Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256)
gp: K = bnfinit(y^32 + 18*y^30 + 203*y^28 + 1492*y^26 + 8025*y^24 + 25714*y^22 + 62137*y^20 + 123428*y^18 + 139579*y^16 - 16686*y^14 + 1661*y^12 + 76916*y^10 + 9756*y^8 - 10192*y^6 + 3152*y^4 - 1088*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256)
\( x^{32} + 18 x^{30} + 203 x^{28} + 1492 x^{26} + 8025 x^{24} + 25714 x^{22} + 62137 x^{20} + 123428 x^{18} + \cdots + 256 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1690320352622233436323740015416175733767719950199015079936\)
\(\medspace = 2^{32}\cdot 3^{16}\cdot 17^{16}\cdot 47^{8}\cdot 53^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(61.43\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2\cdot 3^{1/2}17^{1/2}47^{1/2}53^{1/2}\approx 712.856226738604$ | ||
| Ramified primes: |
\(2\), \(3\), \(17\), \(47\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}+\frac{1}{4}a^{7}-\frac{1}{4}a^{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{3}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}+\frac{3}{8}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{3}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{18}-\frac{1}{8}a^{6}$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{7}$, $\frac{1}{16}a^{20}-\frac{1}{16}a^{16}+\frac{1}{16}a^{14}+\frac{1}{16}a^{12}-\frac{3}{16}a^{10}-\frac{1}{2}a^{7}-\frac{3}{16}a^{6}+\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{21}-\frac{1}{16}a^{19}+\frac{1}{32}a^{17}-\frac{1}{32}a^{15}-\frac{1}{32}a^{13}+\frac{3}{32}a^{11}-\frac{1}{4}a^{10}+\frac{3}{16}a^{9}-\frac{1}{4}a^{8}+\frac{5}{32}a^{7}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{22}+\frac{1}{32}a^{18}+\frac{1}{32}a^{16}-\frac{3}{32}a^{14}+\frac{1}{32}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{32}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{23}+\frac{1}{32}a^{19}+\frac{1}{32}a^{17}+\frac{1}{32}a^{15}-\frac{3}{32}a^{13}-\frac{3}{32}a^{9}+\frac{3}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{32}a^{24}-\frac{1}{32}a^{20}+\frac{1}{32}a^{18}-\frac{1}{32}a^{16}-\frac{1}{32}a^{14}+\frac{1}{16}a^{12}+\frac{7}{32}a^{10}+\frac{1}{8}a^{8}-\frac{1}{2}a^{7}+\frac{3}{16}a^{6}+\frac{1}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{25}-\frac{1}{64}a^{24}-\frac{1}{64}a^{23}-\frac{1}{64}a^{21}-\frac{1}{64}a^{20}+\frac{3}{64}a^{18}+\frac{1}{32}a^{17}-\frac{1}{64}a^{16}-\frac{1}{32}a^{15}+\frac{3}{64}a^{14}-\frac{3}{64}a^{13}+\frac{15}{64}a^{11}+\frac{3}{64}a^{10}+\frac{15}{64}a^{9}-\frac{1}{4}a^{8}-\frac{9}{32}a^{7}+\frac{1}{4}a^{6}+\frac{7}{16}a^{5}-\frac{1}{4}a^{4}$, $\frac{1}{64}a^{26}-\frac{1}{64}a^{23}-\frac{1}{64}a^{22}+\frac{1}{64}a^{20}-\frac{1}{64}a^{19}-\frac{1}{64}a^{18}+\frac{3}{64}a^{17}-\frac{1}{64}a^{16}-\frac{1}{64}a^{15}-\frac{3}{32}a^{14}-\frac{5}{64}a^{13}-\frac{1}{64}a^{12}+\frac{1}{8}a^{11}+\frac{3}{16}a^{10}+\frac{11}{64}a^{9}+\frac{7}{32}a^{8}+\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{64}a^{27}-\frac{1}{64}a^{24}-\frac{1}{64}a^{23}-\frac{1}{64}a^{21}-\frac{1}{64}a^{20}+\frac{3}{64}a^{19}+\frac{3}{64}a^{18}-\frac{3}{64}a^{17}-\frac{1}{64}a^{16}-\frac{1}{16}a^{15}-\frac{5}{64}a^{14}+\frac{1}{64}a^{13}-\frac{1}{8}a^{12}-\frac{5}{32}a^{11}+\frac{11}{64}a^{10}+\frac{1}{32}a^{9}-\frac{1}{8}a^{8}-\frac{7}{32}a^{7}-\frac{3}{8}a^{6}+\frac{1}{16}a^{5}+\frac{3}{8}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{486198592}a^{28}+\frac{2666321}{486198592}a^{26}-\frac{1210911}{243099296}a