# Properties

 Label 1728e1 Conductor $1728$ Discriminant $-14155776$ j-invariant $$-\frac{132651}{2}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -204, 1136])

gp: E = ellinit([0, 0, 0, -204, 1136])

magma: E := EllipticCurve([0, 0, 0, -204, 1136]);

$$y^2=x^3-204x+1136$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(14, 32\right)$$ $\hat{h}(P)$ ≈ $0.28260648052457390838010381208$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(8,\pm 4)$$, $$(14,\pm 32)$$

## 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: $-14155776$ = $-1 \cdot 2^{19} \cdot 3^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{132651}{2}$$ = $-1 \cdot 2^{-1} \cdot 3^{3} \cdot 17^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.17521019840595195140435587873\dots$ Stable Faltings height: $-1.1391636446009934355703036127\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.28260648052457390838010381208\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.2323974242336884741097038118\dots$ 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: $4$  = $2^{2}\cdot1$ 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: $L'(E,1)$ ≈ $2.5235599167792273558906286689755906923$

## 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);

$$q - 3 q^{5} - q^{7} + 3 q^{11} + 4 q^{13} - 2 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 384 $\Gamma_0(N)$-optimal: yes 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$(j)_{-}$
$2$ $4$ $I_{9}^{*}$ Additive 1 6 19 1
$3$ $1$ $II$ Additive 1 3 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B 9.36.0.5

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - - 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

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 and 9.
Its isogeny class 1728e consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/3\Z$$ 2.2.8.1-1458.1-k1 $3$ 3.1.216.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1119744.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.4478976.2 $$\Z/3\Z$$ Not in database $6$ 6.6.3359232.1 $$\Z/9\Z$$ Not in database $6$ 6.2.373248.1 $$\Z/6\Z$$ Not in database $12$ 12.2.1925877696823296.3 $$\Z/4\Z$$ Not in database $12$ 12.0.20061226008576.9 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ 12.0.1624959306694656.14 $$\Z/9\Z$$ Not in database $12$ 12.0.1253826625536.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.191007866638713621225406464.1 $$\Z/9\Z$$ Not in database $18$ 18.0.7278153735662003552256.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.27634239965091669737472.1 $$\Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive.