# Properties

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

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

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")

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

magma: E := EllipticCurve("26284a1");

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

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-4, 2\right)$$ $$\left(-3, 5\right)$$ $$\left(-1, 5\right)$$ $$\hat{h}(P)$$ ≈ 1.7502013367783218 1.206868178899694 1.0183468457691478

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

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

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$26284$$ = $$2^{2} \cdot 6571$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-105136$$ = $$-1 \cdot 2^{4} \cdot 6571$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); 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

 sage: E.rank()  magma: Rank(E); Rank: $$3$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.666468303903$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$3.03520809448$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$3$$  = $$3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 26284.2.a.a

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5256 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

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

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

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

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

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

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

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.

sage: rho = E.galois_representation();

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

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

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 26284.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 3.1.6571.1 $$\Z/2\Z$$ 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.