Properties

Label 14586d1
Conductor $14586$
Discriminant $-2.811\times 10^{12}$
j-invariant \( -\frac{1449073218392281}{2811246362496} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -2358, 91720])
 
gp: E = ellinit([1, 0, 1, -2358, 91720])
 
magma: E := EllipticCurve([1, 0, 1, -2358, 91720]);
 

\(y^2+xy+y=x^3-2358x+91720\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(20, 219\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.28342529960380512758784506454$

Integral points

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

\( \left(-34, 381\right) \), \( \left(-34, -348\right) \), \( \left(20, 219\right) \), \( \left(20, -240\right) \), \( \left(70, 485\right) \), \( \left(70, -556\right) \)  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: $-2811246362496 $  =  $-1 \cdot 2^{7} \cdot 3^{12} \cdot 11 \cdot 13 \cdot 17^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{1449073218392281}{2811246362496} \)  =  $-1 \cdot 2^{-7} \cdot 3^{-12} \cdot 11^{-1} \cdot 13^{-1} \cdot 17^{-2} \cdot 113161^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.0783466935083218265745660358\dots$
Stable Faltings height: $1.0783466935083218265745660358\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.28342529960380512758784506454\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.71810174100427841920394716665\dots$
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: $ 24 $  = $ 1\cdot( 2^{2} \cdot 3 )\cdot1\cdot1\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
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) $ ≈ $ 4.8846768261636404301664336902740633531 $

Modular invariants

Modular form 14586.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 34944
$ \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$(j)_{-}$
$2$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7
$3$ $12$ $I_{12}$ Split multiplicative -1 1 12 12
$11$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$17$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 $\ell$.

$p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type nonsplit split ordinary ordinary nonsplit split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 9 2 3 1 1 2 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 14586d consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.1144.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1497193984.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$8$ Deg 8 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) 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.