Properties

Label 10051.c2
Conductor $10051$
Discriminant $-1.015\times 10^{12}$
j-invariant \( -\frac{89915392}{6859} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -4937, 140415])
 
gp: E = ellinit([0, 1, 1, -4937, 140415])
 
magma: E := EllipticCurve([0, 1, 1, -4937, 140415]);
 

\(y^2+y=x^3+x^2-4937x+140415\)  Toggle raw display

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \(\left(15, 264\right)\)  Toggle raw display\(\left(61, 264\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $0.96440831279472612351461600507$$1.0056250084802039116542993047$

Integral points

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

\( \left(-77, 264\right) \), \( \left(-77, -265\right) \), \( \left(15, 264\right) \), \( \left(15, -265\right) \), \( \left(21, 215\right) \), \( \left(21, -216\right) \), \( \left(33, 120\right) \), \( \left(33, -121\right) \), \( \left(61, 264\right) \), \( \left(61, -265\right) \), \( \left(729, 19608\right) \), \( \left(729, -19609\right) \), \( \left(6363, 507575\right) \), \( \left(6363, -507576\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 10051 \)  =  \(19 \cdot 23^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-1015378162651 \)  =  \(-1 \cdot 19^{3} \cdot 23^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{89915392}{6859} \)  =  \(-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.0518801206218378032258486810\dots\)
Stable Faltings height: \(-0.51586698734273704217752773490\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.94442984883344241395105604338\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.86055825181833166799702650140\dots\)
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: \( 4 \)  = \( 1\cdot2^{2} \)
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: \(1\) (rounded)

Modular invariants

Modular form 10051.2.a.c

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 12672
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 3.2509475987086337902439572489875197408 \)

Local data

This elliptic curve is not semistable. There are 2 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\((j)_{-}\)
\(19\) \(1\) \(I_{3}\) Non-split multiplicative 1 1 3 3
\(23\) \(4\) \(I_0^{*}\) Additive -1 2 6 0

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.

prime Image of Galois representation
\(3\) Cs

$p$-adic data

$p$-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 ss ordinary ordinary ordinary ordinary ordinary ordinary nonsplit add ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 4,7 2 2 2 2 2 2 4 - 2 2 2 2 2 2
$\mu$-invariant(s) 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 \(d=\) 3.
Its isogeny class 10051.c consists of 2 curves linked by isogenies of degrees dividing 9.

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
$2$ \(\Q(\sqrt{69}) \) \(\Z/3\Z\) Not in database
$2$ \(\Q(\sqrt{-23}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.76.1 \(\Z/2\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-23})\) \(\Z/3\Z \times \Z/3\Z\) Not in database
$6$ 6.0.109744.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.2.1897467984.1 \(\Z/6\Z\) Not in database
$6$ 6.0.70276592.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.6.30399582070319365645983310107125622658941.2 \(\Z/9\Z\) Not in database
$18$ 18.0.2906171663649067858916667032847.1 \(\Z/9\Z\) 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.