Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1)
gp: K = bnfinit(y^33 - 3*y^32 - 3*y^31 + 28*y^30 - 58*y^29 + 22*y^28 + 191*y^27 - 526*y^26 + 330*y^25 + 728*y^24 - 1012*y^23 + 413*y^22 - 2524*y^21 + 3437*y^20 + 6311*y^19 - 13555*y^18 - 573*y^17 + 14501*y^16 - 6805*y^15 - 2479*y^14 + 5007*y^13 - 7594*y^12 - 344*y^11 + 8602*y^10 - 626*y^9 - 5041*y^8 + 15*y^7 + 1789*y^6 + 121*y^5 - 352*y^4 - 33*y^3 + 32*y^2 + 2*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1)
\( x^{33} - 3 x^{32} - 3 x^{31} + 28 x^{30} - 58 x^{29} + 22 x^{28} + 191 x^{27} - 526 x^{26} + 330 x^{25} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[11, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1635170022196481349560959748587682926364327\)
\(\medspace = -\,23^{31}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.02\) | 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: | $23^{21/22}\approx 19.944865695037844$ | ||
| Ramified primes: |
\(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $11$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{599}a^{31}-\frac{277}{599}a^{30}-\frac{244}{599}a^{29}+\frac{108}{599}a^{28}+\frac{231}{599}a^{27}+\frac{282}{599}a^{26}-\frac{77}{599}a^{25}+\frac{163}{599}a^{24}+\frac{284}{599}a^{23}+\frac{207}{599}a^{22}+\frac{198}{599}a^{21}+\frac{186}{599}a^{20}-\frac{67}{599}a^{19}-\frac{65}{599}a^{18}-\frac{209}{599}a^{17}+\frac{81}{599}a^{16}+\frac{12}{599}a^{15}-\frac{123}{599}a^{14}-\frac{251}{599}a^{13}+\frac{137}{599}a^{12}+\frac{208}{599}a^{11}+\frac{49}{599}a^{10}+\frac{56}{599}a^{9}+\frac{207}{599}a^{8}+\frac{58}{599}a^{7}+\frac{147}{599}a^{6}+\frac{235}{599}a^{5}+\frac{176}{599}a^{4}-\frac{119}{599}a^{3}+\frac{272}{599}a^{2}-\frac{218}{599}a-\frac{118}{599}$, $\frac{1}{66\!\cdots\!81}a^{32}-\frac{48\!\cdots\!71}{11\!\cdots\!19}a^{31}+\frac{18\!\cdots\!40}{66\!\cdots\!81}a^{30}+\frac{50\!\cdots\!62}{66\!\cdots\!81}a^{29}+\frac{10\!\cdots\!91}{66\!\cdots\!81}a^{28}+\frac{15\!\cdots\!22}{66\!\cdots\!81}a^{27}+\frac{20\!\cdots\!60}{66\!\cdots\!81}a^{26}+\frac{71\!\cdots\!12}{66\!\cdots\!81}a^{25}+\frac{11\!\cdots\!23}{66\!\cdots\!81}a^{24}-\frac{31\!\cdots\!21}{66\!\cdots\!81}a^{23}+\frac{28\!\cdots\!73}{66\!\cdots\!81}a^{22}+\frac{19\!\cdots\!92}{66\!\cdots\!81}a^{21}+\frac{28\!\cdots\!54}{66\!\cdots\!81}a^{20}+\frac{50\!\cdots\!16}{66\!\cdots\!81}a^{19}+\frac{14\!\cdots\!37}{66\!\cdots\!81}a^{18}+\frac{30\!\cdots\!13}{66\!\cdots\!81}a^{17}-\frac{22\!\cdots\!81}{66\!\cdots\!81}a^{16}+\frac{13\!\cdots\!33}{66\!\cdots\!81}a^{15}-\frac{17\!\cdots\!37}{66\!\cdots\!81}a^{14}-\frac{77\!\cdots\!28}{66\!\cdots\!81}a^{13}+\frac{30\!\cdots\!37}{66\!\cdots\!81}a^{12}-\frac{74\!\cdots\!16}{66\!\cdots\!81}a^{11}-\frac{61\!\cdots\!58}{66\!\cdots\!81}a^{10}+\frac{41\!\cdots\!03}{66\!\cdots\!81}a^{9}+\frac{13\!\cdots\!63}{66\!\cdots\!81}a^{8}-\frac{28\!\cdots\!23}{66\!\cdots\!81}a^{7}-\frac{17\!\cdots\!05}{66\!\cdots\!81}a^{6}+\frac{40\!\cdots\!98}{66\!\cdots\!81}a^{5}+\frac{19\!\cdots\!02}{66\!\cdots\!81}a^{4}-\frac{31\!\cdots\!43}{66\!\cdots\!81}a^{3}+\frac{27\!\cdots\!29}{66\!\cdots\!81}a^{2}-\frac{15\!\cdots\!41}{66\!\cdots\!81}a-\frac{92\!\cdots\!11}{66\!\cdots\!81}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{20\!\cdots\!21}{21\!\cdots\!93}a^{32}-\frac{70\!\cdots\!90}{21\!\cdots\!93}a^{31}-\frac{31\!\cdots\!33}{21\!\cdots\!93}a^{30}+\frac{59\!\cdots\!93}{21\!\cdots\!93}a^{29}-\frac{14\!