# Properties

 Label 1728.p1 Conductor $1728$ Discriminant $-322486272$ j-invariant $$0$$ CM yes ($$D=-3$$) Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 0, -864])

gp: E = ellinit([0, 0, 0, 0, -864])

magma: E := EllipticCurve([0, 0, 0, 0, -864]);

$$y^2=x^3-864$$

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1728$$ = $2^{6} \cdot 3^{3}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-322486272$ = $-1 \cdot 2^{14} \cdot 3^{9}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$0$$ = $0$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $0.31151349872631388121000220687\dots$ Stable Faltings height: $-1.3211174284280379149898691959\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.78694809902317917748306142310\dots$ 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: $3$  = $1\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ 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: $L(E,1)$ ≈ $2.3608442970695375324491842692938745131$

## Modular invariants

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);

$$q + 5 q^{7} + 7 q^{13} + q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 576 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive 1 6 14 0
$3$ $3$ $IV^{*}$ Additive -1 3 9 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]

All $p$-adic regulators are identically $1$ since the rank is $0$.

## 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 ordinary ss ordinary ss ordinary ss ss ordinary ordinary ss ordinary ss - - 0,0 0 0,0 0 0,0 2 0,0 0,0 0 2 0,0 0 0,0 - - 0,0 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 $d=$ 3.
Its isogeny class 1728.p consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-6})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.108.1 $$\Z/2\Z$$ Not in database $6$ 6.0.34992.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.1492992.4 $$\Z/6\Z$$ Not in database $6$ 6.0.4478976.2 $$\Z/6\Z$$ Not in database $12$ 12.2.962938848411648.4 $$\Z/4\Z$$ Not in database $12$ 12.0.20061226008576.9 $$\Z/6\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/7\Z$$ Not in database $18$ 18.0.5305774073297600589594624.1 $$\Z/18\Z$$ 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.