Properties

 Label 2760.h1 Conductor $2760$ Discriminant $2.901\times 10^{13}$ j-invariant $$\frac{544328872410114151778}{14166950625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Simplified equation

 $$y^2=x^3+x^2-2160176x-1222750176$$ y^2=x^3+x^2-2160176x-1222750176 (homogenize, simplify) $$y^2z=x^3+x^2z-2160176xz^2-1222750176z^3$$ y^2z=x^3+x^2z-2160176xz^2-1222750176z^3 (dehomogenize, simplify) $$y^2=x^3-174974283x-890859955482$$ y^2=x^3-174974283x-890859955482 (homogenize, minimize)

sage: E = EllipticCurve([0, 1, 0, -2160176, -1222750176])

gp: E = ellinit([0, 1, 0, -2160176, -1222750176])

magma: E := EllipticCurve([0, 1, 0, -2160176, -1222750176]);

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);

 $P$ = $$\left(2515, 96222\right)$$ (2515, 96222) $\hat{h}(P)$ ≈ $7.2968959186229858394793962872$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-849, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-849, 0\right)$$, $$(2515,\pm 96222)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2760$$ = $2^{3} \cdot 3 \cdot 5 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $29013914880000$ = $2^{11} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{544328872410114151778}{14166950625}$$ = $2 \cdot 3^{-4} \cdot 5^{-4} \cdot 11^{3} \cdot 23^{-4} \cdot 589139^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.0985712505147886404928158627\dots$ Stable Faltings height: $1.4631863350015054401936864180\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $7.2968959186229858394793962872\dots$ 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: $0.12453093659617832842627327836\dots$ 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: $16$  = $1\cdot2^{2}\cdot2\cdot2$ 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: $L'(E,1)$ ≈ $3.6347571319638058766810499103$

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);

$$q + q^{3} - q^{5} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 24576 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 4 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 3 11 0
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$5$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$23$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

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 8.24.0.58
sage: gens = [[177, 8, 176, 9], [123, 116, 162, 29], [97, 8, 20, 33], [1, 0, 8, 1], [7, 6, 178, 179], [1, 4, 4, 17], [72, 169, 107, 94], [1, 8, 0, 1]]

sage: GL(2,Integers(184)).subgroup(gens)

magma: Gens := [[177, 8, 176, 9], [123, 116, 162, 29], [97, 8, 20, 33], [1, 0, 8, 1], [7, 6, 178, 179], [1, 4, 4, 17], [72, 169, 107, 94], [1, 8, 0, 1]];

magma: sub<GL(2,Integers(184))|Gens>;

The image of the adelic Galois representation has level $184$, index $48$, genus $0$, and generators

$\left(\begin{array}{rr} 177 & 8 \\ 176 & 9 \end{array}\right),\left(\begin{array}{rr} 123 & 116 \\ 162 & 29 \end{array}\right),\left(\begin{array}{rr} 97 & 8 \\ 20 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 178 & 179 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 72 & 169 \\ 107 & 94 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$

$p$-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 split nonsplit ss ss ord ord ss nonsplit ord ss ord ord ss ord - 4 5 1,1 1,1 1 1 1,1 1 1 1,1 1 3 1,1 1 - 0 0 0,0 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 and 4.
Its isogeny class 2760.h consists of 4 curves linked by isogenies of degrees dividing 4.

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 $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ Not in database $4$ 4.2.2048.1 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ $$\Q(\zeta_{8})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.16777216.2 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.1146228736.4 $$\Z/8\Z$$ Not in database $8$ 8.0.1173738225664.17 $$\Z/8\Z$$ Not in database $8$ 8.2.31726715921280000.4 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.336343120699432370176.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.