Properties

Label 19600be1
Conductor 19600
Discriminant 3073280000
j-invariant \( 2450 \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -408, 1588]); // or
 
magma: E := EllipticCurve("19600be1");
 
sage: E = EllipticCurve([0, 1, 0, -408, 1588]) # or
 
sage: E = EllipticCurve("19600be1")
 
gp: E = ellinit([0, 1, 0, -408, 1588]) \\ or
 
gp: E = ellinit("19600be1")
 

\( y^2 = x^{3} + x^{2} - 408 x + 1588 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-22, 20\right) \)\( \left(-12, 70\right) \)
\(\hat{h}(P)\) ≈  1.187256663170.248685904005

Integral points

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

\((-22,\pm 20)\), \((-12,\pm 70)\), \((2,\pm 28)\), \((3,\pm 20)\), \((4,\pm 6)\), \((18,\pm 20)\), \((23,\pm 70)\), \((44,\pm 266)\), \((58,\pm 420)\), \((114,\pm 1204)\), \((188,\pm 2570)\), \((282,\pm 4732)\), \((668,\pm 17270)\), \((2298,\pm 110180)\)

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: \(3073280000 \)  =  \(2^{11} \cdot 5^{4} \cdot 7^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( 2450 \)  =  \(2 \cdot 5^{2} \cdot 7^{2}\)
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: \(0.11483524367\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(1.28929879872\)
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: \( 36 \)  = \( 2^{2}\cdot3\cdot3 \)
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.l

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)
 

\( q - 2q^{3} + q^{9} - 4q^{11} - 2q^{13} - 3q^{17} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 9792
\( \Gamma_0(N) \)-optimal: yes
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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 5.3300499017 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_3^{*} \) Additive 1 4 11 0
\(5\) \(3\) \( IV \) Additive -1 2 4 0
\(7\) \(3\) \( IV \) Additive 1 2 4 0

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X5.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 5 & 5 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 0 & 1 \\ 1 & 1 \end{array}\right)$ and has index 2.

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 \( p \) .

$p$-adic data

$p$-adic regulators

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

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

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 ss ordinary ordinary ordinary ss ordinary ordinary ordinary
$\lambda$-invariant(s) - 8 - - 2 2 2 2,2 2 2 2 2,2 2 2 2
$\mu$-invariant(s) - 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 no rational isogenies. Its isogeny class 19600be consists of this curve only.

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
3 3.3.9800.1 \(\Z/2\Z\) Not in database
6 6.6.768320000.1 \(\Z/2\Z \times \Z/2\Z\) 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.