$ magma computations/derickx-X1_16_classgroups/order_5.m 
Magma V2.26-10    Fri Jul 26 2024 16:14:55 on Maartens-MacBook-Pro [Seed = 
1688317336]

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Loading file "computations/derickx-X1_16_classgroups/order_5.m"
Loading "elliptic_curve_chabauty.m"
Claim 1 successfully verified
Claim 2: equation for g
(-6*x^2 - 4*x + 2)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)*y + (x^5 + 13*x^4 -
    2*x^3 + 10*x^2 - 7*x + 1)/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)
Claim 3 successfully verified
[ 3 ]
Claim 4 successfully verified
y^5 - 5*y^3 + 5*y + (-2*x^5 - 26*x^4 + 4*x^3 - 20*x^2 + 14*x - 2)/(x^5 - 5*x^4 +
    10*x^3 - 10*x^2 + 5*x - 1)
Hyperelliptic Curve defined by y^2 = x^6 - x^5 + 5*x^3 - x + 1 over Rational 
Field
Claim 5: The equation for Y_{16} in the paper
Hyperelliptic Curve defined by y^2 = x^6 + x^5 - 5*x^3 + x + 1 over Rational 
Field
3/2
Claim 6: equation for h16+1
(2*z^2 - 6*z + 2)/(z^5 - 5*z^4 + 5*z^3 + 5*z^2 - 5*z - 3)*y + (10*z^2 - 10*z - 
    2)/(z^5 - 5*z^4 + 5*z^3 + 5*z^2 - 5*z - 3)
Claim 7 successfully verified
Claim 8: The automorphism group of Y_16:
C2^2
MatrixGroup<2, RationalField() |
    Matrix(RationalField(), 2, 2, [ 0, -1, -1, 0 ]) >
Claim 9: Equation for the elliptic curve E
Elliptic Curve defined by y^2 = x^3 - x^2 + 17*x - 13 over Rational Field
Claim 10 successfully verified
Claim 11: factorisation of z^6 + z^5 - 5z^3 + z + 1
z^2 + (-alpha^2 + alpha - 1)*z + 1
z^4 + (alpha^2 - alpha + 2)*z^3 + (alpha^2 - 3*alpha + 3)*z^2 + (alpha^2 - alpha
    + 2)*z + 1
Starting two cover descent + elliptic curve chabauty
doing hk 1 0
gamma 1
finding points
Time: 9.570
Elliptic curve Elliptic Curve defined by y^2 + 6*x*y + (4*alpha^2 + 10*alpha + 
    10)*y = x^3 + (alpha^2 - 3)*x^2 + (-9*alpha^2 - 32*alpha - 23)*x + 
    (-61*alpha^2 + 6*alpha + 23) over K
Time: 2.640
starting chabauty
13120 8
chabauty result true <1, z^4 + (alpha^2 - alpha + 2)*z^3 + (alpha^2 - 3*alpha + 
    3)*z^2 + (alpha^2 - alpha + 2)*z + 1>
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1) @}
found all points
doing hk 2 $.1
gamma -alpha^2 - 4*alpha - 2
finding points
Time: 9.740
Elliptic curve Elliptic Curve defined by y^2 + 6*x*y + (4*alpha^2 + 10*alpha + 
    10)*y = x^3 + (alpha^2 - 3)*x^2 + (-9*alpha^2 - 32*alpha - 23)*x + 
    (-61*alpha^2 + 6*alpha + 23) over K
Time: 2.610
starting chabauty
13120 8
chabauty result true <-alpha^2 - 4*alpha - 2, z^4 + (alpha^2 - alpha + 2)*z^3 + 
    (alpha^2 - 3*alpha + 3)*z^2 + (alpha^2 - alpha + 2)*z + 1>
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1) @}
found all points
doing hk 3 $.2 + $.3
gamma alpha^2 + 2
finding points
Time: 8.570
Elliptic curve Elliptic Curve defined by y^2 + 6*x*y + (8*alpha^2 + 6*alpha + 
    2)*y = x^3 + (alpha^2 + 4*alpha - 5)*x^2 + (-181*alpha^2 + 262*alpha - 
    521)*x + (-583*alpha^2 + 1030*alpha - 1679) over K
Time: 4.240
starting chabauty
4 8
chabauty result true <alpha^2 + 2, z^4 + (alpha^2 - alpha + 2)*z^3 + (alpha^2 - 
    3*alpha + 3)*z^2 + (alpha^2 - alpha + 2)*z + 1>
{@ (3 : -29 : 1), (3 : 29 : 1), (1/3 : -29/27 : 1), (1/3 : 29/27 : 1) @}
found all points
Claim 12: the rational points on Y_{16}
[ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1), (3 : -29 : 1), (3 : 29 :
1), (1 : -29 : 3), (1 : 29 : 3) ]
Claim 13: table of points where divisibility by 5 could potentially fail
(1 : -1 : 0) -3 -60 -15 2
(1 : 1 : 0) 1 4 1 1
(0 : -1 : 1) 1/3 -20/243 -15 2
(0 : 1 : 1) -1 4 1 1
(3 : -29 : 1) -242/29 -628044748870/20511149 -2030 40
(3 : 29 : 1) Infinity Infinity 1 1
(1 : -29 : 3) 29/242 -75261560815/829997587232 -2030 40
(1 : 29 : 3) 0 0 1 1
Claim 14 successfully verified
> 

Total time: 358.480 [711.267] seconds, Total memory usage: 280.81MB