Properties

Label 5610t1
Conductor $5610$
Discriminant $-6.157\times 10^{15}$
j-invariant \( \frac{9023321954633914439}{6156756739584000} \)
CM no
Rank $0$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 43372, -1467502])
 
gp: E = ellinit([1, 0, 1, 43372, -1467502])
 
magma: E := EllipticCurve([1, 0, 1, 43372, -1467502]);
 

\(y^2+xy+y=x^3+43372x-1467502\)  Toggle raw display

Mordell-Weil group structure

$\Z/{6}\Z$

Torsion generators

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

\( \left(609, 15535\right) \)  Toggle raw display

Integral points

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

\( \left(33, -17\right) \), \( \left(114, 2170\right) \), \( \left(114, -2285\right) \), \( \left(609, 15535\right) \), \( \left(609, -16145\right) \)  Toggle raw display

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: $-6156756739584000 $  =  $-1 \cdot 2^{12} \cdot 3^{12} \cdot 5^{3} \cdot 11^{3} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{9023321954633914439}{6156756739584000} \)  =  $2^{-12} \cdot 3^{-12} \cdot 5^{-3} \cdot 11^{-3} \cdot 17^{-1} \cdot 37^{3} \cdot 56267^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.7183341240180619236995642351\dots$
Stable Faltings height: $1.7183341240180619236995642351\dots$

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: $0.24050500889864283326946327256\dots$
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: $ 216 $  = $ 2\cdot( 2^{2} \cdot 3 )\cdot3\cdot3\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $6$
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) $ ≈ $ 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: 55296
$ \Gamma_0(N) $-optimal: yes
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$(j)_{-}$
$2$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$3$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$5$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$17$ $1$ $I_{1}$ Non-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 $\GL(2,\Z_\ell)$ 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.1.1 3.8.0.1

$p$-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 0 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 5610t consists of 8 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-935}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
$2$ \(\Q(\sqrt{85}) \) \(\Z/12\Z\) Not in database
$2$ \(\Q(\sqrt{-11}) \) \(\Z/12\Z\) Not in database
$4$ \(\Q(\sqrt{-11}, \sqrt{85})\) \(\Z/2\Z \times \Z/12\Z\) Not in database
$6$ 6.0.2255067.2 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ Deg 8 \(\Z/24\Z\) Not in database
$8$ Deg 8 \(\Z/24\Z\) Not in database
$9$ 9.3.33173450251082263546875.3 \(\Z/18\Z\) 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
$16$ Deg 16 \(\Z/4\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/24\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/24\Z\) Not in database
$18$ 18.0.259964449658121650679535108642988367643040452972412109375.1 \(\Z/2\Z \times \Z/18\Z\) Not in database
$18$ 18.6.195315138736379902839620667650629878018813262939453125.1 \(\Z/36\Z\) Not in database
$18$ 18.0.1464735953877730789393777591084107020033935546875.3 \(\Z/36\Z\) 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.