Properties

Label 52371.f2
Conductor 52371
Discriminant -17380328082358131
j-invariant \( -\frac{122023936}{161051} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -49197, -7607417]) # or
 
sage: E = EllipticCurve("52371f2")
 
gp: E = ellinit([0, 0, 1, -49197, -7607417]) \\ or
 
gp: E = ellinit("52371f2")
 
magma: E := EllipticCurve([0, 0, 1, -49197, -7607417]); // or
 
magma: E := EllipticCurve("52371f2");
 

\( y^2 + y = x^{3} - 49197 x - 7607417 \)

Mordell-Weil group structure

Trivial

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 
None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 52371 \)  =  \(3^{2} \cdot 11 \cdot 23^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-17380328082358131 \)  =  \(-1 \cdot 3^{6} \cdot 11^{5} \cdot 23^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{122023936}{161051} \)  =  \(-1 \cdot 2^{12} \cdot 11^{-5} \cdot 31^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.15279484484\)
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: \( 5 \)  = \( 1\cdot5\cdot1 \)
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: \(9\) (exact)

Modular invariants

Modular form 52371.2.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);
 

\( q + 2q^{2} + 2q^{4} + q^{5} + 2q^{7} + 2q^{10} + q^{11} + 4q^{13} + 4q^{14} - 4q^{16} - 2q^{17} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 379500
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 6.8757680178 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(3\) \(1\) \( I_0^{*} \) Additive -1 2 6 0
\(11\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(23\) \(1\) \( I_0^{*} \) 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
\(5\) Cs.4.1

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss add ordinary ordinary split ordinary ordinary ss add ss ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 0,1 - 2 0 1 0 0 0,0 - 0,0 0 0 0 0 0
$\mu$-invariant(s) 0,0 - 1 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 \(d=\) 5.
Its isogeny class 52371.f consists of 3 curves linked by isogenies of degrees dividing 25.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 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 \(\Q(\sqrt{69}) \) \(\Z/5\Z\) 2.2.69.1-121.1-b2
3 3.1.44.1 \(\Z/2\Z\) Not in database
4 4.0.595125.1 \(\Z/5\Z\) Not in database
6 6.0.21296.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.2.635993424.1 \(\Z/10\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.