^{24}-\frac{74143}{60774824}a^{22}+\frac{9509635}{486198592}a^{20}-\frac{14834923}{486198592}a^{18}+\frac{2369111}{121549648}a^{16}+\frac{366559}{15193706}a^{14}-\frac{30481301}{486198592}a^{12}-\frac{1}{4}a^{11}+\frac{73989913}{486198592}a^{10}-\frac{1}{4}a^{9}+\frac{3999877}{243099296}a^{8}-\frac{1}{2}a^{7}-\frac{3323817}{121549648}a^{6}-\frac{1}{4}a^{5}+\frac{1008895}{7596853}a^{4}-\frac{1}{4}a^{3}+\frac{694679}{15193706}a^{2}-\frac{1}{2}a+\frac{2915831}{7596853}$, $\frac{1}{486198592}a^{29}+\frac{2666321}{486198592}a^{27}-\frac{1210911}{243099296}a^{25}-\frac{74143}{60774824}a^{23}-\frac{5684071}{486198592}a^{21}+\frac{15552489}{486198592}a^{19}-\frac{2858631}{243099296}a^{17}+\frac{13461797}{243099296}a^{15}-\frac{15287595}{486198592}a^{13}-\frac{93140853}{486198592}a^{11}-\frac{1}{4}a^{10}-\frac{41581241}{243099296}a^{9}-\frac{1}{4}a^{8}-\frac{44631899}{243099296}a^{7}-\frac{6648239}{121549648}a^{5}-\frac{1}{4}a^{4}-\frac{20011843}{60774824}a^{3}-\frac{1}{4}a^{2}+\frac{2915831}{7596853}a$, $\frac{1}{18\!\cdots\!12}a^{30}+\frac{88\!\cdots\!57}{91\!\cdots\!56}a^{28}+\frac{12\!\cdots\!65}{18\!\cdots\!12}a^{26}+\frac{76\!\cdots\!17}{91\!\cdots\!56}a^{24}+\frac{16\!\cdots\!23}{18\!\cdots\!12}a^{22}-\frac{11\!\cdots\!23}{91\!\cdots\!56}a^{20}-\frac{48\!\cdots\!59}{18\!\cdots\!12}a^{18}-\frac{20\!\cdots\!29}{45\!\cdots\!28}a^{16}-\frac{13\!\cdots\!63}{18\!\cdots\!12}a^{14}-\frac{32\!\cdots\!19}{45\!\cdots\!28}a^{12}-\frac{1}{4}a^{11}-\frac{24\!\cdots\!13}{18\!\cdots\!12}a^{10}-\frac{10\!\cdots\!99}{45\!\cdots\!28}a^{8}-\frac{1}{2}a^{7}+\frac{45\!\cdots\!62}{28\!\cdots\!83}a^{6}-\frac{1}{4}a^{5}-\frac{54\!\cdots\!55}{22\!\cdots\!64}a^{4}-\frac{80\!\cdots\!60}{28\!\cdots\!83}a^{2}-\frac{1}{2}a+\frac{77\!\cdots\!06}{28\!\cdots\!83}$, $\frac{1}{36\!\cdots\!24}a^{31}+\frac{88\!\cdots\!57}{18\!\cdots\!12}a^{29}+\frac{12\!\cdots\!65}{36\!\cdots\!24}a^{27}+\frac{76\!\cdots\!17}{18\!\cdots\!12}a^{25}-\frac{41\!\cdots\!43}{36\!\cdots\!24}a^{23}-\frac{11\!\cdots\!23}{18\!\cdots\!12}a^{21}+\frac{12\!\cdots\!39}{36\!\cdots\!24}a^{19}-\frac{70\!\cdots\!41}{18\!\cdots\!12}a^{17}+\frac{15\!\cdots\!35}{36\!\cdots\!24}a^{15}-\frac{1}{8}a^{14}+\frac{13\!\cdots\!43}{18\!\cdots\!12}a^{13}-\frac{1}{8}a^{12}-\frac{71\!\cdots\!05}{36\!\cdots\!24}a^{11}-\frac{1}{8}a^{10}-\frac{24\!\cdots\!81}{18\!\cdots\!12}a^{9}-\frac{1}{8}a^{8}-\frac{24\!\cdots\!21}{57\!\cdots\!66}a^{7}+\frac{3}{8}a^{6}-\frac{11\!\cdots\!21}{45\!\cdots\!28}a^{5}+\frac{3}{8}a^{4}-\frac{28\!\cdots\!23}{22\!\cdots\!64}a^{3}-\frac{1}{2}a^{2}+\frac{38\!\cdots\!53}{28\!\cdots\!83}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{12}\times C_{84}$, which has order $1008$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $1008$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{66147952685957028463930265}{33294046697832198035335648384} a^{31} + \frac{1638059983290991399933825795}{45779314209519272298586516528} a^{29} + \frac{147873535708918789056791944591}{366234513676154178388692132224} a^{27} + \frac{135964957459492740191956329699}{45779314209519272298586516528} a^{25} + \frac{136212621290181787126026033489}{8517081713398934381132375168} a^{23} + \frac{2352588821035757530555543037685}{45779314209519272298586516528} a^{21} + \frac{45681046909331189019838897330877}{366234513676154178388692132224} a^{19} + \frac{22790166376009496637468600383101}{91558628419038544597173033056} a^{17} + \frac{104652200625641851732665246406629}{366234513676