Properties

Label 19600.o1
Conductor 19600
Discriminant -38416000000000
j-invariant \( -\frac{5452947409}{250} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -196408, 33439188]); // or
 
magma: E := EllipticCurve("19600bx2");
 
sage: E = EllipticCurve([0, 1, 0, -196408, 33439188]) # or
 
sage: E = EllipticCurve("19600bx2")
 
gp: E = ellinit([0, 1, 0, -196408, 33439188]) \\ or
 
gp: E = ellinit("19600bx2")
 

\( y^2 = x^{3} + x^{2} - 196408 x + 33439188 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-397, 7000\right) \)\( \left(-82, 7000\right) \)
\(\hat{h}(P)\) ≈  2.406579433241.57791733978

Integral points

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

\((-397,\pm 7000)\), \((-82,\pm 7000)\), \((-68,\pm 6818)\), \((174,\pm 2136)\), \((228,\pm 750)\), \((247,\pm 238)\), \((254,\pm 56)\), \((268,\pm 350)\), \((318,\pm 1800)\), \((478,\pm 7000)\), \((1262,\pm 42392)\), \((2228,\pm 103250)\), \((974318,\pm 961725800)\)

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 19600 \)  =  \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-38416000000000 \)  =  \(-1 \cdot 2^{13} \cdot 5^{9} \cdot 7^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{5452947409}{250} \)  =  \(-1 \cdot 2^{-1} \cdot 5^{-3} \cdot 7^{2} \cdot 13^{3} \cdot 37^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.180588002078\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.609907114066\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 48 \)  = \( 2^{2}\cdot2^{2}\cdot3 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 19600.2.a.o

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

\( q - 2q^{3} + q^{9} - 3q^{11} - 5q^{13} - 6q^{17} + q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 103680
\( \Gamma_0(N) \)-optimal: no
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 5.28681154474 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(4\) \( I_5^{*} \) Additive -1 4 13 1
\(5\) \(4\) \( I_3^{*} \) Additive 1 2 9 3
\(7\) \(3\) \( IV \) Additive 1 2 4 0

Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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

prime Image of Galois representation
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 add ordinary add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 4 - - 2 2 2 4 2 2 2 2 2 2 2
$\mu$-invariant(s) - 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 19600.o consists of 2 curves linked by isogenies of degree 3.

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 \(\Q(\sqrt{15}) \) \(\Z/3\Z\) Not in database
3 3.1.1960.1 \(\Z/2\Z\) Not in database
6 6.0.153664000.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.2.8297856000.14 \(\Z/6\Z\) Not in database
6.0.56010528000.3 \(\Z/3\Z\) 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.