# Properties

 Label 44.a1 Conductor $44$ Discriminant $-340736$ j-invariant $$-\frac{199794688}{1331}$$ CM no Rank $0$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+x^2-77x-289$$ y^2=x^3+x^2-77x-289 (homogenize, simplify) $$y^2z=x^3+x^2z-77xz^2-289z^3$$ y^2z=x^3+x^2z-77xz^2-289z^3 (dehomogenize, simplify) $$y^2=x^3-6264x-191916$$ y^2=x^3-6264x-191916 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -77, -289])

gp: E = ellinit([0, 1, 0, -77, -289])

magma: E := EllipticCurve([0, 1, 0, -77, -289]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## 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: $-340736$ = $-1 \cdot 2^{8} \cdot 11^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{199794688}{1331}$$ = $-1 \cdot 2^{13} \cdot 11^{-3} \cdot 29^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.10298768145811296210585672847\dots$ Stable Faltings height: $-0.56508580183140983505067814278\dots$

## 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: $0.80464628757465185995669379233\dots$ 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: $L(E,1)$ ≈ $0.80464628757465185995669379233$

## 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: 6 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $1$ $IV^{*}$ Additive -1 2 8 0
$11$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3

## 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.1.2 3.8.0.2

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 - 1 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 44.a 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 $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ 2.0.3.1-1936.1-a1 $3$ 3.1.44.1 $$\Z/2\Z$$ Not in database $3$ 3.1.108.1 $$\Z/3\Z$$ Not in database $6$ 6.0.21296.1 $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $6$ 6.0.34992.1 $$\Z/3\Z \oplus \Z/3\Z$$ Not in database $6$ 6.0.52272.1 $$\Z/6\Z$$ Not in database $9$ 9.1.1676676672.1 $$\Z/6\Z$$ Not in database $12$ 12.2.20433779818496.1 $$\Z/4\Z$$ Not in database $12$ 12.0.330615800064.1 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $18$ 18.0.98027075732830169406618120192.3 $$\Z/9\Z$$ Not in database $18$ 18.0.75903605885582880768.1 $$\Z/3\Z \oplus \Z/6\Z$$ Not in database $18$ 18.0.3741766645692993122304.2 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive.