Height / L-function Decomposition for K(163,163)

Exact rational piece

• E = 163a1
• generator canonical height h(P) = 0.18990923249810324
• exact Heegner image on the elliptic factor: pi_1(z_-163) = -4P
• therefore K_1(163,163) = 16 h(P) = 3.0385477199698

The central derivative on the rational orbit is numerically:

L'(E,1) = 1.0479330218247145

Higher orbits

Untwisted per-embedding derivatives L'(f,1) are available directly in Sage:

[
  {
    "degree": 1,
    "derivatives": [
      1.0479330218247145
    ]
  },
  {
    "degree": 5,
    "derivatives": [
      0.513123266772634,
      0.756909081479706,
      0.7499190474949019,
      1.0559552439998783,
      1.2448928386402103
    ]
  },
  {
    "degree": 7,
    "derivatives": [
      -0.0896080592443103,
      -0.15173393040818722,
      -0.08335216315772423,
      -0.24477209378855294,
      -0.2414176979622005,
      -0.26090460092372497,
      -0.3155987177685356
    ]
  }
]

But the direct twisted route f.twist(chi_-163).lseries().derivative(1) is blocked here:

degree-5: {
  "status": "failed",
  "error": "Timeout after 20s"
}
degree-7: {
  "status": "failed",
  "error": "Timeout after 20s"
}

Heuristic fallback

I also computed a smoothed Dirichlet-series proxy for L'(f, chi_-163, 1) using the twisted q-coefficients and an exponential cutoff. These numbers are exploratory only; they are not a validated approximate functional equation with full conductor/root-number normalization.

degree-5 proxy: {
  "cutoff": 800,
  "analytic_conductor_proxy": 26569,
  "derivative_proxy": [
    36.151221598182694,
    22.298520821005614,
    13.884507946041637,
    7.364224377819543,
    10.943828803927381
  ]
}
degree-7 proxy: {
  "cutoff": 800,
  "analytic_conductor_proxy": 26569,
  "derivative_proxy": [
    11.156121376394214,
    3.3437208436513983,
    8.481988937721761,
    2.441687583118806,
    1.6368149820449602,
    4.149035067213272,
    2.801140305067073
  ]
}

Honest conclusion

• The elliptic-factor contribution K_1(163,163) is exact and equals 3.0385477199698.
• The degree-5 and degree-7 twisted central derivatives are not practically available through the stock Sage high-level API here.
• Without those twisted derivatives and the explicit Gross-Zagier normalization constants for the higher-dimensional factors, the full K(163,163) cannot yet be assembled from this session alone.