154178388692132224} a^{15} - \frac{1681166797212301036725582454131}{91558628419038544597173033056} a^{13} + \frac{6149689042324124287316029195977}{366234513676154178388692132224} a^{11} + \frac{13967800387680419096701034614361}{91558628419038544597173033056} a^{9} + \frac{2174919967633832817223478894117}{91558628419038544597173033056} a^{7} - \frac{41796435504930732719372277949}{2861207138094954518661657283} a^{5} + \frac{156592169744286079346987521967}{22889657104759636149293258264} a^{3} - \frac{13202946218497571654706065843}{5722414276189909037323314566} a \)
(order $12$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{66\!\cdots\!65}{33\!\cdots\!84}a^{31}+\frac{16\!\cdots\!95}{45\!\cdots\!28}a^{29}+\frac{14\!\cdots\!91}{36\!\cdots\!24}a^{27}+\frac{13\!\cdots\!99}{45\!\cdots\!28}a^{25}+\frac{13\!\cdots\!89}{85\!\cdots\!68}a^{23}+\frac{23\!\cdots\!85}{45\!\cdots\!28}a^{21}+\frac{45\!\cdots\!77}{36\!\cdots\!24}a^{19}+\frac{22\!\cdots\!01}{91\!\cdots\!56}a^{17}+\frac{10\!\cdots\!29}{36\!\cdots\!24}a^{15}-\frac{16\!\cdots\!31}{91\!\cdots\!56}a^{13}+\frac{61\!\cdots\!77}{36\!\cdots\!24}a^{11}+\frac{13\!\cdots\!61}{91\!\cdots\!56}a^{9}+\frac{21\!\cdots\!17}{91\!\cdots\!56}a^{7}-\frac{41\!\cdots\!49}{28\!\cdots\!83}a^{5}+\frac{15\!\cdots\!67}{22\!\cdots\!64}a^{3}-\frac{13\!\cdots\!43}{57\!\cdots\!66}a+1$, $\frac{17\!\cdots\!91}{35\!\cdots\!08}a^{31}-\frac{28\!\cdots\!21}{18\!\cdots\!12}a^{30}+\frac{19\!\cdots\!41}{22\!\cdots\!88}a^{29}-\frac{52\!\cdots\!55}{18\!\cdots\!12}a^{28}+\frac{36\!\cdots\!95}{35\!\cdots\!08}a^{27}-\frac{29\!\cdots\!57}{91\!\cdots\!56}a^{26}+\frac{12\!\cdots\!31}{16\!\cdots\!64}a^{25}-\frac{43\!\cdots\!53}{18\!\cdots\!12}a^{24}+\frac{14\!\cdots\!49}{35\!\cdots\!08}a^{23}-\frac{21\!\cdots\!83}{16\!\cdots\!92}a^{22}+\frac{24\!\cdots\!73}{17\!\cdots\!04}a^{21}-\frac{38\!\cdots\!21}{91\!\cdots\!56}a^{20}+\frac{11\!\cdots\!69}{35\!\cdots\!08}a^{19}-\frac{19\!\cdots\!25}{18\!\cdots\!12}a^{18}+\frac{30\!\cdots\!25}{44\!\cdots\!76}a^{17}-\frac{38\!\cdots\!99}{18\!\cdots\!12}a^{16}+\frac{29\!\cdots\!69}{35\!\cdots\!08}a^{15}-\frac{11\!\cdots\!39}{45\!\cdots\!28}a^{14}+\frac{19\!\cdots\!75}{17\!\cdots\!04}a^{13}-\frac{26\!\cdots\!09}{18\!\cdots\!12}a^{12}+\frac{20\!\cdots\!31}{35\!\cdots\!08}a^{11}+\frac{82\!\cdots\!75}{18\!\cdots\!12}a^{10}+\frac{16\!\cdots\!09}{41\!\cdots\!28}a^{9}-\frac{50\!\cdots\!21}{45\!\cdots\!28}a^{8}+\frac{11\!\cdots\!13}{88\!\cdots\!52}a^{7}-\frac{82\!\cdots\!65}{22\!\cdots\!64}a^{6}-\frac{19\!\cdots\!69}{11\!\cdots\!44}a^{5}+\frac{35\!\cdots\!71}{57\!\cdots\!66}a^{4}+\frac{12\!\cdots\!13}{11\!\cdots\!44}a^{3}-\frac{44\!\cdots\!09}{11\!\cdots\!32}a^{2}-\frac{97\!\cdots\!77}{27\!\cdots\!61}a+\frac{43\!\cdots\!71}{28\!\cdots\!83}$, $\frac{55\!\cdots\!05}{33\!\cdots\!84}a^{31}+\frac{16\!\cdots\!93}{18\!\cdots\!12}a^{29}+\frac{50\!\cdots\!37}{36\!\cdots\!24}a^{27}+\frac{12\!\cdots\!61}{91\!\cdots\!56}a^{25}+\frac{35\!\cdots\!75}{36\!\cdots\!24}a^{23}+\frac{86\!\cdots\!99}{18\!\cdots\!12}a^{21}+\frac{49\!\cdots\!03}{36\!\cdots\!24}a^{19}+\frac{14\!\cdots\!61}{45\!\cdots\!28}a^{17}+\frac{20\!\cdots\!89}{36\!\cdots\!24}a^{15}+\frac{84\!\cdots\!43}{18\!\cdots\!12}a^{13}-\frac{15\!\cdots\!89}{36\!\cdots\!24}a^{11}+\frac{30\!\cdots\!81}{91\!\cdots\!56}a^{9}+\frac{20\!\cdots\!23}{91\!\cdots\!56}a^{7}-\frac{96\!\cdots\!39}{11\!\cdots\!32}a^{5}+\frac{21\!\cdots\!53}{22\!\cdots\!64}a^{3}-\frac{47\!\cdots\!35}{57\!\cdots\!66}a$, $\frac{27\!\cdots\!01}{18\!