# Properties

 Label 27773.a1 Conductor 27773 Discriminant -27773 j-invariant $$-\frac{979146657}{27773}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -21, 42]); // or

magma: E := EllipticCurve("27773a1");

sage: E = EllipticCurve([1, -1, 1, -21, 42]) # or

sage: E = EllipticCurve("27773a1")

gp: E = ellinit([1, -1, 1, -21, 42]) \\ or

gp: E = ellinit("27773a1")

$$y^2 + x y + y = x^{3} - x^{2} - 21 x + 42$$

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-5, 3\right)$$ $$\left(3, -1\right)$$ $$\left(-2, 9\right)$$ $$\hat{h}(P)$$ ≈ 2.06638006103 0.847894878575 1.78581677844

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-5, 3\right)$$, $$\left(-5, 1\right)$$, $$\left(-2, 9\right)$$, $$\left(-2, -8\right)$$, $$\left(0, 6\right)$$, $$\left(0, -7\right)$$, $$\left(2, 1\right)$$, $$\left(2, -4\right)$$, $$\left(3, -1\right)$$, $$\left(3, -3\right)$$, $$\left(4, 1\right)$$, $$\left(4, -6\right)$$, $$\left(8, 14\right)$$, $$\left(8, -23\right)$$, $$\left(14, 41\right)$$, $$\left(14, -56\right)$$, $$\left(19, 69\right)$$, $$\left(19, -89\right)$$, $$\left(59, 419\right)$$, $$\left(59, -479\right)$$, $$\left(618, 15044\right)$$, $$\left(618, -15663\right)$$, $$\left(15052, 1839106\right)$$, $$\left(15052, -1854159\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$27773$$ = $$27773$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-27773$$ = $$-1 \cdot 27773$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{979146657}{27773}$$ = $$-1 \cdot 3^{3} \cdot 331^{3} \cdot 27773^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$3$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1.00675571342$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$3.73153661008$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$1$$  = $$1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 27773.2.a.a

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - q^{2} - 3q^{3} - q^{4} - 2q^{5} + 3q^{6} - 4q^{7} + 3q^{8} + 6q^{9} + 2q^{10} - 4q^{11} + 3q^{12} - 6q^{13} + 4q^{14} + 6q^{15} - q^{16} - 4q^{17} - 6q^{18} + q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 5536 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L^{(3)}(E,1)/3!$$ ≈ $$3.75674580203$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$27773$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ .

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 27773 ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit 3 3,7 7 3 3 5 3 3 3 3 3 5 3 3 3 ? 0 0,0 0 0 0 0 0 0 0 0 0 0 0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

## Isogenies

This curve has no rational isogenies. Its isogeny class 27773.a consists of this curve only.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.111092.4 $$\Z/2\Z$$ Not in database
6 $$x^{6} + 166 x^{4} + 6889 x^{2} + 111092$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.