\cdots\!31}{21\!\cdots\!93}a^{28}+\frac{10\!\cdots\!87}{21\!\cdots\!93}a^{27}+\frac{36\!\cdots\!98}{21\!\cdots\!93}a^{26}-\frac{12\!\cdots\!58}{21\!\cdots\!93}a^{25}+\frac{12\!\cdots\!90}{21\!\cdots\!93}a^{24}+\frac{11\!\cdots\!51}{21\!\cdots\!93}a^{23}-\frac{27\!\cdots\!79}{21\!\cdots\!93}a^{22}+\frac{19\!\cdots\!78}{21\!\cdots\!93}a^{21}-\frac{57\!\cdots\!42}{21\!\cdots\!93}a^{20}+\frac{95\!\cdots\!02}{21\!\cdots\!93}a^{19}+\frac{95\!\cdots\!70}{21\!\cdots\!93}a^{18}-\frac{33\!\cdots\!49}{21\!\cdots\!93}a^{17}+\frac{12\!\cdots\!10}{21\!\cdots\!93}a^{16}+\frac{27\!\cdots\!89}{21\!\cdots\!93}a^{15}-\frac{27\!\cdots\!79}{21\!\cdots\!93}a^{14}+\frac{45\!\cdots\!47}{21\!\cdots\!93}a^{13}+\frac{97\!\cdots\!95}{21\!\cdots\!93}a^{12}-\frac{20\!\cdots\!48}{21\!\cdots\!93}a^{11}+\frac{81\!\cdots\!61}{21\!\cdots\!93}a^{10}+\frac{16\!\cdots\!82}{21\!\cdots\!93}a^{9}-\frac{86\!\cdots\!35}{21\!\cdots\!93}a^{8}-\frac{81\!\cdots\!07}{21\!\cdots\!93}a^{7}+\frac{36\!\cdots\!66}{21\!\cdots\!93}a^{6}+\frac{27\!\cdots\!49}{21\!\cdots\!93}a^{5}-\frac{91\!\cdots\!67}{21\!\cdots\!93}a^{4}-\frac{47\!\cdots\!00}{21\!\cdots\!93}a^{3}+\frac{12\!\cdots\!35}{21\!\cdots\!93}a^{2}+\frac{23\!\cdots\!49}{21\!\cdots\!93}a-\frac{72\!\cdots\!94}{21\!\cdots\!93}$, $\frac{59\!\cdots\!92}{66\!\cdots\!81}a^{32}-\frac{22\!\cdots\!72}{11\!\cdots\!19}a^{31}-\frac{26\!\cdots\!65}{66\!\cdots\!81}a^{30}+\frac{14\!\cdots\!52}{66\!\cdots\!81}a^{29}-\frac{24\!\cdots\!54}{66\!\cdots\!81}a^{28}-\frac{23\!\cdots\!76}{66\!\cdots\!81}a^{27}+\frac{10\!\cdots\!26}{66\!\cdots\!81}a^{26}-\frac{23\!\cdots\!22}{66\!\cdots\!81}a^{25}+\frac{42\!\cdots\!84}{66\!\cdots\!81}a^{24}+\frac{42\!\cdots\!96}{66\!\cdots\!81}a^{23}-\frac{28\!\cdots\!81}{66\!\cdots\!81}a^{22}+\frac{97\!\cdots\!85}{66\!\cdots\!81}a^{21}-\frac{14\!\cdots\!58}{66\!\cdots\!81}a^{20}+\frac{10\!\cdots\!85}{66\!\cdots\!81}a^{19}+\frac{42\!\cdots\!95}{66\!\cdots\!81}a^{18}-\frac{48\!\cdots\!84}{66\!\cdots\!81}a^{17}-\frac{32\!\cdots\!10}{66\!\cdots\!81}a^{16}+\frac{55\!\cdots\!01}{66\!\cdots\!81}a^{15}-\frac{35\!\cdots\!27}{66\!\cdots\!81}a^{14}-\frac{91\!\cdots\!88}{66\!\cdots\!81}a^{13}+\frac{20\!\cdots\!05}{66\!\cdots\!81}a^{12}-\frac{31\!\cdots\!40}{66\!\cdots\!81}a^{11}-\frac{21\!\cdots\!08}{66\!\cdots\!81}a^{10}+\frac{31\!\cdots\!50}{66\!\cdots\!81}a^{9}+\frac{16\!\cdots\!25}{66\!\cdots\!81}a^{8}-\frac{13\!\cdots\!12}{66\!\cdots\!81}a^{7}-\frac{83\!\cdots\!53}{66\!\cdots\!81}a^{6}+\frac{25\!\cdots\!37}{66\!\cdots\!81}a^{5}+\frac{19\!\cdots\!27}{66\!\cdots\!81}a^{4}-\frac{15\!\cdots\!13}{66\!\cdots\!81}a^{3}-\frac{18\!\cdots\!94}{66\!\cdots\!81}a^{2}-\frac{27\!\cdots\!40}{66\!\cdots\!81}a+\frac{42\!\cdots\!70}{66\!\cdots\!81}$, $\frac{33\!\cdots\!37}{66\!\cdots\!81}a^{32}-\frac{14\!\cdots\!22}{66\!\cdots\!81}a^{31}+\frac{37\!\cdots\!70}{66\!\cdots\!81}a^{30}+\frac{10\!\cdots\!98}{66\!\cdots\!81}a^{29}-\frac{31\!\cdots\!07}{66\!\cdots\!81}a^{28}+\frac{34\!\cdots\!59}{66\!\cdots\!81}a^{27}+\frac{47\!\cdots\!48}{66\!\cdots\!81}a^{26}-\frac{24\!\cdots\!43}{66\!\cdots\!81}a^{25}+\frac{33\!\cdots\!02}{66\!\cdots\!81}a^{24}+\frac{60\!\cdots\!42}{66\!\cdots\!81}a^{23}-\frac{52\!\cdots\!81}{66\!\cdots\!81}a^{22}+\frac{43\!\cdots\!23}{66\!\cdots\!81}a^{21}-\frac{11\!\cdots\!05}{66\!\cdots\!81}a^{20}+\frac{26\!\cdots\!91}{66\!\cdots\!81}a^{19}+\frac{47\!\cdots\!50}{66\!\cdots\!81}a^{18}-\frac{67\!\cdots\!34}{66\!