Properties

Label 14586h2
Conductor $14586$
Discriminant $-2.900\times 10^{18}$
j-invariant \( \frac{2773679829880629422375}{2899504554614368272} \)
CM no
Rank $1$
Torsion structure \(\Z/{6}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, 292714, 54762104])
 
gp: E = ellinit([1, 0, 1, 292714, 54762104])
 
magma: E := EllipticCurve([1, 0, 1, 292714, 54762104]);
 

\(y^2+xy+y=x^3+292714x+54762104\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{6}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(-63, 6037\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.93211303552130654193357309234$

Torsion generators

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

\( \left(1224, 46792\right) \)  Toggle raw display

Integral points

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

\( \left(-63, 6037\right) \), \( \left(-63, -5975\right) \), \( \left(102, 9205\right) \), \( \left(102, -9308\right) \), \( \left(234, 11548\right) \), \( \left(234, -11783\right) \), \( \left(561, 19609\right) \), \( \left(561, -20171\right) \), \( \left(1224, 46792\right) \), \( \left(1224, -48017\right) \), \( \left(3369, 196513\right) \), \( \left(3369, -199883\right) \), \( \left(8517, 783385\right) \), \( \left(8517, -791903\right) \), \( \left(24786, 3890764\right) \), \( \left(24786, -3915551\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 14586 \)  =  $2 \cdot 3 \cdot 11 \cdot 13 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-2899504554614368272 $  =  $-1 \cdot 2^{4} \cdot 3^{6} \cdot 11^{6} \cdot 13^{4} \cdot 17^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{2773679829880629422375}{2899504554614368272} \)  =  $2^{-4} \cdot 3^{-6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-6} \cdot 13^{-4} \cdot 17^{-3} \cdot 229^{3} \cdot 1753^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.2293332654650940733062542130\dots$
Stable Faltings height: $2.2293332654650940733062542130\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.93211303552130654193357309234\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.16808338249122457901756708971\dots$
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: $ 864 $  = $ 2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2^{2}\cdot3 $
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) $ ≈ $ 3.7601450849900200823438435907303750000 $

Modular invariants

Modular form 14586.2.a.e

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 304128
$ \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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$11$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$13$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$17$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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
$3$ 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]
 

$p$-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 nonsplit split ss ordinary split split split ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 7 4 1,1 1 2 2 2 1 1,1 1 1 1 1 1 1
$\mu$-invariant(s) 1 0 0,0 0 0 0 0 0 0,0 0 0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 14586h consists of 4 curves linked by isogenies of degrees dividing 6.

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
$2$ \(\Q(\sqrt{-17}) \) \(\Z/2\Z \times \Z/6\Z\) Not in database
$4$ 4.2.74052.4 \(\Z/12\Z\) Not in database
$6$ 6.0.12338352.2 \(\Z/3\Z \times \Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/12\Z\) Not in database
$8$ 8.0.25356622807296.4 \(\Z/2\Z \times \Z/12\Z\) Not in database
$9$ 9.3.280389568253455762358173488.1 \(\Z/18\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) Not in database
$18$ 18.0.6328348786001025039997966997346099142744003920746925678133248.1 \(\Z/2\Z \times \Z/18\Z\) 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.