# Properties

 Label 2736q2 Conductor $2736$ Discriminant $-20480864256$ j-invariant $$-\frac{89915392}{6859}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -1344, -20176])

gp: E = ellinit([0, 0, 0, -1344, -20176])

magma: E := EllipticCurve([0, 0, 0, -1344, -20176]);

$$y^2=x^3-1344x-20176$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(97, 873\right)$$ $$\hat{h}(P)$$ ≈ $3.6757007349520942507364355173$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(97,\pm 873)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2736$$ = $$2^{4} \cdot 3^{2} \cdot 19$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-20480864256$$ = $$-1 \cdot 2^{12} \cdot 3^{6} \cdot 19^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{89915392}{6859}$$ = $$-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.72658633755126311293732700504\dots$$ Stable Faltings height: $$-0.51586698734273704217752773488\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.6757007349520942507364355173\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.39252882408134501697290691736\dots$$ 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: $$2$$  = $$1\cdot2\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

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);

$$q - 3q^{5} + q^{7} + 3q^{11} - 4q^{13} + 3q^{17} - q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1728 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$2.8856369743313623825014774095515339966$$

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$1$$ $$II^{*}$$ Additive -1 4 12 0
$$3$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$19$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## 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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ Cs

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 add add ordinary ordinary ordinary ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary - - 1 1 1 1 1 1 1,1 1 1 1 1 1 1 - - 0 0 0 0 0 0 0,0 0 0 0 0 0 0

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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 2736q consists of 2 curves linked by isogenies of degrees dividing 9.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{3})$$ $$\Z/3\Z$$ 2.2.12.1-361.1-a2 $2$ $$\Q(\sqrt{-1})$$ $$\Z/3\Z$$ 2.0.4.1-29241.1-b2 $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\zeta_{12})$$ $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.2495232.1 $$\Z/6\Z$$ Not in database $6$ 6.0.92416.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.6226182733824.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ 12.0.2247651966910464.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.3083198857216.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.11420218832186047715868672.1 $$\Z/9\Z$$ Not in database $18$ 18.0.1474808921484175848353057988673536.1 $$\Z/9\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.