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7077 bytes, as of the version from 2026-09-23 19:17 (current). Recorded here, not run.
"""Values of the elliptic nome q(m) -- numberdb.org/T425.
This generator fills T425 with q(m) = exp(-pi*K(1-m)/K(m)), where K takes the
elliptic parameter m = k^2. The table uses every rational m = a/b in lowest terms with b <= 12 and
0 < m < 1 -- the arguments a reader is likely to have written down, rather
than the ones base ten makes short. m = 1/2 is the distinguished one, where
q = exp(-pi), and the script checks it.
Run it with SageMath:
$ sage -pip install numberdb # once
$ sage -python generate.py # check the table against this code
$ sage -python generate.py --publish # fill the draft, with NUMBERDB_API_KEY set
Before publishing, the script checks q(1/2) = exp(-pi) and recovers m from the
theta constants theta_2(0,q) and theta_3(0,q) by a q-series with an explicit
geometric tail bound.
"""
import os
import sys
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
TABLE = os.environ.get("NUMBERDB_TABLE", "T425")
DIGITS = 100
WORKING_GUARD = 80
CHECK_GUARD = 256
#: Rationals of small height, not a decimal grid. The parameter is a modulus
#: on (0,1) and nobody arrives holding q(m) for m = 137/1000; the values that
#: turn up are the ones with names -- m = 1/2 above all, where q = exp(-pi) --
#: and otherwise halves, thirds, quarters and their neighbours. Twelve admits
#: every denominator through a twelfth and stops: 45 arguments, each one a
#: number somebody would write.
DENOMINATOR_BOUND = 12
THETA_TERMS = 80
def _key_from_stdin():
if os.environ.get("NUMBERDB_KEY_FROM_STDIN") != "1":
return
token = sys.stdin.read().strip()
if "=" in token and token.split("=", 1)[0].isupper():
token = token.split("=", 1)[1].strip().strip("'\"")
if token:
os.environ["NUMBERDB_API_KEY"] = token
def _complex_field(digits, guard=WORKING_GUARD):
return ComplexBallField(numberdb.bits(digits, losing=guard))
def _real_field(digits, guard=WORKING_GUARD):
return RealBallField(numberdb.bits(digits, losing=guard))
def _nome(m, digits, guard=WORKING_GUARD):
field = _complex_field(digits, guard)
parameter = field(QQ(m))
value = (-field.pi() * (field(1) - parameter).elliptic_k()
/ parameter.elliptic_k()).exp()
if not (value.real().is_finite() and value.imag().is_finite()):
raise ArithmeticError("q(%s) produced a non-finite ball: %s" % (m, value))
if not value.imag().contains_zero():
raise ArithmeticError("q(%s) has nonzero imaginary part: %s" % (m, value.imag()))
real = value.real()
if not (real > 0 and real < 1):
raise ArithmeticError("q(%s) is not contained in (0, 1): %s" % (m, real))
return real
def _theta_parameter(q, terms=THETA_TERMS):
"""Recover m from q by theta q-series, with explicit tail bounds."""
field = q.parent()
q = field(q)
theta3 = field(1)
for n in range(1, terms):
theta3 += 2 * q ** (n * n)
tail3 = 2 * q ** (terms * terms) / (1 - q ** (2 * terms + 1))
theta3 = theta3.add_error(tail3)
theta2_sum = field(0)
for n in range(terms):
theta2_sum += q ** (n * (n + 1))
leading = q.sqrt().sqrt()
tail2 = 2 * leading * q ** (terms * (terms + 1)) / (1 - q ** (2 * terms + 2))
theta2 = (2 * leading * theta2_sum).add_error(tail2)
return (theta2 / theta3) ** 4
class EllipticNomeValues(numberdb.Generator):
"""Generator for T425, the elliptic nome q(m)."""
table = TABLE
parameters = ("m",)
type = "R"
digits = DIGITS
rigour = "proven"
def enumerate(self, bound=DENOMINATOR_BOUND):
#Every a/b in lowest terms with b <= bound and 0 < a/b < 1.
seen = set()
for denominator in range(2, int(bound) + 1):
for numerator in range(1, denominator):
seen.add(QQ(numerator) / QQ(denominator))
for value in sorted(seen):
yield {"m": str(value)}
def value(self, params, digits):
return _nome(params["m"], digits)
def run_integrity_checks():
field = _real_field(DIGITS, CHECK_GUARD)
generator = EllipticNomeValues()
half_seen = False
worst_radius = field(0)
worst_at = None
for params in generator.enumerate():
m = QQ(params["m"])
q = _nome(m, DIGITS, CHECK_GUARD)
theta_m = _theta_parameter(q)
if not (theta_m - field(m)).contains_zero():
raise ArithmeticError(
"theta inversion failed at m=%s: %s vs %s" % (m, theta_m, m))
if m == QQ(1) / QQ(2):
half_seen = True
expected = (-field.pi()).exp()
if not (field(q) - expected).contains_zero():
raise ArithmeticError("q(1/2) did not contain exp(-pi): %s" % q)
radius = field(q.rad())
if radius > worst_radius:
worst_radius = radius
worst_at = m
if not half_seen:
raise ArithmeticError("the m=1/2 control was not enumerated")
print("integrity checks passed")
print("widest value ball radius: %s at m=%s" % (worst_radius, worst_at))
def fill_draft_once(generator, message):
"""Fill a fresh prose draft without the client's empty upsert probe."""
from numberdb._generate import (
_check_precision,
_check_rigour,
_producer,
_run_name,
_source_files,
)
from numberdb._write import Entries, attach, submit_entries, to_text
table = generator.table
run = _run_name(generator)
entries = Entries(*generator.parameters)
for params in generator.enumerate():
params = dict(params)
wanted = generator.digits_for(params)
entry = generator._entry(params, wanted)
value = entry["number"]
identity = ",".join(str(params[name]) for name in generator.parameters)
_check_rigour(generator, table, identity, value)
written = to_text(value, wanted, generator.format)
_check_precision(table, identity, written, wanted, lowering=False)
record = dict(entry)
record.pop("digits", None)
entries.add(**params, **record, digits=wanted)
answer = submit_entries(
table,
entries,
message=message,
produced_by=_producer(generator, os.environ.get("NUMBERDB_ASSISTED_BY", "")),
upsert=False,
run=run,
rigour=generator.rigour,
)
for name, body in sorted(_source_files(generator).items()):
attach(table, name, body, run=run, message=message, rigour=generator.rigour)
return answer
if __name__ == "__main__":
_key_from_stdin()
generator = EllipticNomeValues()
run_integrity_checks()
if os.environ.get("NUMBERDB_PUBLISH") == "1" or "--publish" in sys.argv:
print(fill_draft_once(
generator,
message="elliptic nome values on the m=j/1000 grid"))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)