Properties

Label 450840.g6
Conductor $450840$
Discriminant $-8.742\times 10^{23}$
j-invariant \( \frac{27980756504588158}{17683545112935} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, 23212384, 13054135020])
 
gp: E = ellinit([0, -1, 0, 23212384, 13054135020])
 
magma: E := EllipticCurve([0, -1, 0, 23212384, 13054135020]);
 

\(y^2=x^3-x^2+23212384x+13054135020\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(34789, 6551612\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $10.525629655606868700140226812$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-555, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-555, 0\right) \), \((34789,\pm 6551612)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 450840 \)  =  $2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-874163794591910615070720 $  =  $-1 \cdot 2^{11} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17^{14} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{27980756504588158}{17683545112935} \)  =  $2 \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-2} \cdot 17^{-8} \cdot 240959^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.2829809194962946232181642955\dots$
Stable Faltings height: $1.2309893319549033827942675419\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $10.525629655606868700140226812\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.055168283295262817873562795910\dots$
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: $ 8 $  = $ 1\cdot1\cdot1\cdot2\cdot2^{2} $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $4$ = $2^2$ (exact)
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);
 
Special value: $ L'(E,1) $ ≈ $ 4.6454473496123147296096043206164982890 $

Modular invariants

Modular form 450840.2.a.g

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);
 

\( q - q^{3} - q^{5} + q^{9} - 4q^{11} + q^{13} + q^{15} - 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 51904512
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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 3 11 0
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$5$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$17$ $4$ $I_8^{*}$ Additive 1 2 14 8

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 16.24.0.8

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 450840.g consists of 4 curves linked by isogenies of degrees dividing 8.