\cdots\!12}a^{31}+\frac{15\!\cdots\!83}{16\!\cdots\!92}a^{30}+\frac{51\!\cdots\!07}{18\!\cdots\!12}a^{29}+\frac{15\!\cdots\!53}{91\!\cdots\!56}a^{28}+\frac{59\!\cdots\!23}{18\!\cdots\!12}a^{27}+\frac{17\!\cdots\!79}{91\!\cdots\!56}a^{26}+\frac{28\!\cdots\!91}{11\!\cdots\!32}a^{25}+\frac{25\!\cdots\!05}{18\!\cdots\!12}a^{24}+\frac{24\!\cdots\!59}{18\!\cdots\!12}a^{23}+\frac{86\!\cdots\!31}{11\!\cdots\!32}a^{22}+\frac{53\!\cdots\!81}{11\!\cdots\!32}a^{21}+\frac{22\!\cdots\!07}{91\!\cdots\!56}a^{20}+\frac{11\!\cdots\!75}{91\!\cdots\!56}a^{19}+\frac{11\!\cdots\!49}{18\!\cdots\!12}a^{18}+\frac{23\!\cdots\!93}{91\!\cdots\!56}a^{17}+\frac{55\!\cdots\!41}{45\!\cdots\!28}a^{16}+\frac{28\!\cdots\!71}{83\!\cdots\!96}a^{15}+\frac{33\!\cdots\!99}{22\!\cdots\!64}a^{14}+\frac{27\!\cdots\!93}{18\!\cdots\!12}a^{13}+\frac{12\!\cdots\!01}{18\!\cdots\!12}a^{12}+\frac{55\!\cdots\!89}{91\!\cdots\!56}a^{11}+\frac{31\!\cdots\!83}{91\!\cdots\!56}a^{10}+\frac{25\!\cdots\!93}{18\!\cdots\!12}a^{9}+\frac{85\!\cdots\!61}{11\!\cdots\!32}a^{8}+\frac{72\!\cdots\!45}{91\!\cdots\!56}a^{7}+\frac{13\!\cdots\!25}{45\!\cdots\!28}a^{6}+\frac{58\!\cdots\!05}{41\!\cdots\!48}a^{5}-\frac{12\!\cdots\!09}{57\!\cdots\!66}a^{4}+\frac{33\!\cdots\!69}{52\!\cdots\!06}a^{3}+\frac{43\!\cdots\!39}{11\!\cdots\!32}a^{2}+\frac{38\!\cdots\!11}{28\!\cdots\!83}a-\frac{40\!\cdots\!78}{28\!\cdots\!83}$, $\frac{66\!\cdots\!81}{36\!\cdots\!24}a^{31}+\frac{13\!\cdots\!63}{83\!\cdots\!96}a^{30}+\frac{75\!\cdots\!53}{22\!\cdots\!64}a^{29}+\frac{12\!\cdots\!65}{41\!\cdots\!48}a^{28}+\frac{13\!\cdots\!27}{36\!\cdots\!24}a^{27}+\frac{67\!\cdots\!23}{20\!\cdots\!24}a^{26}+\frac{50\!\cdots\!95}{18\!\cdots\!12}a^{25}+\frac{19\!\cdots\!87}{83\!\cdots\!96}a^{24}+\frac{55\!\cdots\!13}{36\!\cdots\!24}a^{23}+\frac{32\!\cdots\!53}{26\!\cdots\!53}a^{22}+\frac{11\!\cdots\!59}{22\!\cdots\!64}a^{21}+\frac{99\!\cdots\!28}{26\!\cdots\!53}a^{20}+\frac{44\!\cdots\!41}{36\!\cdots\!24}a^{19}+\frac{74\!\cdots\!67}{83\!\cdots\!96}a^{18}+\frac{45\!\cdots\!03}{18\!\cdots\!12}a^{17}+\frac{36\!\cdots\!47}{20\!\cdots\!24}a^{16}+\frac{99\!\cdots\!33}{33\!\cdots\!84}a^{15}+\frac{16\!\cdots\!21}{96\!\cdots\!36}a^{14}+\frac{38\!\cdots\!03}{22\!\cdots\!64}a^{13}-\frac{64\!\cdots\!09}{83\!\cdots\!96}a^{12}-\frac{58\!\cdots\!43}{36\!\cdots\!24}a^{11}+\frac{18\!\cdots\!13}{41\!\cdots\!48}a^{10}+\frac{28\!\cdots\!71}{18\!\cdots\!12}a^{9}+\frac{45\!\cdots\!69}{41\!\cdots\!48}a^{8}+\frac{52\!\cdots\!31}{10\!\cdots\!96}a^{7}-\frac{24\!\cdots\!21}{20\!\cdots\!24}a^{6}-\frac{11\!\cdots\!87}{41\!\cdots\!48}a^{5}-\frac{24\!\cdots\!03}{20\!\cdots\!24}a^{4}+\frac{19\!\cdots\!61}{20\!\cdots\!24}a^{3}+\frac{21\!\cdots\!64}{26\!\cdots\!53}a^{2}-\frac{19\!\cdots\!04}{28\!\cdots\!83}a-\frac{31\!\cdots\!86}{26\!\cdots\!53}$, $\frac{11\!\cdots\!97}{36\!\cdots\!24}a^{31}-\frac{71\!\cdots\!79}{18\!\cdots\!12}a^{30}+\frac{11\!\cdots\!39}{21\!\cdots\!92}a^{29}-\frac{68\!\cdots\!19}{91\!\cdots\!56}a^{28}+\frac{22\!\cdots\!73}{36\!\cdots\!24}a^{27}-\frac{19\!\cdots\!75}{22\!\cdots\!64}a^{26}+\frac{42\!\cdots\!97}{91\!\cdots\!56}a^{25}-\frac{55\!\cdots\!01}{83\!\cdots\!96}a^{24}+\frac{90\!\cdots\!67}{36\!\cdots\!24}a^{23}-\frac{34\!\cdots\!89}{91\!\cdots\!56}a^{22}+\frac{91\!\cdots\!09}{11\!\cdots\!32}a^{21}-\frac{24\!\cdots\!63}{18\!\cdots\!12}a^{20}+\frac{71\!\cdots\!17}{36\!\cdots\!24}a^{19}-\frac{15\!\cdots\!23}{45\!\cdots\!28}a^{18}+\frac{71\!\cdots\!47}{18\!\cdots\!12}a^{17}-\frac{12\!