# Properties

 Label 44a1 Conductor $44$ Discriminant $-2816$ j-invariant $$\frac{8192}{11}$$ CM no Rank $0$ Torsion structure $$\Z/{3}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2+3x-1$$ y^2=x^3+x^2+3x-1 (homogenize, simplify) $$y^2z=x^3+x^2z+3xz^2-z^3$$ y^2z=x^3+x^2z+3xz^2-z^3 (dehomogenize, simplify) $$y^2=x^3+216x-1404$$ y^2=x^3+216x-1404 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, 3, -1])

gp: E = ellinit([0, 1, 0, 3, -1])

magma: E := EllipticCurve([0, 1, 0, 3, -1]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1, 2\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(1,\pm 2)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$44$$ = $2^{2} \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-2816$ = $-1 \cdot 2^{8} \cdot 11$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{8192}{11}$$ = $2^{13} \cdot 11^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.65229382579216780780347934693\dots$ Stable Faltings height: $-1.1143919461654646807483007612\dots$

## 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: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $2.4139388627239555798700813770\dots$ 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: $3$  = $3\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $3$ 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: $L(E,1)$ ≈ $0.80464628757465185995669379233$

## 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);

$$q + q^{3} - 3 q^{5} + 2 q^{7} - 2 q^{9} - q^{11} - 4 q^{13} - 3 q^{15} + 6 q^{17} + 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $3$ $IV^{*}$ Additive -1 2 8 0
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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.1.1 3.8.0.1

The image of the adelic Galois representation has level $66$, index $16$, and genus $0$.

## $p$-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) 2 3 11 add ord nonsplit - 4 0 - 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 44a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.44.1 $$\Z/6\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $6$ 6.0.6324912.1 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $9$ 9.3.18443443392.1 $$\Z/9\Z$$ Not in database $12$ 12.2.20433779818496.1 $$\Z/12\Z$$ Not in database $18$ 18.0.30616027031219947941888.1 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive.