cdot 3^{-3} \cdot 7^{-1} \cdot 11 \cdot 23^{-6} \cdot 5807^{3}$$ 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); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.6126374703338473131278090356$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.74813969950021129524700400163$$ 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: $$6$$  = $$3\cdot1\cdot1\cdot2$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 58443.2.a.k

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);

$$q + q^{3} - 2q^{4} + 3q^{5} - q^{7} + q^{9} - 2q^{12} - 2q^{13} + 3q^{15} + 4q^{16} - 3q^{17} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 186624 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$7.2388686747500726169296613336141305241$$

Local data

This elliptic curve is 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$$(j)_{-}$$
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$11$$ $$1$$ $$II$$ Additive -1 2 2 0
$$23$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$3$$ 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 ss split ordinary nonsplit add ordinary ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary 2,3 2 1 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 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 $$d=$$ 3.
Its isogeny class 58443.k consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{33})$$ $$\Z/3\Z$$ Not in database $3$ 3.3.10164.1 $$\Z/2\Z$$ Not in database $6$ 6.6.2169444816.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.3480151059.1 $$\Z/3\Z$$ Not in database $6$ 6.6.3409127568.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.4024227338732521240694660207082762255184359591892977.3 $$\Z/9\Z$$ Not in database $18$ 18.0.76136485082320813151221339689529344.1 $$\Z/6\Z$$ 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.