# Properties

 Label 26198.a1 Conductor 26198 Discriminant 209584 j-invariant $$\frac{469097433}{209584}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

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

magma: E := EllipticCurve("26198a1");

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

sage: E = EllipticCurve("26198a1")

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

gp: E = ellinit("26198a1")

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

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-4, 4\right)$$ $$\left(0, -4\right)$$ $$\left(-1, 6\right)$$ $$\hat{h}(P)$$ ≈ 1.15179015538 0.781497240671 1.58782432757

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-4, 4\right)$$, $$\left(-4, 0\right)$$, $$\left(-3, 7\right)$$, $$\left(-3, -4\right)$$, $$\left(-1, 6\right)$$, $$\left(-1, -5\right)$$, $$\left(0, 4\right)$$, $$\left(0, -4\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(4, 0\right)$$, $$\left(4, -4\right)$$, $$\left(5, 4\right)$$, $$\left(5, -9\right)$$, $$\left(7, 11\right)$$, $$\left(7, -18\right)$$, $$\left(12, 32\right)$$, $$\left(12, -44\right)$$, $$\left(20, 76\right)$$, $$\left(20, -96\right)$$, $$\left(35, 186\right)$$, $$\left(35, -221\right)$$, $$\left(64, 476\right)$$, $$\left(64, -540\right)$$, $$\left(92, 832\right)$$, $$\left(92, -924\right)$$, $$\left(99, 931\right)$$, $$\left(99, -1030\right)$$, $$\left(209, 2911\right)$$, $$\left(209, -3120\right)$$, $$\left(2295, 108779\right)$$, $$\left(2295, -111074\right)$$, $$\left(70300, 18604196\right)$$, $$\left(70300, -18674496\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$26198$$ = $$2 \cdot 13099$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$209584$$ = $$2^{4} \cdot 13099$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{469097433}{209584}$$ = $$2^{-4} \cdot 3^{3} \cdot 7^{3} \cdot 37^{3} \cdot 13099^{-1}$$ 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: $$0.698734427437$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$2.84179979513$$ 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: $$2$$  = $$2\cdot1$$ 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 26198.2.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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 12704 $$\Gamma_0(N)$$-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])

$$L^{(3)}(E,1)/3!$$ ≈ $$3.97132670549$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13099$$ $$1$$ $$I_{1}$$ 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 13099 nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit 3 3,3 7 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 ?

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 26198.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 3.3.52396.1 $$\Z/2\Z$$ Not in database
6 6.6.143844877395136.1 $$\Z/2\Z \times \Z/2\Z$$ 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.