Properties

Label 466752.bu2
Conductor $466752$
Discriminant $4.735\times 10^{32}$
j-invariant \( \frac{4474676144192042711273397261697}{1806328356954994499451382272} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -21971478017, -689374449575007])
 
gp: E = ellinit([0, -1, 0, -21971478017, -689374449575007])
 
magma: E := EllipticCurve([0, -1, 0, -21971478017, -689374449575007]);
 

\(y^2=x^3-x^2-21971478017x-689374449575007\)  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);
 

$P$ =  \(\left(-\frac{1671853145480931853157009431905471369532279869359180225581799301232082756093096}{38338734352097138395760660390592998269265332593497914065043810478077205625}, \frac{3235948302608982317495477181088399899361635716522717400877793131229814236012363541350483026894796467299966622778832583}{237386849992953740896302467390752607374365993183027921906272161613516539551776906943295853454165075366885421875}\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $181.28383830254959768364872434$

Torsion generators

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

\( \left(-128937, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-128937, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 466752 \)  =  $2^{6} \cdot 3 \cdot 11 \cdot 13 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $473518140805610078064183154311168 $  =  $2^{27} \cdot 3 \cdot 11 \cdot 13^{3} \cdot 17^{16} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{4474676144192042711273397261697}{1806328356954994499451382272} \)  =  $2^{-9} \cdot 3^{-1} \cdot 11^{-1} \cdot 13^{-3} \cdot 17^{-16} \cdot 10859^{3} \cdot 1517507^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $4.9677181225501196579932679549\dots$
Stable Faltings height: $3.9279973517102016938674197727\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $181.28383830254959768364872434\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.012843213740983095668226583484\dots$
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: $ 4 $  = $ 2\cdot1\cdot1\cdot1\cdot2 $
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) $ ≈ $ 9.3130683324218505046273375053240205848 $

Modular invariants

Modular form 466752.2.a.bu

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);
 

\( q - q^{3} + 2q^{5} + q^{9} + q^{11} - q^{13} - 2q^{15} - q^{17} + 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1815478272
$ \Gamma_0(N) $-optimal: no
Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 466752.bu4 is optimal.

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$(j)_{-}$
$2$ $2$ $I_{17}^{*}$ Additive -1 6 27 9
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$13$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$17$ $2$ $I_{16}$ Non-split multiplicative 1 1 16 16

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 8.12.0.12

$p$-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 and 4.
Its isogeny class 466752.bu consists of 3 curves linked by isogenies of degrees dividing 4.