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sagemathgh-38561: Period lattice: fix documentation error
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That's what the code appear to be doing.

The existing example in `_compute_periods_real` also show it:

```sage
sage:             sage: # needs sage.rings.number_field
....:             sage: x = polygen(ZZ, 'x')
....:             sage: K.<a> = NumberField(x^3 - 2)
....:             sage: E = EllipticCurve([0,1,0,a,a])
....:             sage: embs = K.embeddings(CC)
....:             sage: Ls = [E.period_lattice(e) for e in embs]
....:             sage: [L.is_real() for L in Ls]
....:
[False, False, True]
sage:             sage: Ls[2]._compute_periods_real(100)
....:
(3.8145297721785450936365098936,
 1.9072648860892725468182549468 + 1.3404778596244020196600112394*I)

```
Then do
```sage
sage: ω1, ω2 = _
sage: ω2/ω1
0.50000000000000000000000000000 + 0.35141365769412556448697319411*I
sage: ω1/ω2
1.3387183410189190672591156778 - 0.94088781767934015952275257142*I
```



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- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

### ⌛ Dependencies

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URL: sagemath#38561
Reported by: user202729
Reviewer(s): Kwankyu Lee
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Release Manager committed Aug 26, 2024
2 parents a1e31c0 + 2215319 commit d28c681
Showing 1 changed file with 3 additions and 3 deletions.
6 changes: 3 additions & 3 deletions src/sage/schemes/elliptic_curves/period_lattice.py
Original file line number Diff line number Diff line change
Expand Up @@ -413,7 +413,7 @@ def basis(self, prec=None, algorithm='sage'):
(tuple of Complex) `(\omega_1,\omega_2)` where the lattice is
`\ZZ\omega_1 + \ZZ\omega_2`. If the lattice is real then
`\omega_1` is real and positive, `\Im(\omega_2)>0` and
`\Re(\omega_1/\omega_2)` is either `0` (for rectangular
`\Re(\omega_2/\omega_1)` is either `0` (for rectangular
lattices) or `\frac{1}{2}` (for non-rectangular lattices).
Otherwise, `\omega_1/\omega_2` is in the fundamental region of
the upper half-plane. If the latter normalisation is required
Expand Down Expand Up @@ -498,7 +498,7 @@ def gens(self, prec=None, algorithm='sage'):
(tuple of Complex) `(\omega_1,\omega_2)` where the lattice is
`\ZZ\omega_1 + \ZZ\omega_2`. If the lattice is real then
`\omega_1` is real and positive, `\Im(\omega_2)>0` and
`\Re(\omega_1/\omega_2)` is either `0` (for rectangular
`\Re(\omega_2/\omega_1)` is either `0` (for rectangular
lattices) or `\frac{1}{2}` (for non-rectangular lattices).
Otherwise, `\omega_1/\omega_2` is in the fundamental region of
the upper half-plane. If the latter normalisation is required
Expand Down Expand Up @@ -650,7 +650,7 @@ def _compute_periods_real(self, prec=None, algorithm='sage'):
(tuple of Complex) `(\omega_1,\omega_2)` where the lattice has
the form `\ZZ\omega_1 + \ZZ\omega_2`, `\omega_1` is real and
`\omega_1/\omega_2` has real part either `0` or `frac{1}{2}`.
`\omega_2/\omega_1` has real part either `0` or `frac{1}{2}`.
EXAMPLES::
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