# Properties

 Label 1728v4 Conductor $1728$ Discriminant $-11337408$ j-invariant $$-12288000$$ CM yes ($$D=-27$$) Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -1080, 13662])

gp: E = ellinit([0, 0, 0, -1080, 13662])

magma: E := EllipticCurve([0, 0, 0, -1080, 13662]);

$$y^2=x^3-1080x+13662$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(19, 1\right)$$ $\hat{h}(P)$ ≈ $1.3495768356801180454777611856$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(19,\pm 1)$$

## 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: $-11337408$ = $-1 \cdot 2^{6} \cdot 3^{11}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-12288000$$ = $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-27})/2]$$ (potential complex multiplication) Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $0.39872152268707185396280341103\dots$ Stable Faltings height: $-0.95491333220533468452478745021\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.3495768356801180454777611856\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.1636817490137510094722028257\dots$ 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: $1$ 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: $L'(E,1)$ ≈ $2.9200547682528014610166866824285120714$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 432 $\Gamma_0(N)$-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 6 0
$3$ $1$ $II^{*}$ Additive -1 3 11 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 ordinary ss ordinary ss ordinary ss ss ordinary ordinary ss ordinary ss - - 1,1 5 1,1 1 1,1 3 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 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, 9 and 27.
Its isogeny class 1728v consists of 4 curves linked by isogenies of degrees dividing 27.

## 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 $K$ $2$ $$\Q(\sqrt{6})$$ $$\Z/3\Z$$ 2.2.24.1-81.1-b2 $3$ 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 $6$ 6.0.30233088.2 $$\Z/3\Z$$ Not in database $6$ 6.6.10077696.1 $$\Z/9\Z$$ Not in database $6$ 6.2.4478976.4 $$\Z/6\Z$$ Not in database $12$ 12.2.962938848411648.4 $$\Z/4\Z$$ Not in database $12$ 12.0.8226356490141696.17 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ 12.0.8226356490141696.35 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/7\Z$$ Not in database $12$ 12.0.20061226008576.4 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.18.396521139274783615537700143104.1 $$\Z/27\Z$$ Not in database $18$ 18.0.1289303099811316943271493632.4 $$\Z/6\Z$$ Not in database $18$ 18.6.5305774073297600589594624.1 $$\Z/18\Z$$ 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.