# Properties

 Label 44880.bh1 Conductor $44880$ Discriminant $2.263\times 10^{15}$ j-invariant $$\frac{901247067798311192691198986281}{552431869440}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-3219783720x+70322579406960$$ y^2=x^3-x^2-3219783720x+70322579406960 (homogenize, simplify) $$y^2z=x^3-x^2z-3219783720xz^2+70322579406960z^3$$ y^2z=x^3-x^2z-3219783720xz^2+70322579406960z^3 (dehomogenize, simplify) $$y^2=x^3-260802481347x+51264377980229826$$ y^2=x^3-260802481347x+51264377980229826 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -3219783720, 70322579406960])

gp: E = ellinit([0, -1, 0, -3219783720, 70322579406960])

magma: E := EllipticCurve([0, -1, 0, -3219783720, 70322579406960]);

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);

 $P$ = $$\left(\frac{398536258058}{10452289}, \frac{58401192145401014}{33792250337}\right)$$ (398536258058/10452289, 58401192145401014/33792250337) $\hat{h}(P)$ ≈ $24.502521496642669394172974313$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(32761, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(32761, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$44880$$ = $2^{4} \cdot 3 \cdot 5 \cdot 11 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $2262760937226240$ = $2^{21} \cdot 3 \cdot 5 \cdot 11^{4} \cdot 17^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{901247067798311192691198986281}{552431869440}$$ = $2^{-9} \cdot 3^{-1} \cdot 5^{-1} \cdot 11^{-4} \cdot 17^{-3} \cdot 181^{3} \cdot 1163^{3} \cdot 45887^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.6539346294720073882316510965\dots$ Stable Faltings height: $2.9607874489120620788144189750\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $24.502521496642669394172974313\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.13376725590652858994413144136\dots$ 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: $8$  = $2^{2}\cdot1\cdot1\cdot2\cdot1$ 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: $L'(E,1)$ ≈ $6.5552701267932357263165600597$

## Modular invariants

Modular form 44880.2.a.bh

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);

$$q - q^{3} + q^{5} + 4 q^{7} + q^{9} - q^{11} + 2 q^{13} - q^{15} - q^{17} + 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 15925248 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $4$ $I_{13}^{*}$ Additive -1 4 21 9
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$11$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$17$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3

## 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 4.6.0.1
$3$ 3B 3.4.0.1

## $p$-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.

## 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 add nonsplit split ord nonsplit ord nonsplit ord ss ord ord ord ord ord ord - 1 2 5 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

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 $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 44880.bh consists of 8 curves linked by isogenies of degrees dividing 12.

## 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 $K$ $2$ $$\Q(\sqrt{510})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{6})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{85})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{3})$$ $$\Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{85})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{170})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{3})$$ $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{3}, \sqrt{85})$$ $$\Z/12\Z$$ Not in database $6$ 6.0.34581456360000.2 $$\Z/6\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/8\Z$$ Not in database $8$ 8.8.277102632960000.10 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $16$ Deg 16 $$\Z/24\Z$$ Not in database $18$ 18.6.7899277863494247069268000737426718681804800000000.1 $$\Z/18\Z$$ 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.