# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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()

 $$P$$ = $$\left(\frac{3}{4}, -\frac{1}{8}\right)$$ $$\left(1, 1\right)$$ $$\left(2, 3\right)$$ $$\hat{h}(P)$$ ≈ 2.42944128491 1.10512976483 1.3396515553

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(1, 1\right)$$, $$\left(1, -2\right)$$, $$\left(2, 3\right)$$, $$\left(2, -5\right)$$, $$\left(3, 5\right)$$, $$\left(3, -8\right)$$, $$\left(5, 10\right)$$, $$\left(5, -15\right)$$, $$\left(6, 13\right)$$, $$\left(6, -19\right)$$, $$\left(23, 100\right)$$, $$\left(23, -123\right)$$, $$\left(27, 128\right)$$, $$\left(27, -155\right)$$, $$\left(47, 300\right)$$, $$\left(47, -347\right)$$, $$\left(82, 703\right)$$, $$\left(82, -785\right)$$, $$\left(105, 1025\right)$$, $$\left(105, -1130\right)$$, $$\left(801, 22273\right)$$, $$\left(801, -23074\right)$$

## 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)

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

 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 569 ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary nonsplit nonsplit 4 5 3 3 3 3 3 3 3 3 3 3 3,3 3 3 3 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.