\cdots\!81}a^{17}+\frac{44\!\cdots\!81}{66\!\cdots\!81}a^{16}+\frac{39\!\cdots\!69}{66\!\cdots\!81}a^{15}-\frac{49\!\cdots\!98}{66\!\cdots\!81}a^{14}+\frac{15\!\cdots\!15}{66\!\cdots\!81}a^{13}-\frac{46\!\cdots\!91}{66\!\cdots\!81}a^{12}-\frac{32\!\cdots\!86}{66\!\cdots\!81}a^{11}+\frac{26\!\cdots\!93}{66\!\cdots\!81}a^{10}+\frac{27\!\cdots\!93}{66\!\cdots\!81}a^{9}-\frac{17\!\cdots\!98}{66\!\cdots\!81}a^{8}-\frac{16\!\cdots\!45}{66\!\cdots\!81}a^{7}+\frac{41\!\cdots\!24}{66\!\cdots\!81}a^{6}+\frac{57\!\cdots\!39}{66\!\cdots\!81}a^{5}+\frac{21\!\cdots\!61}{66\!\cdots\!81}a^{4}-\frac{77\!\cdots\!68}{66\!\cdots\!81}a^{3}-\frac{18\!\cdots\!64}{66\!\cdots\!81}a^{2}+\frac{13\!\cdots\!43}{66\!\cdots\!81}a+\frac{12\!\cdots\!95}{66\!\cdots\!81}$, $\frac{57\!\cdots\!65}{66\!\cdots\!81}a^{32}-\frac{12\!\cdots\!23}{66\!\cdots\!81}a^{31}-\frac{28\!\cdots\!93}{66\!\cdots\!81}a^{30}+\frac{13\!\cdots\!18}{66\!\cdots\!81}a^{29}-\frac{21\!\cdots\!42}{66\!\cdots\!81}a^{28}-\frac{67\!\cdots\!96}{66\!\cdots\!81}a^{27}+\frac{10\!\cdots\!78}{66\!\cdots\!81}a^{26}-\frac{21\!\cdots\!59}{66\!\cdots\!81}a^{25}+\frac{84\!\cdots\!47}{66\!\cdots\!81}a^{24}+\frac{43\!\cdots\!37}{66\!\cdots\!81}a^{23}-\frac{21\!\cdots\!04}{66\!\cdots\!81}a^{22}+\frac{37\!\cdots\!10}{66\!\cdots\!81}a^{21}-\frac{13\!\cdots\!29}{66\!\cdots\!81}a^{20}+\frac{73\!\cdots\!93}{66\!\cdots\!81}a^{19}+\frac{43\!\cdots\!01}{66\!\cdots\!81}a^{18}-\frac{40\!\cdots\!69}{66\!\cdots\!81}a^{17}-\frac{40\!\cdots\!78}{66\!\cdots\!81}a^{16}+\frac{51\!\cdots\!87}{66\!\cdots\!81}a^{15}+\frac{47\!\cdots\!56}{66\!\cdots\!81}a^{14}-\frac{11\!\cdots\!50}{66\!\cdots\!81}a^{13}+\frac{19\!\cdots\!76}{66\!\cdots\!81}a^{12}-\frac{27\!\cdots\!02}{66\!\cdots\!81}a^{11}-\frac{25\!\cdots\!56}{66\!\cdots\!81}a^{10}+\frac{28\!\cdots\!11}{66\!\cdots\!81}a^{9}+\frac{20\!\cdots\!34}{66\!\cdots\!81}a^{8}-\frac{11\!\cdots\!72}{66\!\cdots\!81}a^{7}-\frac{10\!\cdots\!41}{66\!\cdots\!81}a^{6}+\frac{19\!\cdots\!63}{66\!\cdots\!81}a^{5}+\frac{24\!\cdots\!50}{66\!\cdots\!81}a^{4}-\frac{32\!\cdots\!47}{66\!\cdots\!81}a^{3}-\frac{23\!\cdots\!03}{66\!\cdots\!81}a^{2}-\frac{75\!\cdots\!11}{66\!\cdots\!81}a+\frac{69\!\cdots\!49}{66\!\cdots\!81}$, $\frac{71\!\cdots\!93}{66\!\cdots\!81}a^{32}-\frac{16\!\cdots\!04}{66\!\cdots\!81}a^{31}-\frac{32\!\cdots\!28}{66\!\cdots\!81}a^{30}+\frac{17\!\cdots\!14}{66\!\cdots\!81}a^{29}-\frac{29\!\cdots\!05}{66\!\cdots\!81}a^{28}-\frac{28\!\cdots\!63}{66\!\cdots\!81}a^{27}+\frac{12\!\cdots\!46}{66\!\cdots\!81}a^{26}-\frac{27\!\cdots\!36}{66\!\cdots\!81}a^{25}+\frac{49\!\cdots\!80}{66\!\cdots\!81}a^{24}+\frac{50\!\cdots\!98}{66\!\cdots\!81}a^{23}-\frac{32\!\cdots\!35}{66\!\cdots\!81}a^{22}+\frac{12\!\cdots\!25}{66\!\cdots\!81}a^{21}-\frac{17\!\cdots\!91}{66\!\cdots\!81}a^{20}+\frac{11\!\cdots\!60}{66\!\cdots\!81}a^{19}+\frac{51\!\cdots\!74}{66\!\cdots\!81}a^{18}-\frac{56\!\cdots\!88}{66\!\cdots\!81}a^{17}-\frac{39\!\cdots\!81}{66\!\cdots\!81}a^{16}+\frac{64\!\cdots\!92}{66\!\cdots\!81}a^{15}-\frac{33\!\cdots\!20}{66\!\cdots\!81}a^{14}-\frac{99\!\cdots\!36}{66\!\cdots\!81}a^{13}+\frac{24\!\cdots\!14}{66\!\cdots\!81}a^{12}-\frac{36\!\cdots\!33}{66\!\cdots\!81}a^{11}-\frac{25\!\cdots\!25}{66\!\cdots\!81}a^{10}+\frac{36\!\cdots\!09}{66\!\cdots\!81}a^{9}+\frac{20\!\cdots\!74}{66\!\cdots\!81}a^{8}-\frac{15\!\cdots\!60}{66\!\cdots\!81}a^{7}-\frac{10\!\cdots\!82}{66\!\cdots\!81}a^{6}+\frac{27\!