KleinInvariantJ

Klein's absolute invariant $J(\tau) = j(\tau)/1728$, normalised so $J(i) = 1$ and $J(\rho) = 0$ at the elliptic points. A thin wrapper over this package's own ModularJ (the un-normalised $j$).

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageKleinInvariantJ!↗

KleinInvariantJ(tau)$J(\tau) = j(\tau)/1728$.

Typed signatures

Type	Package	Applies
(number) -> number	analytic	always

Domain: Special functions

Details

• All the analytic work — fundamental-domain reduction, the $\eta^{24}$ discriminant — is ModularJ's; this only rescales.
• $\tau = i$ is returned exactly (1): $j(i) = 1728$ is a textbook identity, kept exact even under plain evaluate(), not just N().

Examples

1 more kept as data, checked against the oracles

Basic #

$J(i) = 1$

live ↗

In$N(\mathrm{KleinInvariantJ}(i))$Out$1.0$

Known value · Wolfram 15.0.0 KleinInvariantJ

$J(2i) = 66^3/1728 = 1331/8$

live ↗

In$N(\mathrm{KleinInvariantJ}(2i))$Out$166.375$

Known value · Wolfram 15.0.0 KleinInvariantJ

Scope #

Exact at the elliptic point $i$

live ↗

In$\mathrm{KleinInvariantJ}(i)$Out$1$

Possible issues #

Re(τ) = π·10^22 spans far more than 10^6 periods of T (period 1): τ this far out has no reduction against the real period at double precision, so tauTooFarForDouble (modular.ts, atop periodsExceedDouble/tauTooCloseToRealAxis in ce-patches/support/precise.ts) declines rather than return a wrong value.

live ↗

In$N(\mathrm{KleinInvariantJ}(i+π×10^{22}))$Out$\mathrm{KleinInvariantJ}(3.141\,592\,653\,589\,793\,238\,462\,643\cdot10^{22}+i)$

The true value (mpmath kleinj, dps=30) — still not computed here. Reducing τ's fundamental-domain step at bignum precision before collapsing to a double is deferred; this example only pins the gap, not a fix.

live ↗

Known value · mpmath 1.3.0 kleinj

Neat examples #

Heegner number 163: $J\bigl(\tfrac{1+i\sqrt{163}}2\bigr) = -640320^3/1728 = -151931373056000$; the double $\tau$ already carries a relative error of $2^{-53}$, which $J$ magnifies by $2\pi\,\mathrm{Im}\,\tau \approx 40$, so about 14 digits agree

live ↗

In$N(\mathrm{KleinInvariantJ}(\frac{1}{2}(1+163^{1/2}i)))$Out$-151\,931\,373\,055\,998.97$

About

Examples	7
Checked against	Wolfram Language 5 agree1 differ
Crosswalk	Wolfram Language

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/modular.ts

j(τ)/1728, delegating entirely to ModularJ.

See also: ModularJ, ModularLambda