\cdots\!61}{18\!\cdots\!12}a^{16}+\frac{16\!\cdots\!37}{36\!\cdots\!24}a^{15}-\frac{17\!\cdots\!99}{18\!\cdots\!12}a^{14}-\frac{38\!\cdots\!81}{18\!\cdots\!12}a^{13}-\frac{49\!\cdots\!25}{18\!\cdots\!12}a^{12}-\frac{40\!\cdots\!33}{36\!\cdots\!24}a^{11}+\frac{49\!\cdots\!43}{18\!\cdots\!12}a^{10}+\frac{45\!\cdots\!81}{18\!\cdots\!12}a^{9}-\frac{26\!\cdots\!67}{45\!\cdots\!28}a^{8}+\frac{43\!\cdots\!61}{91\!\cdots\!56}a^{7}-\frac{11\!\cdots\!23}{57\!\cdots\!66}a^{6}-\frac{38\!\cdots\!21}{11\!\cdots\!32}a^{5}+\frac{39\!\cdots\!69}{28\!\cdots\!83}a^{4}+\frac{27\!\cdots\!57}{57\!\cdots\!66}a^{3}-\frac{67\!\cdots\!53}{11\!\cdots\!32}a^{2}+\frac{38\!\cdots\!59}{28\!\cdots\!83}a+\frac{14\!\cdots\!70}{28\!\cdots\!83}$, $\frac{12\!\cdots\!45}{18\!\cdots\!12}a^{30}+\frac{26\!\cdots\!35}{18\!\cdots\!12}a^{28}+\frac{41\!\cdots\!43}{22\!\cdots\!64}a^{26}+\frac{68\!\cdots\!17}{45\!\cdots\!28}a^{24}+\frac{16\!\cdots\!79}{18\!\cdots\!12}a^{22}+\frac{59\!\cdots\!65}{16\!\cdots\!92}a^{20}+\frac{86\!\cdots\!65}{91\!\cdots\!56}a^{18}+\frac{36\!\cdots\!03}{20\!\cdots\!24}a^{16}+\frac{32\!\cdots\!07}{18\!\cdots\!12}a^{14}-\frac{43\!\cdots\!53}{18\!\cdots\!12}a^{12}-\frac{25\!\cdots\!71}{20\!\cdots\!24}a^{10}-\frac{36\!\cdots\!51}{22\!\cdots\!64}a^{8}-\frac{53\!\cdots\!61}{11\!\cdots\!32}a^{6}+\frac{15\!\cdots\!05}{11\!\cdots\!32}a^{4}+\frac{44\!\cdots\!91}{57\!\cdots\!66}a^{2}-\frac{13\!\cdots\!97}{66\!\cdots\!81}$, $\frac{18\!\cdots\!83}{91\!\cdots\!56}a^{31}-\frac{18\!\cdots\!27}{18\!\cdots\!12}a^{30}+\frac{67\!\cdots\!35}{18\!\cdots\!12}a^{29}-\frac{53\!\cdots\!89}{28\!\cdots\!83}a^{28}+\frac{76\!\cdots\!57}{18\!\cdots\!12}a^{27}-\frac{39\!\cdots\!09}{18\!\cdots\!12}a^{26}+\frac{57\!\cdots\!15}{18\!\cdots\!12}a^{25}-\frac{29\!\cdots\!31}{18\!\cdots\!12}a^{24}+\frac{31\!\cdots\!49}{18\!\cdots\!12}a^{23}-\frac{15\!\cdots\!73}{18\!\cdots\!12}a^{22}+\frac{58\!\cdots\!95}{10\!\cdots\!12}a^{21}-\frac{47\!\cdots\!71}{16\!\cdots\!92}a^{20}+\frac{25\!\cdots\!97}{18\!\cdots\!12}a^{19}-\frac{64\!\cdots\!23}{91\!\cdots\!56}a^{18}+\frac{23\!\cdots\!55}{83\!\cdots\!96}a^{17}-\frac{23\!\cdots\!67}{16\!\cdots\!92}a^{16}+\frac{10\!\cdots\!09}{28\!\cdots\!83}a^{15}-\frac{49\!\cdots\!95}{28\!\cdots\!83}a^{14}+\frac{47\!\cdots\!23}{91\!\cdots\!56}a^{13}-\frac{17\!\cdots\!27}{91\!\cdots\!56}a^{12}+\frac{22\!\cdots\!51}{20\!\cdots\!24}a^{11}-\frac{33\!\cdots\!27}{10\!\cdots\!12}a^{10}+\frac{28\!\cdots\!49}{18\!\cdots\!12}a^{9}-\frac{44\!\cdots\!21}{57\!\cdots\!66}a^{8}+\frac{54\!\cdots\!89}{91\!\cdots\!56}a^{7}-\frac{30\!\cdots\!89}{11\!\cdots\!32}a^{6}-\frac{34\!\cdots\!29}{45\!\cdots\!28}a^{5}+\frac{12\!\cdots\!79}{22\!\cdots\!64}a^{4}+\frac{93\!\cdots\!44}{28\!\cdots\!83}a^{3}-\frac{37\!\cdots\!95}{28\!\cdots\!83}a^{2}-\frac{92\!\cdots\!57}{57\!\cdots\!66}a+\frac{27\!\cdots\!17}{28\!\cdots\!83}$, $\frac{33\!\cdots\!75}{41\!\cdots\!48}a^{31}+\frac{16\!\cdots\!71}{22\!\cdots\!64}a^{30}+\frac{13\!\cdots\!97}{91\!\cdots\!56}a^{29}+\frac{61\!\cdots\!41}{45\!\cdots\!28}a^{28}+\frac{15\!\cdots\!25}{91\!\cdots\!56}a^{27}+\frac{28\!\cdots\!85}{18\!\cdots\!12}a^{26}+\frac{23\!\cdots\!09}{18\!\cdots\!12}a^{25}+\frac{21\!\cdots\!91}{18\!\cdots\!12}a^{24}+\frac{65\!\cdots\!69}{91\!\cdots\!56}a^{23}+\frac{11\!\cdots\!95}{18\!\cdots\!12}a^{22}+\frac{44\!\cdots\!05}{18\!\cdots\!12}a^{21}+\frac{24\!\cdots\!75}{11\!\cdots\!32}a^{20}+\frac{11\!\cdots\!81}{18\!\cdots\!12}a^{19}+\frac{48\!\cdots\!67}{91\!\cdots\!56}a^{18}+\frac{23\!\cdots\!85}{18\!