# Properties

 Label 450840z1 Conductor $450840$ Discriminant $1.027\times 10^{23}$ j-invariant $$\frac{2396726313900986596}{4154072495625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-81213720x+281308387932$$ y^2=x^3-x^2-81213720x+281308387932 (homogenize, simplify) $$y^2z=x^3-x^2z-81213720xz^2+281308387932z^3$$ y^2z=x^3-x^2z-81213720xz^2+281308387932z^3 (dehomogenize, simplify) $$y^2=x^3-6578311347x+205054079868414$$ y^2=x^3-6578311347x+205054079868414 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -81213720, 281308387932])

gp: E = ellinit([0, -1, 0, -81213720, 281308387932])

magma: E := EllipticCurve([0, -1, 0, -81213720, 281308387932]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(4749, 52020\right)$$ (4749, 52020) $\hat{h}(P)$ ≈ $2.7864619892929687343110415092$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5038, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(4749,\pm 52020)$$, $$\left(5038, 0\right)$$, $$(22614,\pm 3163680)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$450840$$ = $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $102675672570010250880000$ = $2^{10} \cdot 3^{4} \cdot 5^{4} \cdot 13^{6} \cdot 17^{7}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2396726313900986596}{4154072495625}$$ = $2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 44371^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.3093706740063191569863294043\dots$ Stable Faltings height: $1.3151413515115900256805353275\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.7864619892929687343110415092\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.10617926872649788077063132255\dots$ 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: $64$  = $2\cdot2\cdot2^{2}\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $4.7338319417169598052046759069$

## Modular invariants

Modular form 450840.2.a.z

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);

$$q - q^{3} + q^{5} + q^{9} - 2 q^{11} - q^{13} - q^{15} - 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 53084160 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive -1 3 10 0
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$17$ $2$ $I_{1}^{*}$ Additive 1 2 7 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

## $p$-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.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 450840z consists of 2 curves linked by isogenies of degree 2.