# Properties

 Label 4998.m1 Conductor $4998$ Discriminant $1.762\times 10^{20}$ j-invariant $$\frac{1617840527930521321}{623760113664}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -16047134, 24732961040])

gp: E = ellinit([1, 0, 1, -16047134, 24732961040])

magma: E := EllipticCurve([1, 0, 1, -16047134, 24732961040]);

$$y^2+xy+y=x^3-16047134x+24732961040$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(2085, 17389\right)$$ $\hat{h}(P)$ ≈ $1.2487906773733077108274326443$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2085, 17389\right)$$, $$\left(2085, -19475\right)$$, $$\left(2472, 11872\right)$$, $$\left(2472, -14345\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$4998$$ = $2 \cdot 3 \cdot 7^{2} \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $176196793423506702336$ = $2^{24} \cdot 3^{7} \cdot 7^{10} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1617840527930521321}{623760113664}$$ = $2^{-24} \cdot 3^{-7} \cdot 7^{2} \cdot 17^{-1} \cdot 53^{3} \cdot 6053^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.8503571039230101022918676594\dots$ Stable Faltings height: $1.2287653130435823480374070399\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.2487906773733077108274326443\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.17730093701738397456834870536\dots$ 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: $14$  = $2\cdot7\cdot1\cdot1$ 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) 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: $L'(E,1)$ ≈ $3.0997646013160554181394834238855968277$

## Modular invariants

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 282240 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not 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$(j)_{-}$
$2$ $2$ $I_{24}$ Non-split multiplicative 1 1 24 24
$3$ $7$ $I_{7}$ Split multiplicative -1 1 7 7
$7$ $1$ $II^{*}$ Additive -1 2 10 0
$17$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 $\ell$.

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 split ordinary add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 1 6 1 - 3 1 1 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 no rational isogenies. Its isogeny class 4998.m consists of this curve only.

## 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 $K$ $3$ 3.3.9996.1 $$\Z/2\Z$$ Not in database $6$ 6.6.20383683264.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.1740675142665963.10 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.