\cdots\!51}{66\!\cdots\!81}a^{5}+\frac{24\!\cdots\!46}{66\!\cdots\!81}a^{4}-\frac{14\!\cdots\!59}{66\!\cdots\!81}a^{3}-\frac{22\!\cdots\!13}{66\!\cdots\!81}a^{2}+\frac{41\!\cdots\!18}{66\!\cdots\!81}a+\frac{67\!\cdots\!79}{66\!\cdots\!81}$, $\frac{14\!\cdots\!53}{66\!\cdots\!81}a^{32}-\frac{33\!\cdots\!41}{66\!\cdots\!81}a^{31}-\frac{61\!\cdots\!32}{66\!\cdots\!81}a^{30}+\frac{35\!\cdots\!98}{66\!\cdots\!81}a^{29}-\frac{60\!\cdots\!76}{66\!\cdots\!81}a^{28}-\frac{35\!\cdots\!17}{66\!\cdots\!81}a^{27}+\frac{25\!\cdots\!81}{66\!\cdots\!81}a^{26}-\frac{57\!\cdots\!68}{66\!\cdots\!81}a^{25}+\frac{14\!\cdots\!81}{66\!\cdots\!81}a^{24}+\frac{98\!\cdots\!66}{66\!\cdots\!81}a^{23}-\frac{71\!\cdots\!84}{66\!\cdots\!81}a^{22}+\frac{31\!\cdots\!77}{66\!\cdots\!81}a^{21}-\frac{36\!\cdots\!97}{66\!\cdots\!81}a^{20}+\frac{26\!\cdots\!44}{66\!\cdots\!81}a^{19}+\frac{98\!\cdots\!51}{66\!\cdots\!81}a^{18}-\frac{11\!\cdots\!15}{66\!\cdots\!81}a^{17}-\frac{67\!\cdots\!53}{66\!\cdots\!81}a^{16}+\frac{13\!\cdots\!23}{66\!\cdots\!81}a^{15}-\frac{16\!\cdots\!94}{66\!\cdots\!81}a^{14}-\frac{16\!\cdots\!12}{66\!\cdots\!81}a^{13}+\frac{47\!\cdots\!53}{66\!\cdots\!81}a^{12}-\frac{76\!\cdots\!90}{66\!\cdots\!81}a^{11}-\frac{44\!\cdots\!40}{66\!\cdots\!81}a^{10}+\frac{73\!\cdots\!02}{66\!\cdots\!81}a^{9}+\frac{35\!\cdots\!51}{66\!\cdots\!81}a^{8}-\frac{31\!\cdots\!31}{66\!\cdots\!81}a^{7}-\frac{17\!\cdots\!46}{66\!\cdots\!81}a^{6}+\frac{61\!\cdots\!26}{66\!\cdots\!81}a^{5}+\frac{42\!\cdots\!23}{66\!\cdots\!81}a^{4}-\frac{41\!\cdots\!08}{66\!\cdots\!81}a^{3}-\frac{37\!\cdots\!27}{66\!\cdots\!81}a^{2}+\frac{66\!\cdots\!27}{66\!\cdots\!81}a+\frac{10\!\cdots\!30}{66\!\cdots\!81}$, $\frac{11\!\cdots\!68}{66\!\cdots\!81}a^{32}-\frac{12\!\cdots\!87}{66\!\cdots\!81}a^{31}-\frac{90\!\cdots\!89}{66\!\cdots\!81}a^{30}+\frac{23\!\cdots\!03}{66\!\cdots\!81}a^{29}-\frac{10\!\cdots\!97}{66\!\cdots\!81}a^{28}-\frac{80\!\cdots\!91}{66\!\cdots\!81}a^{27}+\frac{24\!\cdots\!57}{66\!\cdots\!81}a^{26}-\frac{21\!\cdots\!53}{66\!\cdots\!81}a^{25}-\frac{59\!\cdots\!22}{66\!\cdots\!81}a^{24}+\frac{12\!\cdots\!98}{66\!\cdots\!81}a^{23}+\frac{25\!\cdots\!06}{66\!\cdots\!81}a^{22}-\frac{94\!\cdots\!31}{66\!\cdots\!81}a^{21}-\frac{21\!\cdots\!33}{66\!\cdots\!81}a^{20}-\frac{17\!\cdots\!48}{66\!\cdots\!81}a^{19}+\frac{12\!\cdots\!81}{66\!\cdots\!81}a^{18}-\frac{16\!\cdots\!93}{66\!\cdots\!81}a^{17}-\frac{22\!\cdots\!80}{66\!\cdots\!81}a^{16}+\frac{10\!\cdots\!86}{66\!\cdots\!81}a^{15}+\frac{13\!\cdots\!28}{66\!\cdots\!81}a^{14}-\frac{97\!\cdots\!70}{66\!\cdots\!81}a^{13}+\frac{44\!\cdots\!79}{66\!\cdots\!81}a^{12}-\frac{13\!\cdots\!15}{66\!\cdots\!81}a^{11}-\frac{13\!\cdots\!73}{66\!\cdots\!81}a^{10}+\frac{59\!\cdots\!30}{66\!\cdots\!81}a^{9}+\frac{11\!\cdots\!44}{66\!\cdots\!81}a^{8}-\frac{28\!\cdots\!26}{66\!\cdots\!81}a^{7}-\frac{56\!\cdots\!09}{66\!\cdots\!81}a^{6}+\frac{34\!\cdots\!37}{66\!\cdots\!81}a^{5}+\frac{15\!\cdots\!46}{66\!\cdots\!81}a^{4}+\frac{51\!\cdots\!39}{66\!\cdots\!81}a^{3}-\frac{19\!\cdots\!36}{66\!\cdots\!81}a^{2}-\frac{92\!\cdots\!70}{66\!\cdots\!81}a+\frac{76\!\cdots\!22}{66\!\cdots\!81}$, $\frac{65\!\cdots\!13}{66\!\cdots\!81}a^{32}-\frac{15\!\cdots\!14}{66\!\cdots\!81}a^{31}-\frac{28\!\cdots\!00}{66\!\cdots\!81}a^{30}+\frac{16\!\cdots\!48}{66\!\cdots\!81}a^{29}-\frac{28\!\cdots\!27}{66\!\cdots\!81}a^{28}+\frac{42\!\cdots\!41}{66\!\cdots\!81}a^{27}+\frac{11\!\cdots\!35}{66\!\cdots\!81}a^{26}-\frac{26\!\cdots\!32}{66\!\cdots\!81}a^{25}+\frac{73\!\cdots\!73}{66\!\cdots\!81}a^{24}+\frac{45\!\cdots\!73}{66\!\cdots\!81}a^{23}-\frac{33\!\cdots\!37}{66\!\cdots\!81}a^{22}+\frac{15\!\cdots\!18}{66\!\cdots\!81}a^{21}-\frac{16\!\cdots\!72}{66\!\cdots\!81}a^{20}+\frac{12\!\cdots\!59}{66\!\cdots\!81}a^{19}+\frac{45\!\cdots\!28}{66\!\cdots\!81}a^{18}-\frac{55\!\cdots\!63}{66\!\cdots\!81}a^{17}-\frac{29\!\cdots\!70}{66\!\cdots\!81}a^{16}+\frac{60\!\cdots\!45}{66\!\cdots\!81}a^{15}-\frac{86\!\cdots\!23}{66\!\cdots\!81}a^{14}-\frac{69\!\cdots\!33}{66\!\cdots\!81}a^{13}+\frac{22\!\cdots\!38}{66\!\cdots\!81}a^{12}-\frac{35\!\cdots\!84}{66\!\cdots\!81}a^{11}-\frac{19\!\cdots\!71}{66\!\cdots\!81}a^{10}+\frac{33\!\cdots\!78}{66\!\cdots\!81}a^{9}+\frac{15\!\cdots\!21}{66\!\cdots\!81}a^{8}-\frac{14\!\cdots\!44}{66\!\cdots\!81}a^{7}-\frac{78\!\cdots\!65}{66\!\cdots\!81}a^{6}+\frac{28\!\cdots\!67}{66\!\cdots\!81}a^{5}+\frac{18\!\cdots\!59}{66\!\cdots\!81}a^{4}-\frac{21\!\cdots\!82}{66\!\cdots\!81}a^{3}-\frac{15\!\cdots\!15}{66\!\cdots\!81}a^{2}+\frac{59\!\cdots\!55}{66\!\cdots\!81}a+\frac{40\!\cdots\!53}{66\!\cdots\!81}$, $\frac{40\!\cdots\!76}{66\!\cdots\!81}a^{32}-\frac{89\!\cdots\!52}{66\!\cdots\!81}a^{31}-\frac{18\!\cdots\!96}{66\!\cdots\!81}a^{30}+\frac{96\!\cdots\!73}{66\!\cdots\!81}a^{29}-\frac{15\!\cdots\!39}{66\!\cdots\!81}a^{28}-\frac{17\!\cdots\!97}{66\!\cdots\!81}a^{27}+\frac{71\!\cdots\!71}{66\!\cdots\!81}a^{26}-\frac{15\!\cdots\!12}{66\!\cdots\!81}a^{25}+\frac{24\!\cdots\!50}{66\!\cdots\!81}a^{24}+\frac{27\!\cdots\!90}{66\!\cdots\!81}a^{23}-\frac{15\!\cdots\!31}{66\!\cdots\!81}a^{22}+\frac{66\!\cdots\!58}{66\!\cdots\!81}a^{21}-\frac{10\!\cdots\!87}{66\!\cdots\!81}a^{20}+\frac{64\!\cdots\!94}{66\!\cdots\!81}a^{19}+\frac{28\!\cdots\!44}{66\!\cdots\!81}a^{18}-\frac{29\!\cdots\!66}{66\!\cdots\!81}a^{17}-\frac{22\!\cdots\!14}{66\!\cdots\!81}a^{16}+\frac{32\!\cdots\!01}{66\!\cdots\!81}a^{15}+\frac{70\!\cdots\!54}{66\!\cdots\!81}a^{14}-\frac{44\!\cdots\!50}{66\!\cdots\!81}a^{13}+\frac{11\!\cdots\!19}{66\!\cdots\!81}a^{12}-\frac{19\!\cdots\!92}{66\!\cdots\!81}a^{11}-\frac{15\!\cdots\!69}{66\!\cdots\!81}a^{10}+\frac{18\!\cdots\!39}{66\!\cdots\!81}a^{9}+\frac{12\!\cdots\!09}{66\!\cdots\!81}a^{8}-\frac{71\!\cdots\!95}{66\!\cdots\!81}a^{7}-\frac{62\!\cdots\!15}{66\!\cdots\!81}a^{6}+\frac{93\!\cdots\!46}{66\!\cdots\!81}a^{5}+\frac{14\!\cdots\!32}{66\!\cdots\!81}a^{4}+\frac{57\!\cdots\!06}{66\!\cdots\!81}a^{3}-\frac{12\!\cdots\!72}{66\!\cdots\!81}a^{2}-\frac{92\!\cdots\!65}{66\!\cdots\!81}a+\frac{37\!\cdots\!07}{66\!\cdots\!81}$, $\frac{15\!\cdots\!92}{66\!\cdots\!81}a^{32}-\frac{52\!\cdots\!65}{66\!\cdots\!81}a^{31}-\frac{26\!\cdots\!37}{66\!\cdots\!81}a^{30}+\frac{43\!\cdots\!40}{66\!\cdots\!81}a^{29}-\frac{10\!\cdots\!71}{66\!\cdots\!81}a^{28}+\frac{75\!\cdots\!32}{66\!\cdots\!81}a^{27}+\frac{26\!\cdots\!20}{66\!\cdots\!81}a^{26}-\frac{89\!\cdots\!67}{66\!\cdots\!81}a^{25}+\frac{84\!\cdots\!41}{66\!\cdots\!81}a^{24}+\frac{75\!\cdots\!19}{66\!\cdots\!81}a^{23}-\frac{17\!\cdots\!81}{66\!\cdots\!81}a^{22}+\frac{12\!\cdots\!61}{66\!\cdots\!81}a^{21}-\frac{45\!\cdots\!11}{66\!\cdots\!81}a^{20}+\frac{70\!\cdots\!40}{66\!\cdots\!81}a^{19}+\frac{68\!\cdots\!08}{66\!\cdots\!81}a^{18}-\frac{22\!\cdots\!85}{66\!\cdots\!81}a^{17}+\frac{74\!\cdots\!24}{66\!\cdots\!81}a^{16}+\frac{17\!\cdots\!65}{66\!\cdots\!81}a^{15}-\frac{15\!\cdots\!69}{66\!\cdots\!81}a^{14}+\frac{28\!\cdots\!32}{66\!\cdots\!81}a^{13}+\frac{50\!\cdots\!66}{66\!\cdots\!81}a^{12}-\frac{13\!\cdots\!80}{66\!\cdots\!81}a^{11}+\frac{43\!\cdots\!45}{66\!\cdots\!81}a^{10}+\frac{10\!\cdots\!23}{66\!\cdots\!81}a^{9}-\frac{40\!\cdots\!11}{66\!\cdots\!81}a^{8}-\frac{56\!\cdots\!02}{66\!\cdots\!81}a^{7}+\frac{13\!\cdots\!43}{66\!\cdots\!81}a^{6}+\frac{18\!\cdots\!57}{66\!\cdots\!81}a^{5}-\frac{19\!\cdots\!55}{66\!\cdots\!81}a^{4}-\frac{30\!\cdots\!24}{66\!\cdots\!81}a^{3}+\frac{55\!\cdots\!05}{66\!\cdots\!81}a^{2}+\frac{14\!\cdots\!09}{66\!\cdots\!81}a-\frac{11\!\cdots\!47}{66\!\cdots\!81}$, $\frac{40\!\cdots\!83}{66\!\cdots\!81}a^{32}-\frac{72\!\cdots\!56}{66\!\cdots\!81}a^{31}-\frac{23\!\cdots\!92}{66\!\cdots\!81}a^{30}+\frac{90\!\cdots\!47}{66\!\cdots\!81}a^{29}-\frac{11\!\cdots\!19}{66\!\cdots\!81}a^{28}-\frac{10\!\cdots\!73}{66\!\cdots\!81}a^{27}+\frac{74\!\cdots\!71}{66\!\cdots\!81}a^{26}-\frac{12\!\cdots\!25}{66\!\cdots\!81}a^{25}-\frac{57\!\cdots\!46}{66\!\cdots\!81}a^{24}+\frac{32\!\cdots\!01}{66\!\cdots\!81}a^{23}-\frac{50\!\cdots\!34}{66\!\cdots\!81}a^{22}-\frac{60\!\cdots\!32}{66\!\cdots\!81}a^{21}-\frac{94\!\cdots\!66}{66\!\cdots\!81}a^{20}+\frac{18\!\cdots\!32}{66\!\cdots\!81}a^{19}+\frac{33\!\cdots\!64}{66\!\cdots\!81}a^{18}-\frac{19\!\cdots\!39}{66\!\cdots\!81}a^{17}-\frac{41\!\cdots\!94}{66\!\cdots\!81}a^{16}+\frac{30\!\cdots\!32}{66\!\cdots\!81}a^{15}+\frac{18\!\cdots\!70}{66\!\cdots\!81}a^{14}-\frac{11\!\cdots\!97}{66\!\cdots\!81}a^{13}+\frac{11\!\cdots\!24}{66\!\cdots\!81}a^{12}-\frac{13\!\cdots\!85}{66\!\cdots\!81}a^{11}-\frac{26\!\cdots\!67}{66\!\cdots\!81}a^{10}+\frac{16\!\cdots\!13}{66\!\cdots\!81}a^{9}+\frac{23\!\cdots\!27}{66\!\cdots\!81}a^{8}-\frac{58\!\cdots\!61}{66\!\cdots\!81}a^{7}-\frac{11\!\cdots\!12}{66\!\cdots\!81}a^{6}+\frac{44\!\cdots\!30}{66\!\cdots\!81}a^{5}+\frac{26\!\cdots\!98}{66\!\cdots\!81}a^{4}+\frac{29\!\cdots\!51}{66\!\cdots\!81}a^{3}-\frac{26\!\cdots\!09}{66\!\cdots\!81}a^{2}-\frac{26\!\cdots\!17}{66\!\cdots\!81}a+\frac{96\!\cdots\!12}{66\!\cdots\!81}$, $\frac{21\!\cdots\!04}{66\!\cdots\!81}a^{32}-\frac{55\!\cdots\!18}{66\!\cdots\!81}a^{31}-\frac{81\!\cdots\!51}{66\!\cdots\!81}a^{30}+\frac{54\!\cdots\!62}{66\!\cdots\!81}a^{29}-\frac{10\!\cdots\!58}{66\!\cdots\!81}a^{28}+\frac{25\!\cdots\!74}{66\!\cdots\!81}a^{27}+\frac{37\!\cdots\!75}{66\!\cdots\!81}a^{26}-\frac{95\!\cdots\!42}{66\!\cdots\!81}a^{25}+\frac{44\!\cdots\!67}{66\!\cdots\!81}a^{24}+\frac{13\!\cdots\!82}{66\!\cdots\!81}a^{23}-\frac{12\!\cdots\!00}{66\!\cdots\!81}a^{22}+\frac{77\!\cdots\!64}{66\!\cdots\!81}a^{21}-\frac{58\!\cdots\!68}{66\!\cdots\!81}a^{20}+\frac{54\!\cdots\!72}{66\!\cdots\!81}a^{19}+\frac{13\!\cdots\!90}{66\!\cdots\!81}a^{18}-\frac{20\!\cdots\!16}{66\!\cdots\!81}a^{17}-\frac{57\!\cdots\!17}{66\!\cdots\!81}a^{16}+\frac{19\!\cdots\!85}{66\!\cdots\!81}a^{15}-\frac{53\!\cdots\!18}{66\!\cdots\!81}a^{14}-\frac{77\!\cdots\!31}{66\!\cdots\!81}a^{13}+\frac{59\!\cdots\!08}{66\!\cdots\!81}a^{12}-\frac{12\!\cdots\!00}{66\!\cdots\!81}a^{11}-\frac{41\!\cdots\!51}{66\!\cdots\!81}a^{10}+\frac{11\!\cdots\!65}{66\!\cdots\!81}a^{9}+\frac{37\!\cdots\!58}{66\!\cdots\!81}a^{8}-\frac{46\!\cdots\!24}{66\!\cdots\!81}a^{7}-\frac{21\!\cdots\!32}{66\!\cdots\!81}a^{6}+\frac{92\!\cdots\!97}{66\!\cdots\!81}a^{5}+\frac{52\!\cdots\!37}{66\!\cdots\!81}a^{4}-\frac{53\!\cdots\!60}{66\!\cdots\!81}a^{3}-\frac{47\!\cdots\!32}{66\!\cdots\!81}a^{2}-\frac{41\!\cdots\!52}{66\!\cdots\!81}a+\frac{17\!\cdots\!74}{66\!\cdots\!81}$, $\frac{22\!\cdots\!87}{66\!\cdots\!81}a^{32}-\frac{58\!\cdots\!98}{66\!\cdots\!81}a^{31}-\frac{89\!\cdots\!42}{66\!\cdots\!81}a^{30}+\frac{58\!\cdots\!83}{66\!\cdots\!81}a^{29}-\frac{10\!\cdots\!62}{66\!\cdots\!81}a^{28}+\frac{15\!\cdots\!34}{66\!\cdots\!81}a^{27}+\frac{41\!\cdots\!23}{66\!\cdots\!81}a^{26}-\frac{10\!\cdots\!91}{66\!\cdots\!81}a^{25}+\frac{38\!\cdots\!13}{66\!\cdots\!81}a^{24}+\frac{15\!\cdots\!55}{66\!\cdots\!81}a^{23}-\frac{14\!\cdots\!91}{66\!\cdots\!81}a^{22}+\frac{57\!\cdots\!53}{66\!\cdots\!81}a^{21}-\frac{58\!\cdots\!60}{66\!\cdots\!81}a^{20}+\frac{54\!\cdots\!36}{66\!\cdots\!81}a^{19}+\frac{15\!\cdots\!39}{66\!\cdots\!81}a^{18}-\frac{22\!\cdots\!63}{66\!\cdots\!81}a^{17}-\frac{84\!\cdots\!64}{66\!\cdots\!81}a^{16}+\frac{23\!\cdots\!63}{66\!\cdots\!81}a^{15}-\frac{50\!\cdots\!04}{66\!\cdots\!81}a^{14}-\frac{38\!\cdots\!25}{66\!\cdots\!81}a^{13}+\frac{78\!\cdots\!02}{66\!\cdots\!81}a^{12}-\frac{13\!\cdots\!38}{66\!\cdots\!81}a^{11}-\frac{56\!\cdots\!96}{66\!\cdots\!81}a^{10}+\frac{13\!\cdots\!20}{66\!\cdots\!81}a^{9}+\frac{43\!\cdots\!30}{66\!\cdots\!81}a^{8}-\frac{70\!\cdots\!13}{66\!\cdots\!81}a^{7}-\frac{25\!\cdots\!18}{66\!\cdots\!81}a^{6}+\frac{19\!\cdots\!58}{66\!\cdots\!81}a^{5}+\frac{80\!\cdots\!64}{66\!\cdots\!81}a^{4}-\frac{26\!\cdots\!08}{66\!\cdots\!81}a^{3}-\frac{11\!\cdots\!06}{66\!\cdots\!81}a^{2}+\frac{12\!\cdots\!80}{66\!\cdots\!81}a+\frac{47\!\cdots\!90}{66\!\cdots\!81}$, $\frac{93\!\cdots\!66}{66\!\cdots\!81}a^{32}-\frac{20\!\cdots\!20}{66\!\cdots\!81}a^{31}-\frac{43\!\cdots\!61}{66\!\cdots\!81}a^{30}+\frac{22\!\cdots\!94}{66\!\cdots\!81}a^{29}-\frac{36\!\cdots\!29}{66\!\cdots\!81}a^{28}-\frac{67\!\cdots\!72}{66\!\cdots\!81}a^{27}+\frac{16\!\cdots\!48}{66\!\cdots\!81}a^{26}-\frac{35\!\cdots\!72}{66\!\cdots\!81}a^{25}+\frac{39\!\cdots\!99}{66\!\cdots\!81}a^{24}+\frac{68\!\cdots\!43}{66\!\cdots\!81}a^{23}-\frac{40\!\cdots\!26}{66\!\cdots\!81}a^{22}+\frac{12\!\cdots\!54}{66\!\cdots\!81}a^{21}-\frac{22\!\cdots\!42}{66\!\cdots\!81}a^{20}+\frac{13\!\cdots\!82}{66\!\cdots\!81}a^{19}+\frac{68\!\cdots\!31}{66\!\cdots\!81}a^{18}-\frac{71\!\cdots\!52}{66\!\cdots\!81}a^{17}-\frac{56\!\cdots\!98}{66\!\cdots\!81}a^{16}+\frac{83\!\cdots\!55}{66\!\cdots\!81}a^{15}-\frac{44\!\cdots\!61}{66\!\cdots\!81}a^{14}-\frac{15\!\cdots\!55}{66\!\cdots\!81}a^{13}+\frac{32\!\cdots\!03}{66\!\cdots\!81}a^{12}-\frac{46\!\cdots\!46}{66\!\cdots\!81}a^{11}-\frac{36\!\cdots\!63}{66\!\cdots\!81}a^{10}+\frac{46\!\cdots\!88}{66\!\cdots\!81}a^{9}+\frac{29\!\cdots\!38}{66\!\cdots\!81}a^{8}-\frac{19\!\cdots\!08}{66\!\cdots\!81}a^{7}-\frac{14\!\cdots\!38}{66\!\cdots\!81}a^{6}+\frac{33\!\cdots\!76}{66\!\cdots\!81}a^{5}+\frac{32\!\cdots\!23}{66\!\cdots\!81}a^{4}-\frac{12\!\cdots\!15}{66\!\cdots\!81}a^{3}-\frac{28\!\cdots\!88}{66\!\cdots\!81}a^{2}-\frac{36\!\cdots\!81}{66\!\cdots\!81}a+\frac{72\!\cdots\!72}{66\!\cdots\!81}$, $\frac{46\!\cdots\!71}{66\!\cdots\!81}a^{32}-\frac{10\!\cdots\!85}{66\!\cdots\!81}a^{31}-\frac{22\!\cdots\!59}{66\!\cdots\!81}a^{30}+\frac{11\!\cdots\!47}{66\!\cdots\!81}a^{29}-\frac{18\!\cdots\!47}{66\!\cdots\!81}a^{28}-\frac{38\!\cdots\!57}{66\!\cdots\!81}a^{27}+\frac{86\!\cdots\!51}{66\!\cdots\!81}a^{26}-\frac{18\!\cdots\!10}{66\!