Properties

Label 5610.q2
Conductor $5610$
Discriminant $6.152\times 10^{21}$
j-invariant \( \frac{224494757451893010998773801}{6152490825146276160000} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -12661763, -16927055362])
 
gp: E = ellinit([1, 0, 1, -12661763, -16927055362])
 
magma: E := EllipticCurve([1, 0, 1, -12661763, -16927055362]);
 

\(y^2+xy+y=x^3-12661763x-16927055362\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(\frac{16391}{4}, -\frac{16395}{8}\right) \)  Toggle raw display

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 5610 \)  =  $2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $6152490825146276160000 $  =  $2^{9} \cdot 3 \cdot 5^{4} \cdot 11 \cdot 17^{12} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{224494757451893010998773801}{6152490825146276160000} \)  =  $2^{-9} \cdot 3^{-1} \cdot 5^{-4} \cdot 11^{-1} \cdot 17^{-12} \cdot 107^{3} \cdot 347^{3} \cdot 16369^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.9607874489120620788144189750\dots$
Stable Faltings height: $2.9607874489120620788144189750\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.080168336299547611089821090855\dots$
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 $  = $ 1\cdot1\cdot2^{2}\cdot1\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: $9$ = $3^2$ (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) $ ≈ $ 1.4430300533918569996167796353867365261 $

Modular invariants

Modular form   5610.2.a.q

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);
 

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + 2q^{13} + 4q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 663552
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 5 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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$17$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12

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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 4.6.0.1
$3$ 3B.1.2 3.8.0.2

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 5 11 17
Reduction type nonsplit split split split nonsplit
$\lambda$-invariant(s) 5 3 1 3 0
$\mu$-invariant(s) 0 1 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3, 4, 6 and 12.
Its isogeny class 5610.q consists of 6 curves linked by isogenies of degrees dividing 12.

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
$2$ \(\Q(\sqrt{66}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{6}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{11}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-3}) \) \(\Z/6\Z\) Not in database
$3$ 3.1.735075.4 \(\Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{6}, \sqrt{11})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-22})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-3})\) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{11})\) \(\Z/12\Z\) Not in database
$6$ 6.0.1621005766875.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$6$ 6.2.9129504479040000.8 \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.2.829954952640000.6 \(\Z/12\Z\) Not in database
$6$ 6.2.380396019960000.3 \(\Z/12\Z\) Not in database
$8$ 8.0.5416809268248576.19 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/8\Z\) Not in database
$8$ 8.0.77720518656.6 \(\Z/2\Z \times \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.30133353666283596303054812383372187354296875.1 \(\Z/18\Z\) 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.