# Properties

 Label 28224bi3 Conductor $28224$ Discriminant $-4.935\times 10^{17}$ j-invariant $$\frac{9938375}{21952}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 126420, -29037008])

gp: E = ellinit([0, 0, 0, 126420, -29037008])

magma: E := EllipticCurve([0, 0, 0, 126420, -29037008]);

$$y^2=x^3+126420x-29037008$$

## Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{104742}{169}, \frac{37244416}{2197}\right)$$ $\hat{h}(P)$ ≈ $8.7677299772200736822263542326$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(182, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(182, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$28224$$ = $2^{6} \cdot 3^{2} \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-493548440962203648$ = $-1 \cdot 2^{24} \cdot 3^{6} \cdot 7^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{9938375}{21952}$$ = $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0792032355106864195506908480\dots$ Stable Faltings height: $-0.48277875419094304282545632437\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $8.7677299772200736822263542326\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.15286385062078200920857866785\dots$ 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: $16$  = $2^{2}\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $5.3610758620844871672230403749565818147$

## Modular invariants

Modular form 28224.2.a.dh

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);

$$q - 4 q^{13} + 6 q^{17} + 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 221184 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $4$ $I_{14}^{*}$ Additive 1 6 24 6
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$7$ $2$ $I_{3}^{*}$ Additive -1 2 9 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 8.6.0.1
$3$ 3Cs 3.12.0.1

## $p$-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 add ss add ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary - - 3,1 - 1,1 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 and 6.
Its isogeny class 28224bi consists of 4 curves linked by isogenies of degrees dividing 18.

## 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 $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{42})$$ $$\Z/6\Z$$ 2.2.168.1-14.1-j3 $2$ $$\Q(\sqrt{-14})$$ $$\Z/6\Z$$ Not in database $4$ 4.2.4032.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-14})$$ $$\Z/3\Z \times \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{-6}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-7})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $8$ 8.0.156132458496.10 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.796594176.12 $$\Z/2\Z \times \Z/12\Z$$ Not in database $8$ 8.0.796594176.2 $$\Z/6\Z \times \Z/6\Z$$ Not in database $8$ 8.4.796594176.2 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ 16.0.634562281237118976.5 $$\Z/6\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $18$ 18.6.6665409440369750708186945421312.1 $$\Z/18\Z$$ Not in database $18$ 18.0.103660447616630363013723375192244224.2 $$\Z/18\Z$$ 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.