# Properties

 Label 28224.dw3 Conductor $28224$ Discriminant $65548320768$ j-invariant $$16581375$$ CM yes ($$D=-28$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -21420, 1206576])

gp: E = ellinit([0, 0, 0, -21420, 1206576])

magma: E := EllipticCurve([0, 0, 0, -21420, 1206576]);

$$y^2=x^3-21420x+1206576$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(70, 224\right)$$ $\hat{h}(P)$ ≈ $0.69190873220748374292040284330$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(84, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-168,\pm 252)$$, $$(70,\pm 224)$$, $$\left(84, 0\right)$$, $$(85,\pm 1)$$, $$(102,\pm 288)$$

## 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: $65548320768$ = $2^{18} \cdot 3^{6} \cdot 7^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$16581375$$ = $3^{3} \cdot 5^{3} \cdot 17^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-7}]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $1.1364621163983230474077751517\dots$ Stable Faltings height: $-0.93904233603947808869203383481\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.69190873220748374292040284330\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $1.0441076544415988952242884345\dots$ 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: $32$  = $2^{2}\cdot2^{2}\cdot2$ 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: $L'(E,1)$ ≈ $5.7794176277825297904209902354670756865$

## Modular invariants

Modular form 28224.2.a.dw

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);

$$q + 4 q^{11} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 32768 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $4$ $I_{8}^{*}$ Additive 1 6 18 0
$3$ $4$ $I_0^{*}$ Additive -1 2 6 0
$7$ $2$ $III$ Additive -1 2 3 0

## 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]

$p$-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 ordinary ss ss ss ordinary ordinary ss ordinary ss ordinary ss - - 1,1 - 1 1,1 1,1 1,1 1 1 3,1 1 1,1 1 1,1 - - 0,0 - 0 0,0 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, 7 and 14.
Its isogeny class 28224.dw consists of 4 curves linked by isogenies of degrees dividing 14.

## 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 $4$ 4.0.197568.2 $$\Z/4\Z$$ Not in database $6$ 6.6.232339968.1 $$\Z/14\Z$$ Not in database $8$ 8.0.624529833984.16 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.624529833984.14 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.156132458496.3 $$\Z/8\Z$$ Not in database $8$ 8.0.156132458496.28 $$\Z/8\Z$$ Not in database $8$ 8.2.1053894094848.2 $$\Z/6\Z$$ Not in database $12$ 12.12.3454839086735425536.2 $$\Z/2\Z \times \Z/14\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/3\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $20$ 20.0.709769256018536283137282494313791488.2 $$\Z/22\Z$$ 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.