\cdots\!12}a^{17}+\frac{10\!\cdots\!07}{91\!\cdots\!56}a^{16}+\frac{31\!\cdots\!49}{18\!\cdots\!12}a^{15}+\frac{25\!\cdots\!89}{18\!\cdots\!12}a^{14}+\frac{58\!\cdots\!75}{91\!\cdots\!56}a^{13}+\frac{65\!\cdots\!91}{18\!\cdots\!12}a^{12}+\frac{36\!\cdots\!89}{18\!\cdots\!12}a^{11}+\frac{18\!\cdots\!27}{18\!\cdots\!12}a^{10}+\frac{62\!\cdots\!05}{91\!\cdots\!56}a^{9}+\frac{53\!\cdots\!21}{91\!\cdots\!56}a^{8}+\frac{37\!\cdots\!47}{91\!\cdots\!56}a^{7}+\frac{12\!\cdots\!09}{45\!\cdots\!28}a^{6}+\frac{31\!\cdots\!55}{45\!\cdots\!28}a^{5}+\frac{13\!\cdots\!49}{11\!\cdots\!32}a^{4}+\frac{67\!\cdots\!97}{22\!\cdots\!64}a^{3}+\frac{21\!\cdots\!79}{11\!\cdots\!32}a^{2}+\frac{46\!\cdots\!53}{57\!\cdots\!66}a-\frac{88\!\cdots\!53}{28\!\cdots\!83}$, $\frac{11\!\cdots\!03}{85\!\cdots\!68}a^{31}-\frac{76\!\cdots\!53}{45\!\cdots\!28}a^{30}+\frac{10\!\cdots\!63}{42\!\cdots\!84}a^{29}-\frac{69\!\cdots\!09}{22\!\cdots\!64}a^{28}+\frac{24\!\cdots\!07}{85\!\cdots\!68}a^{27}-\frac{62\!\cdots\!07}{18\!\cdots\!12}a^{26}+\frac{94\!\cdots\!79}{42\!\cdots\!84}a^{25}-\frac{58\!\cdots\!15}{22\!\cdots\!64}a^{24}+\frac{10\!\cdots\!37}{85\!\cdots\!68}a^{23}-\frac{25\!\cdots\!13}{18\!\cdots\!12}a^{22}+\frac{17\!\cdots\!09}{42\!\cdots\!84}a^{21}-\frac{81\!\cdots\!59}{18\!\cdots\!12}a^{20}+\frac{90\!\cdots\!67}{85\!\cdots\!68}a^{19}-\frac{19\!\cdots\!89}{18\!\cdots\!12}a^{18}+\frac{23\!\cdots\!27}{10\!\cdots\!96}a^{17}-\frac{39\!\cdots\!65}{18\!\cdots\!12}a^{16}+\frac{24\!\cdots\!95}{85\!\cdots\!68}a^{15}-\frac{22\!\cdots\!79}{91\!\cdots\!56}a^{14}+\frac{84\!\cdots\!05}{10\!\cdots\!96}a^{13}+\frac{28\!\cdots\!99}{18\!\cdots\!12}a^{12}-\frac{33\!\cdots\!03}{85\!\cdots\!68}a^{11}+\frac{31\!\cdots\!25}{22\!\cdots\!64}a^{10}+\frac{21\!\cdots\!49}{21\!\cdots\!92}a^{9}-\frac{12\!\cdots\!83}{91\!\cdots\!56}a^{8}+\frac{18\!\cdots\!69}{21\!\cdots\!92}a^{7}-\frac{14\!\cdots\!05}{45\!\cdots\!28}a^{6}-\frac{10\!\cdots\!48}{66\!\cdots\!81}a^{5}+\frac{61\!\cdots\!73}{22\!\cdots\!64}a^{4}-\frac{52\!\cdots\!43}{53\!\cdots\!48}a^{3}-\frac{10\!\cdots\!62}{28\!\cdots\!83}a^{2}+\frac{66\!\cdots\!77}{13\!\cdots\!62}a-\frac{42\!\cdots\!71}{28\!\cdots\!83}$, $\frac{13\!\cdots\!31}{36\!\cdots\!24}a^{31}+\frac{77\!\cdots\!13}{45\!\cdots\!28}a^{30}+\frac{10\!\cdots\!41}{16\!\cdots\!92}a^{29}+\frac{51\!\cdots\!61}{18\!\cdots\!12}a^{28}+\frac{23\!\cdots\!87}{33\!\cdots\!84}a^{27}+\frac{34\!\cdots\!77}{11\!\cdots\!32}a^{26}+\frac{93\!\cdots\!47}{18\!\cdots\!12}a^{25}+\frac{37\!\cdots\!29}{18\!\cdots\!12}a^{24}+\frac{98\!\cdots\!55}{36\!\cdots\!24}a^{23}+\frac{18\!\cdots\!15}{18\!\cdots\!12}a^{22}+\frac{37\!\cdots\!11}{45\!\cdots\!28}a^{21}+\frac{47\!\cdots\!37}{18\!\cdots\!12}a^{20}+\frac{16\!\cdots\!43}{85\!\cdots\!68}a^{19}+\frac{22\!\cdots\!49}{42\!\cdots\!84}a^{18}+\frac{67\!\cdots\!31}{18\!\cdots\!12}a^{17}+\frac{84\!\cdots\!79}{91\!\cdots\!56}a^{16}+\frac{13\!\cdots\!99}{36\!\cdots\!24}a^{15}+\frac{31\!\cdots\!15}{18\!\cdots\!12}a^{14}-\frac{38\!\cdots\!75}{16\!\cdots\!92}a^{13}-\frac{19\!\cdots\!09}{91\!\cdots\!56}a^{12}-\frac{51\!\cdots\!91}{36\!\cdots\!24}a^{11}+\frac{48\!\cdots\!95}{22\!\cdots\!64}a^{10}+\frac{21\!\cdots\!73}{83\!\cdots\!96}a^{9}+\frac{88\!\cdots\!67}{91\!\cdots\!56}a^{8}-\frac{70\!\cdots\!95}{11\!\cdots\!32}a^{7}-\frac{54\!\cdots\!27}{41\!\cdots\!48}a^{6}-\frac{32\!\cdots\!37}{45\!\cdots\!28}a^{5}+\frac{33\!\cdots\!43}{22\!\cdots\!64}a^{4}+\frac{12\!\cdots\!13}{57\!\cdots\!66}a^{3}+\frac{11\!\cdots\!45}{11\!