Properties

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

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-459x+378\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-459xz^2+378z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-459x+378\) Copy content Toggle raw display (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);
 

$P$ =  \(\left(-6, 54\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $1.5517182044964501659920625154$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(21, 0\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\((-11,\pm 64)\), \((-6,\pm 54)\), \( \left(21, 0\right) \) Copy content Toggle raw display

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

Modular form   2736.2.a.r

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);
 

\( q + 2 q^{5} - 2 q^{11} - 4 q^{13} + q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add ord ss ord ord ss split ord ord ord ord ord ord ord
$\lambda$-invariant(s) - - 1 1,1 1 1 1,1 2 1 1 1 1 1 1 1
$\mu$-invariant(s) - - 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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{57}) \) \(\Z/2\Z \oplus \Z/2\Z\) 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.