Properties

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

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+x^2-2160176x-1222750176\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+x^2z-2160176xz^2-1222750176z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-174974283x-890859955482\) Copy content Toggle raw display (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);
 

$P$ =  \(\left(2515, 96222\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $7.2968959186229858394793962872$

Torsion generators

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

\( \left(-849, 0\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-849, 0\right) \), \((2515,\pm 96222)\) Copy content Toggle raw display

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

Modular form   2760.2.a.h

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);
 

\( q + q^{3} - q^{5} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + O(q^{20}) \) Copy content Toggle raw display

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

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add split nonsplit ss ss ord ord ss nonsplit ord ss ord ord ss ord
$\lambda$-invariant(s) - 4 5 1,1 1,1 1 1 1,1 1 1 1,1 1 3 1,1 1
$\mu$-invariant(s) - 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:

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