# 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 / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3-21420x+1206576$$ y^2=x^3-21420x+1206576 (homogenize, simplify) $$y^2z=x^3-21420xz^2+1206576z^3$$ y^2z=x^3-21420xz^2+1206576z^3 (dehomogenize, simplify) $$y^2=x^3-21420x+1206576$$ y^2=x^3-21420x+1206576 (homogenize, minimize)

comment: Define the curve

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

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

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

oscar: E = elliptic_curve([0, 0, 0, -21420, 1206576])

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

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

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

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

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

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

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$28224$$ = $2^{6} \cdot 3^{2} \cdot 7^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $65548320768$ = $2^{18} \cdot 3^{6} \cdot 7^{3}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$16581375$$ = $3^{3} \cdot 5^{3} \cdot 17^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-7}]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $1.1364621163983230474077751517\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.93904233603947808869203383481\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.1980441775334598\dots$ Szpiro ratio: $4.052511320499709\dots$

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.69190873220748374292040284330\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $1.0441076544415988952242884345\dots$ 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: $32$  = $2^{2}\cdot2^{2}\cdot2$ 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: $1$ ( rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L'(E,1)$ ≈ $5.7794176277825297904209902355$ 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

$\displaystyle 5.779417628 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.044108 \cdot 0.691909 \cdot 32}{2^2} \approx 5.779417628$

# 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

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

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: 32768
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 3 primes $p$ of bad reduction:

$p$ Tamagawa number Kodaira symbol Reduction type Root number $v_p(N)$ $v_p(\Delta)$ $v_p(\mathrm{den}(j))$
$2$ $4$ $I_{8}^{*}$ additive 1 6 18 0
$3$ $4$ $I_0^{*}$ additive -1 2 6 0
$7$ $2$ $III$ additive -1 2 3 0

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 $\ell$.

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)];

The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.

$\ell$ Reduction type Serre weight Serre conductor
$2$ additive $2$ $$63 = 3^{2} \cdot 7$$
$3$ additive $6$ $$3136 = 2^{6} \cdot 7^{2}$$
$7$ additive $20$ $$576 = 2^{6} \cdot 3^{2}$$

## Isogenies

gp: ellisomat(E)

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.

## Twists

The minimal quadratic twist of this elliptic curve is 49.a3, its twist by $-24$.

## 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 $K$ $2$ $$\Q(\sqrt{7})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 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 $$\Z/2\Z \oplus \Z/4\Z$$ not in database $8$ 8.4.624529833984.14 $$\Z/2\Z \oplus \Z/4\Z$$ 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 $$\Z/2\Z \oplus \Z/14\Z$$ 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 $$\Z/16\Z$$ not in database $16$ deg 16 $$\Z/16\Z$$ not in database $16$ deg 16 $$\Z/3\Z \oplus \Z/6\Z$$ not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.

## 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 ord ss ss ss ord ord ss ord ss ord 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.

## $p$-adic regulators

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