Properties

Label 26743.a1
Conductor 26743
Discriminant -26743
j-invariant \( \frac{23639903}{26743} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, 6, -5]); // or
magma: E := EllipticCurve("26743a1");
sage: E = EllipticCurve([1, 0, 0, 6, -5]) # or
sage: E = EllipticCurve("26743a1")
gp: E = ellinit([1, 0, 0, 6, -5]) \\ or
gp: E = ellinit("26743a1")

\( y^2 + x y = x^{3} + 6 x - 5 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(\frac{3}{4}, -\frac{1}{8}\right) \)\( \left(1, 1\right) \)\( \left(2, 3\right) \)
\(\hat{h}(P)\) ≈  2.429441284911.105129764831.3396515553

Integral points

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

\( \left(1, 1\right) \), \( \left(2, 3\right) \), \( \left(3, 5\right) \), \( \left(5, 10\right) \), \( \left(6, 13\right) \), \( \left(23, 100\right) \), \( \left(27, 128\right) \), \( \left(47, 300\right) \), \( \left(82, 703\right) \), \( \left(105, 1025\right) \), \( \left(801, 22273\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 26743 \)  =  \(47 \cdot 569\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-26743 \)  =  \(-1 \cdot 47 \cdot 569 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{23639903}{26743} \)  =  \(7^{3} \cdot 41^{3} \cdot 47^{-1} \cdot 569^{-1}\)
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: \(2.08297216415\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(2.02392471236\)
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: \( 1 \)  = \( 1\cdot1 \)
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 26743.2.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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 2480
\( \Gamma_0(N) \)-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])

\( L^{(3)}(E,1)/3! \) ≈ \( 4.21577883818 \)

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\((j)_{-}\)
\(47\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(569\) \(1\) \( I_{1} \) 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 47 569
Reduction type ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary nonsplit nonsplit
$\lambda$-invariant(s) 4 5 3 3 3 3 3 3 3 3 3 3 3,3 3 3 3
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0,0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 26743.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 3.1.26743.1 \(\Z/2\Z\) Not in database
6 6.0.19126273994407.1 \(\Z/2\Z \times \Z/2\Z\) 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.