# Properties

 Label 4998.be3 Conductor $4998$ Discriminant $-1.477\times 10^{16}$ j-invariant $$-\frac{491411892194497}{125563633938}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3+x^2-80557x+10532549$$ y^2+xy+y=x^3+x^2-80557x+10532549 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3+x^2z-80557xz^2+10532549z^3$$ y^2z+xyz+yz^2=x^3+x^2z-80557xz^2+10532549z^3 (dehomogenize, simplify) $$y^2=x^3-104401899x+492972642918$$ y^2=x^3-104401899x+492972642918 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 1, 1, -80557, 10532549])

gp: E = ellinit([1, 1, 1, -80557, 10532549])

magma: E := EllipticCurve([1, 1, 1, -80557, 10532549]);

oscar: E = EllipticCurve([1, 1, 1, -80557, 10532549])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(\frac{259}{4}, \frac{18657}{8}\right)$$ (259/4, 18657/8) $\hat{h}(P)$ ≈ $7.0204900572046214257702756916$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(-\frac{1341}{4}, \frac{1337}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$4998$$ = $2 \cdot 3 \cdot 7^{2} \cdot 17$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-14772435969171762$ = $-1 \cdot 2 \cdot 3^{2} \cdot 7^{6} \cdot 17^{8}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$-\frac{491411892194497}{125563633938}$$ = $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 47^{3} \cdot 73^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.8200333601691289067905034284\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.84707828564147225423782705668\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $7.0204900572046214257702756916\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $0.37550084709445294722396237344\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $8$  = $1\cdot2\cdot2\cdot2$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $2$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $5.2723999269970395491011736824$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 5.272399927 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.375501 \cdot 7.020490 \cdot 8}{2^2} \approx 5.272399927$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 49152
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $I_0^{*}$ Additive -1 2 6 0
$17$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

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 8.48.0.224

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[5, 4, 1900, 1901], [1, 16, 0, 1], [543, 0, 0, 1903], [1716, 1701, 567, 1506], [1, 0, 16, 1], [785, 560, 840, 673], [1889, 16, 1888, 17], [1198, 1701, 1337, 1506], [15, 2, 1806, 1891]]

GL(2,Integers(1904)).subgroup(gens)

Gens := [[5, 4, 1900, 1901], [1, 16, 0, 1], [543, 0, 0, 1903], [1716, 1701, 567, 1506], [1, 0, 16, 1], [785, 560, 840, 673], [1889, 16, 1888, 17], [1198, 1701, 1337, 1506], [15, 2, 1806, 1891]];

sub<GL(2,Integers(1904))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$1904 = 2^{4} \cdot 7 \cdot 17$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 5 & 4 \\ 1900 & 1901 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 1903 \end{array}\right),\left(\begin{array}{rr} 1716 & 1701 \\ 567 & 1506 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 785 & 560 \\ 840 & 673 \end{array}\right),\left(\begin{array}{rr} 1889 & 16 \\ 1888 & 17 \end{array}\right),\left(\begin{array}{rr} 1198 & 1701 \\ 1337 & 1506 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1806 & 1891 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E[1904])$ is a degree-$20214448128$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1904\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 4998.be consists of 6 curves linked by isogenies of degrees dividing 8.

## Twists

The minimal quadratic twist of this elliptic curve is 102.c3, its twist by $-7$.

## 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 $K$ $2$ $$\Q(\sqrt{-2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{7})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-14})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{7})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{7})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.815712436224.17 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.815712436224.91 $$\Z/8\Z$$ Not in database $8$ 8.0.157351936.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ deg 8 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.

## 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 ord add ord ord nonsplit ord ss ord ord ord ord ord ss 7 1 1 - 1 1 3 1 1,1 1 1 1 1 3 1,1 1 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.

## $p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.