# Properties

 Label 14586k1 Conductor $14586$ Discriminant $-5.539\times 10^{23}$ j-invariant $$\frac{79374649975090937760383}{553856914190911653543936}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 895336, -35804233639])

gp: E = ellinit([1, 1, 1, 895336, -35804233639])

magma: E := EllipticCurve([1, 1, 1, 895336, -35804233639]);

$$y^2+xy+y=x^3+x^2+895336x-35804233639$$

## Mordell-Weil group structure

$$\Z/{4}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(8837, 809405\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(3205, -1603\right)$$, $$\left(8837, 809405\right)$$, $$\left(8837, -818243\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$14586$$ = $$2 \cdot 3 \cdot 11 \cdot 13 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-553856914190911653543936$$ = $$-1 \cdot 2^{36} \cdot 3 \cdot 11^{4} \cdot 13^{3} \cdot 17^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{79374649975090937760383}{553856914190911653543936}$$ = $$2^{-36} \cdot 3^{-1} \cdot 11^{-4} \cdot 13^{-3} \cdot 17^{-4} \cdot 42976127^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$3.2348501711502563844501876512\dots$$ Stable Faltings height: $$3.2348501711502563844501876512\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.042660519323931063890125167834\dots$$ 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: $$864$$  = $$( 2^{2} \cdot 3^{2} )\cdot1\cdot2^{2}\cdot3\cdot2$$ 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: $$1$$ (exact)

## Modular invariants

Modular form 14586.2.a.i

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);

$$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{8} + q^{9} - 2q^{10} + q^{11} - q^{12} + q^{13} + 2q^{15} + q^{16} - q^{17} + q^{18} + 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2363904 $$\Gamma_0(N)$$-optimal: yes 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);

$$L(E,1)$$ ≈ $$2.3036680434922774500667590630137941578$$

## Local data

This elliptic curve is 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$$(j)_{-}$$
$$2$$ $$36$$ $$I_{36}$$ Split multiplicative -1 1 36 36
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$11$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$13$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$17$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4

## 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 X13h.

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

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 11 13 17 split nonsplit split split nonsplit 5 2 1 1 0 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 14586k consists of 4 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.302016.1 $$\Z/8\Z$$ Not in database $8$ 8.0.900798402816.3 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.138735983333376.6 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.216620836248267.6 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.