\cdots\!32}a^{2}-\frac{15\!\cdots\!12}{28\!\cdots\!83}a-\frac{55\!\cdots\!06}{28\!\cdots\!83}$, $\frac{21\!\cdots\!05}{45\!\cdots\!28}a^{31}+\frac{39\!\cdots\!35}{45\!\cdots\!28}a^{30}+\frac{38\!\cdots\!01}{45\!\cdots\!28}a^{29}+\frac{29\!\cdots\!45}{18\!\cdots\!12}a^{28}+\frac{17\!\cdots\!79}{18\!\cdots\!12}a^{27}+\frac{33\!\cdots\!61}{18\!\cdots\!12}a^{26}+\frac{62\!\cdots\!41}{91\!\cdots\!56}a^{25}+\frac{25\!\cdots\!75}{18\!\cdots\!12}a^{24}+\frac{14\!\cdots\!71}{38\!\cdots\!44}a^{23}+\frac{35\!\cdots\!85}{45\!\cdots\!28}a^{22}+\frac{21\!\cdots\!37}{18\!\cdots\!12}a^{21}+\frac{27\!\cdots\!55}{10\!\cdots\!96}a^{20}+\frac{50\!\cdots\!89}{18\!\cdots\!12}a^{19}+\frac{59\!\cdots\!15}{91\!\cdots\!56}a^{18}+\frac{99\!\cdots\!13}{18\!\cdots\!12}a^{17}+\frac{24\!\cdots\!85}{18\!\cdots\!12}a^{16}+\frac{53\!\cdots\!93}{91\!\cdots\!56}a^{15}+\frac{31\!\cdots\!89}{18\!\cdots\!12}a^{14}-\frac{31\!\cdots\!41}{18\!\cdots\!12}a^{13}+\frac{57\!\cdots\!15}{18\!\cdots\!12}a^{12}-\frac{30\!\cdots\!05}{91\!\cdots\!56}a^{11}-\frac{27\!\cdots\!45}{91\!\cdots\!56}a^{10}+\frac{80\!\cdots\!09}{22\!\cdots\!64}a^{9}+\frac{57\!\cdots\!35}{91\!\cdots\!56}a^{8}+\frac{73\!\cdots\!97}{45\!\cdots\!28}a^{7}+\frac{95\!\cdots\!65}{22\!\cdots\!64}a^{6}-\frac{85\!\cdots\!13}{11\!\cdots\!32}a^{5}-\frac{13\!\cdots\!45}{11\!\cdots\!32}a^{4}+\frac{12\!\cdots\!39}{11\!\cdots\!32}a^{3}-\frac{81\!\cdots\!53}{11\!\cdots\!32}a^{2}-\frac{11\!\cdots\!23}{28\!\cdots\!83}a-\frac{16\!\cdots\!20}{28\!\cdots\!83}$, $\frac{26\!\cdots\!47}{36\!\cdots\!24}a^{31}+\frac{85\!\cdots\!61}{18\!\cdots\!12}a^{30}+\frac{23\!\cdots\!23}{18\!\cdots\!12}a^{29}+\frac{15\!\cdots\!93}{18\!\cdots\!12}a^{28}+\frac{52\!\cdots\!95}{36\!\cdots\!24}a^{27}+\frac{22\!\cdots\!71}{22\!\cdots\!64}a^{26}+\frac{96\!\cdots\!45}{91\!\cdots\!56}a^{25}+\frac{13\!\cdots\!75}{18\!\cdots\!12}a^{24}+\frac{20\!\cdots\!69}{36\!\cdots\!24}a^{23}+\frac{72\!\cdots\!59}{18\!\cdots\!12}a^{22}+\frac{16\!\cdots\!61}{91\!\cdots\!56}a^{21}+\frac{11\!\cdots\!41}{91\!\cdots\!56}a^{20}+\frac{16\!\cdots\!05}{36\!\cdots\!24}a^{19}+\frac{59\!\cdots\!39}{18\!\cdots\!12}a^{18}+\frac{16\!\cdots\!15}{18\!\cdots\!12}a^{17}+\frac{12\!\cdots\!99}{18\!\cdots\!12}a^{16}+\frac{41\!\cdots\!33}{36\!\cdots\!24}a^{15}+\frac{37\!\cdots\!75}{45\!\cdots\!28}a^{14}+\frac{16\!\cdots\!23}{45\!\cdots\!28}a^{13}+\frac{22\!\cdots\!53}{18\!\cdots\!12}a^{12}+\frac{37\!\cdots\!47}{36\!\cdots\!24}a^{11}+\frac{25\!\cdots\!77}{18\!\cdots\!12}a^{10}+\frac{37\!\cdots\!13}{45\!\cdots\!28}a^{9}+\frac{32\!\cdots\!29}{91\!\cdots\!56}a^{8}-\frac{69\!\cdots\!65}{91\!\cdots\!56}a^{7}+\frac{86\!\cdots\!73}{57\!\cdots\!66}a^{6}+\frac{17\!\cdots\!11}{57\!\cdots\!66}a^{5}-\frac{51\!\cdots\!05}{22\!\cdots\!64}a^{4}+\frac{58\!\cdots\!57}{22\!\cdots\!64}a^{3}+\frac{31\!\cdots\!37}{11\!\cdots\!32}a^{2}-\frac{10\!\cdots\!53}{57\!\cdots\!66}a-\frac{52\!\cdots\!51}{28\!\cdots\!83}$, $\frac{25\!\cdots\!33}{36\!\cdots\!24}a^{31}-\frac{96\!\cdots\!12}{28\!\cdots\!83}a^{30}+\frac{56\!\cdots\!75}{45\!\cdots\!28}a^{29}-\frac{11\!\cdots\!15}{18\!\cdots\!12}a^{28}+\frac{50\!\cdots\!27}{36\!\cdots\!24}a^{27}-\frac{64\!\cdots\!17}{91\!\cdots\!56}a^{26}+\frac{18\!\cdots\!69}{18\!\cdots\!12}a^{25}-\frac{24\!\cdots\!47}{45\!\cdots\!28}a^{24}+\frac{19\!\cdots\!79}{36\!\cdots\!24}a^{23}-\frac{52\!\cdots\!65}{18\!\cdots\!12}a^{22}+\frac{19\!\cdots\!23}{11\!\cdots\!32}a^{21}-\frac{88\!\cdots\!91}{91\!\cdots\!56}a^{20}+\frac{14\!\cdots\!59}{36\!