# Properties

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

# Related objects

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]);

$$y^2=x^3-x^2-21971478017x-689374449575007$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-128937, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-128937, 0\right)$$

## 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);

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

 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.