Properties

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

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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()

\(P\) =  \( \left(-5, 3\right) \)\( \left(3, -1\right) \)\( \left(-2, 9\right) \)
\(\hat{h}(P)\) ≈  2.066380061030.8478948785751.78581677844

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

\( \left(-5, 3\right) \), \( \left(-2, 9\right) \), \( \left(0, 6\right) \), \( \left(2, 1\right) \), \( \left(3, -1\right) \), \( \left(4, 1\right) \), \( \left(8, 14\right) \), \( \left(14, 41\right) \), \( \left(19, 69\right) \), \( \left(59, 419\right) \), \( \left(618, 15044\right) \), \( \left(15052, 1839106\right) \)

Note: only one of each pair $\pm P$ is listed.

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)

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

For more coefficients, see the Downloads section to the right.

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 27773
Reduction type ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 3 3,7 7 3 3 5 3 3 3 3 3 5 3 3 3 ?
$\mu$-invariant(s) 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 \(x^{6} \) \(\mathstrut +\mathstrut 166 x^{4} \) \(\mathstrut +\mathstrut 6889 x^{2} \) \(\mathstrut +\mathstrut 111092 \) \(\Z/2\Z \times \Z/2\Z\) 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.