Properties

Label 23104bz3
Conductor $23104$
Discriminant $-57207791296$
j-invariant \( -\frac{50357871050752}{19} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -1110917, -450312229])
 
gp: E = ellinit([0, -1, 0, -1110917, -450312229])
 
magma: E := EllipticCurve([0, -1, 0, -1110917, -450312229]);
 

\(y^2=x^3-x^2-1110917x-450312229\)  Toggle raw display

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(\frac{860257601401936222}{682211616110761}, \frac{219911539875530199402232911}{17818808681499482606741}\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $41.689614909892388097581266225$

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 23104 \)  =  \(2^{6} \cdot 19^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-57207791296 \)  =  \(-1 \cdot 2^{6} \cdot 19^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{50357871050752}{19} \)  =  \(-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(1.8522322368545106882332246603\dots\)
Stable Faltings height: \(0.033439156991317803520094883627\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(41.689614909892388097581266225\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.073527385454286900217375542445\dots\)
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: \( 2 \)  = \( 1\cdot2 \)
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)

Modular invariants

Modular form 23104.2.a.bs

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 155520
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 6.1306567698408877137827406166205018627 \)

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\((j)_{-}\)
\(2\) \(1\) \(II\) Additive -1 6 6 0
\(19\) \(2\) \(I_1^{*}\) Additive -1 2 7 1

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
\(3\) B

$p$-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 add ordinary ordinary ordinary ordinary ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 1 1 1 1 1 1 - 1,1 3 1 1 1 1 1
$\mu$-invariant(s) - 2 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 and 9.
Its isogeny class 23104bz consists of 3 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
$2$ \(\Q(\sqrt{-114}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.76.1 \(\Z/2\Z\) Not in database
$6$ 6.0.109744.2 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.2.11409864192.2 \(\Z/3\Z\) Not in database
$6$ 6.0.24953372987904.2 \(\Z/9\Z\) Not in database
$6$ 6.0.379275264.2 \(\Z/6\Z\) Not in database
$12$ 12.2.59986716965994496.21 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/9\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.2.95065163511991987809112484216832.1 \(\Z/6\Z\) Not in database
$18$ 18.0.994415187636384118589386726864825557712896.2 \(\Z/18\Z\) 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.