\cdots\!24}a^{19}-\frac{21\!\cdots\!07}{91\!\cdots\!56}a^{18}+\frac{72\!\cdots\!51}{91\!\cdots\!56}a^{17}-\frac{89\!\cdots\!35}{18\!\cdots\!12}a^{16}+\frac{30\!\cdots\!29}{36\!\cdots\!24}a^{15}-\frac{27\!\cdots\!09}{45\!\cdots\!28}a^{14}-\frac{47\!\cdots\!87}{18\!\cdots\!12}a^{13}-\frac{78\!\cdots\!41}{91\!\cdots\!56}a^{12}+\frac{23\!\cdots\!69}{36\!\cdots\!24}a^{11}+\frac{10\!\cdots\!19}{18\!\cdots\!12}a^{10}+\frac{51\!\cdots\!95}{91\!\cdots\!56}a^{9}-\frac{14\!\cdots\!45}{57\!\cdots\!66}a^{8}-\frac{30\!\cdots\!21}{11\!\cdots\!32}a^{7}-\frac{29\!\cdots\!71}{22\!\cdots\!64}a^{6}-\frac{35\!\cdots\!53}{45\!\cdots\!28}a^{5}+\frac{82\!\cdots\!19}{22\!\cdots\!64}a^{4}+\frac{54\!\cdots\!75}{11\!\cdots\!32}a^{3}+\frac{10\!\cdots\!83}{11\!\cdots\!32}a^{2}-\frac{64\!\cdots\!87}{57\!\cdots\!66}a-\frac{16\!\cdots\!08}{66\!\cdots\!81}$, $\frac{15\!\cdots\!61}{91\!\cdots\!56}a^{31}+\frac{81\!\cdots\!73}{18\!\cdots\!12}a^{30}+\frac{58\!\cdots\!81}{18\!\cdots\!12}a^{29}+\frac{14\!\cdots\!21}{18\!\cdots\!12}a^{28}+\frac{34\!\cdots\!11}{91\!\cdots\!56}a^{27}+\frac{16\!\cdots\!59}{18\!\cdots\!12}a^{26}+\frac{53\!\cdots\!81}{18\!\cdots\!12}a^{25}+\frac{63\!\cdots\!11}{91\!\cdots\!56}a^{24}+\frac{27\!\cdots\!87}{16\!\cdots\!92}a^{23}+\frac{86\!\cdots\!97}{22\!\cdots\!64}a^{22}+\frac{11\!\cdots\!87}{18\!\cdots\!12}a^{21}+\frac{14\!\cdots\!35}{11\!\cdots\!32}a^{20}+\frac{30\!\cdots\!49}{18\!\cdots\!12}a^{19}+\frac{57\!\cdots\!47}{18\!\cdots\!12}a^{18}+\frac{84\!\cdots\!51}{22\!\cdots\!64}a^{17}+\frac{11\!\cdots\!95}{18\!\cdots\!12}a^{16}+\frac{10\!\cdots\!03}{18\!\cdots\!12}a^{15}+\frac{14\!\cdots\!93}{18\!\cdots\!12}a^{14}+\frac{98\!\cdots\!35}{22\!\cdots\!64}a^{13}+\frac{14\!\cdots\!21}{91\!\cdots\!56}a^{12}+\frac{16\!\cdots\!27}{10\!\cdots\!96}a^{11}+\frac{96\!\cdots\!15}{11\!\cdots\!32}a^{10}+\frac{15\!\cdots\!43}{91\!\cdots\!56}a^{9}+\frac{40\!\cdots\!53}{11\!\cdots\!32}a^{8}+\frac{17\!\cdots\!67}{91\!\cdots\!56}a^{7}+\frac{64\!\cdots\!91}{45\!\cdots\!28}a^{6}+\frac{41\!\cdots\!17}{45\!\cdots\!28}a^{5}+\frac{13\!\cdots\!07}{22\!\cdots\!64}a^{4}+\frac{32\!\cdots\!15}{22\!\cdots\!64}a^{3}+\frac{97\!\cdots\!55}{57\!\cdots\!66}a^{2}+\frac{15\!\cdots\!51}{57\!\cdots\!66}a-\frac{10\!\cdots\!57}{28\!\cdots\!83}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 182938558023068.47 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 182938558023068.47 \cdot 1008}{12\cdot\sqrt{1690320352622233436323740015416175733767719950199015079936}}\cr\approx \mathstrut & 2.20535201595411 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 18*x^30 + 203*x^28 + 1492*x^26 + 8025*x^24 + 25714*x^22 + 62137*x^20 + 123428*x^18 + 139579*x^16 - 16686*x^14 + 1661*x^12 + 76916*x^10 + 9756*x^8 - 10192*x^6 + 3152*x^4 - 1088*x^2 + 256);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
$D_4^2:C_2^3$ (as 32T12882):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.8.0.1}{8} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.8.0.1}{8} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{16}$ | R | R | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(17\)
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(47\)
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(53\)
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.2.0.1 | $x^{2} + 49 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 53.4.2.1 | $x^{4} + 4974 x^{3} + 6304741 x^{2} + 297375564 x + 18587952$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |