# Properties

 Label 2736.r2 Conductor $2736$ Discriminant $6127239168$ j-invariant $$\frac{132651}{76}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-459x+378$$ y^2=x^3-459x+378 (homogenize, simplify) $$y^2z=x^3-459xz^2+378z^3$$ y^2z=x^3-459xz^2+378z^3 (dehomogenize, simplify) $$y^2=x^3-459x+378$$ y^2=x^3-459x+378 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -459, 378])

gp: E = ellinit([0, 0, 0, -459, 378])

magma: E := EllipticCurve([0, 0, 0, -459, 378]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

$$\Z \oplus \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-6, 54\right)$$ (-6, 54) $\hat{h}(P)$ ≈ $1.5517182044964501659920625154$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(21, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-11,\pm 64)$$, $$(-6,\pm 54)$$, $$\left(21, 0\right)$$

## 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: $6127239168$ = $2^{14} \cdot 3^{9} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{132651}{76}$$ = $2^{-2} \cdot 3^{3} \cdot 17^{3} \cdot 19^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.56769457232367721870672506321\dots$ Stable Faltings height: $-0.94941182473735035925694098594\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.5517182044964501659920625154\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $1.1487389207708002286001043010\dots$ 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: $8$  = $2^{2}\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) 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); Special value: $L'(E,1)$ ≈ $3.5650381911473121085286996083$

## 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);

$$q + 2 q^{5} - 2 q^{11} - 4 q^{13} + q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1152 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $4$ $I_{6}^{*}$ Additive -1 4 14 2
$3$ $2$ $III^{*}$ Additive 1 2 9 0
$19$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1
sage: gens = [[1, 0, 4, 1], [308, 1, 151, 0], [3, 4, 8, 11], [194, 1, 359, 0], [1, 2, 2, 5], [1, 4, 0, 1], [453, 4, 452, 5], [229, 4, 2, 9], [289, 172, 56, 399]]

sage: GL(2,Integers(456)).subgroup(gens)

magma: Gens := [[1, 0, 4, 1], [308, 1, 151, 0], [3, 4, 8, 11], [194, 1, 359, 0], [1, 2, 2, 5], [1, 4, 0, 1], [453, 4, 452, 5], [229, 4, 2, 9], [289, 172, 56, 399]];

magma: sub<GL(2,Integers(456))|Gens>;

The image of the adelic Galois representation has level $456$, index $12$, genus $0$, and generators

$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 151 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 194 & 1 \\ 359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 229 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 289 & 172 \\ 56 & 399 \end{array}\right)$

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 add add ord ss ord ord ss split ord ord ord ord ord ord ord - - 1 1,1 1 1 1,1 2 1 1 1 1 1 1 1 - - 0 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 $d=$ 2.
Its isogeny class 2736.r consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{57})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.4.131328.1 $$\Z/4\Z$$ Not in database $8$ 8.8.6226182733824.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.8779890495744.12 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.2.291852315648.12 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.