Properties

 Label 26284a1 Conductor 26284 Discriminant -105136 j-invariant $$-\frac{5619712}{6571}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -9, 16]); // or

magma: E := EllipticCurve("26284a1");

sage: E = EllipticCurve([0, 1, 0, -9, 16]) # or

sage: E = EllipticCurve("26284a1")

gp: E = ellinit([0, 1, 0, -9, 16]) \\ or

gp: E = ellinit("26284a1")

$$y^2 = x^{3} + x^{2} - 9 x + 16$$

Mordell-Weil group structure

$$\Z^3$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-4, 2\right)$$ $$\left(-3, 5\right)$$ $$\left(-1, 5\right)$$ $$\hat{h}(P)$$ ≈ 1.75020133678 1.2068681789 1.01834684577

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(-4,\pm 2)$$, $$(-3,\pm 5)$$, $$(-1,\pm 5)$$, $$(0,\pm 4)$$, $$(1,\pm 3)$$, $$(3,\pm 5)$$, $$(5,\pm 11)$$, $$(11,\pm 37)$$, $$(15,\pm 59)$$, $$(28,\pm 150)$$, $$(47,\pm 325)$$, $$(55,\pm 411)$$, $$(81,\pm 733)$$, $$(549,\pm 12875)$$, $$(1799,\pm 76325)$$, $$(8361,\pm 764563)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$26284$$ = $$2^{2} \cdot 6571$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-105136$$ = $$-1 \cdot 2^{4} \cdot 6571$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{5619712}{6571}$$ = $$-1 \cdot 2^{14} \cdot 7^{3} \cdot 6571^{-1}$$ 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: $$0.666468303903$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$3.03520809448$$ 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: $$3$$  = $$3\cdot1$$ 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 26284.2.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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 5256 $$\Gamma_0(N)$$-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])

$$L^{(3)}(E,1)/3!$$ ≈ $$6.06860997217$$

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$$(j)_{-}$$
$$2$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$6571$$ $$1$$ $$I_{1}$$ 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 6571 add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split - 5 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 ?

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

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 26284a 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 3.1.6571.1 $$\Z/2\Z$$ Not in database
6 6.0.283722907411.1 $$\Z/2\Z \times \Z/2\Z$$ 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.