# Properties

 Label 27262.a1 Conductor 27262 Discriminant 111665152 j-invariant $$\frac{1466390638881}{111665152}$$ 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, 1, -237, 1365]); // or

magma: E := EllipticCurve("27262c1");

sage: E = EllipticCurve([1, -1, 1, -237, 1365]) # or

sage: E = EllipticCurve("27262c1")

gp: E = ellinit([1, -1, 1, -237, 1365]) \\ or

gp: E = ellinit("27262c1")

$$y^2 + x y + y = x^{3} - x^{2} - 237 x + 1365$$

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(15, -40\right)$$ $$\left(-15, 44\right)$$ $$\left(-13, 52\right)$$ $$\hat{h}(P)$$ ≈ 0.478352716181 2.57871770231 1.99704614029

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-17, 24\right)$$, $$\left(-15, 44\right)$$, $$\left(-13, 52\right)$$, $$\left(-9, 56\right)$$, $$\left(-1, 40\right)$$, $$\left(3, 24\right)$$, $$\left(5, 14\right)$$, $$\left(7, 0\right)$$, $$\left(11, -4\right)$$, $$\left(13, 12\right)$$, $$\left(15, 24\right)$$, $$\left(21, 62\right)$$, $$\left(25, 90\right)$$, $$\left(31, 136\right)$$, $$\left(47, 280\right)$$, $$\left(49, 300\right)$$, $$\left(87, 752\right)$$, $$\left(99, 920\right)$$, $$\left(143, 1624\right)$$, $$\left(271, 4312\right)$$, $$\left(455, 9464\right)$$, $$\left(623, 15224\right)$$, $$\left(735, 19544\right)$$, $$\left(1751, 72376\right)$$, $$\left(98815, 31012824\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$27262$$ = $$2 \cdot 43 \cdot 317$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$111665152$$ = $$2^{13} \cdot 43 \cdot 317$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{1466390638881}{111665152}$$ = $$2^{-13} \cdot 3^{3} \cdot 7^{3} \cdot 43^{-1} \cdot 317^{-1} \cdot 541^{3}$$ 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: $$0.297872072084$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$1.83426171198$$ 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: $$13$$  = $$13\cdot1\cdot1$$ 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 27262.2.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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 65312 $$\Gamma_0(N)$$-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])

$$L^{(3)}(E,1)/3!$$ ≈ $$7.10287937958$$

## 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$$(j)_{-}$$
$$2$$ $$13$$ $$I_{13}$$ Split multiplicative -1 1 13 13
$$43$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$317$$ $$1$$ $$I_{1}$$ 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 47 317 split ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split ordinary nonsplit 4 5,5 3 3 3 3 3 3 3 3 3 3 3 4 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 27262.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 3.3.109048.1 $$\Z/2\Z$$ Not in database
6 $$x^{6} - 290 x^{4} + 21025 x^{2} - 436192$$ $$\Z/2\Z \times \Z/2\Z$$ 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.