\cdots\!81}a^{25}+\frac{17\!\cdots\!69}{66\!\cdots\!81}a^{24}+\frac{35\!\cdots\!33}{66\!\cdots\!81}a^{23}-\frac{21\!\cdots\!49}{66\!\cdots\!81}a^{22}+\frac{53\!\cdots\!00}{66\!\cdots\!81}a^{21}-\frac{11\!\cdots\!63}{66\!\cdots\!81}a^{20}+\frac{70\!\cdots\!61}{66\!\cdots\!81}a^{19}+\frac{35\!\cdots\!49}{66\!\cdots\!81}a^{18}-\frac{36\!\cdots\!23}{66\!\cdots\!81}a^{17}-\frac{29\!\cdots\!81}{66\!\cdots\!81}a^{16}+\frac{44\!\cdots\!40}{66\!\cdots\!81}a^{15}-\frac{54\!\cdots\!77}{66\!\cdots\!81}a^{14}-\frac{89\!\cdots\!44}{66\!\cdots\!81}a^{13}+\frac{17\!\cdots\!19}{66\!\cdots\!81}a^{12}-\frac{24\!\cdots\!97}{66\!\cdots\!81}a^{11}-\frac{18\!\cdots\!59}{66\!\cdots\!81}a^{10}+\frac{24\!\cdots\!24}{66\!\cdots\!81}a^{9}+\frac{14\!\cdots\!10}{66\!\cdots\!81}a^{8}-\frac{10\!\cdots\!27}{66\!\cdots\!81}a^{7}-\frac{73\!\cdots\!93}{66\!\cdots\!81}a^{6}+\frac{20\!\cdots\!07}{66\!\cdots\!81}a^{5}+\frac{17\!\cdots\!11}{66\!\cdots\!81}a^{4}-\frac{10\!\cdots\!42}{66\!\cdots\!81}a^{3}-\frac{16\!\cdots\!19}{66\!\cdots\!81}a^{2}-\frac{15\!\cdots\!76}{66\!\cdots\!81}a+\frac{53\!\cdots\!31}{66\!\cdots\!81}$, $\frac{16\!\cdots\!98}{66\!\cdots\!81}a^{32}-\frac{50\!\cdots\!47}{66\!\cdots\!81}a^{31}-\frac{34\!\cdots\!96}{66\!\cdots\!81}a^{30}+\frac{43\!\cdots\!19}{66\!\cdots\!81}a^{29}-\frac{10\!\cdots\!50}{66\!\cdots\!81}a^{28}+\frac{68\!\cdots\!28}{66\!\cdots\!81}a^{27}+\frac{25\!\cdots\!68}{66\!\cdots\!81}a^{26}-\frac{86\!\cdots\!11}{66\!\cdots\!81}a^{25}+\frac{77\!\cdots\!71}{66\!\cdots\!81}a^{24}+\frac{70\!\cdots\!71}{66\!\cdots\!81}a^{23}-\frac{14\!\cdots\!01}{66\!\cdots\!81}a^{22}+\frac{12\!\cdots\!84}{66\!\cdots\!81}a^{21}-\frac{48\!\cdots\!47}{66\!\cdots\!81}a^{20}+\frac{65\!\cdots\!76}{66\!\cdots\!81}a^{19}+\frac{71\!\cdots\!26}{66\!\cdots\!81}a^{18}-\frac{20\!\cdots\!79}{66\!\cdots\!81}a^{17}+\frac{58\!\cdots\!77}{66\!\cdots\!81}a^{16}+\frac{14\!\cdots\!20}{66\!\cdots\!81}a^{15}-\frac{12\!\cdots\!49}{66\!\cdots\!81}a^{14}+\frac{45\!\cdots\!23}{66\!\cdots\!81}a^{13}+\frac{35\!\cdots\!19}{66\!\cdots\!81}a^{12}-\frac{11\!\cdots\!06}{66\!\cdots\!81}a^{11}+\frac{31\!\cdots\!00}{66\!\cdots\!81}a^{10}+\frac{84\!\cdots\!56}{66\!\cdots\!81}a^{9}-\frac{22\!\cdots\!29}{66\!\cdots\!81}a^{8}-\frac{37\!\cdots\!29}{66\!\cdots\!81}a^{7}+\frac{63\!\cdots\!98}{66\!\cdots\!81}a^{6}+\frac{97\!\cdots\!16}{66\!\cdots\!81}a^{5}-\frac{11\!\cdots\!66}{66\!\cdots\!81}a^{4}-\frac{95\!\cdots\!74}{66\!\cdots\!81}a^{3}+\frac{16\!\cdots\!42}{11\!\cdots\!19}a^{2}-\frac{23\!\cdots\!31}{66\!\cdots\!81}a+\frac{11\!\cdots\!28}{66\!\cdots\!81}$, $\frac{10\!\cdots\!94}{66\!\cdots\!81}a^{32}-\frac{24\!\cdots\!20}{66\!\cdots\!81}a^{31}-\frac{47\!\cdots\!59}{66\!\cdots\!81}a^{30}+\frac{26\!\cdots\!13}{66\!\cdots\!81}a^{29}-\frac{44\!\cdots\!62}{66\!\cdots\!81}a^{28}-\frac{31\!\cdots\!76}{66\!\cdots\!81}a^{27}+\frac{19\!\cdots\!27}{66\!\cdots\!81}a^{26}-\frac{42\!\cdots\!31}{66\!\cdots\!81}a^{25}+\frac{84\!\cdots\!85}{66\!\cdots\!81}a^{24}+\frac{75\!\cdots\!78}{66\!\cdots\!81}a^{23}-\frac{50\!\cdots\!25}{66\!\cdots\!81}a^{22}+\frac{19\!\cdots\!05}{66\!\cdots\!81}a^{21}-\frac{26\!\cdots\!23}{66\!\cdots\!81}a^{20}+\frac{18\!\cdots\!53}{66\!\cdots\!81}a^{19}+\frac{75\!\cdots\!64}{66\!\cdots\!81}a^{18}-\frac{86\!\cdots\!97}{66\!\cdots\!81}a^{17}-\frac{56\!\cdots\!95}{66\!\cdots\!81}a^{16}+\frac{96\!\cdots\!98}{66\!\cdots\!81}a^{15}-\frac{70\!\cdots\!09}{66\!\cdots\!81}a^{14}-\frac{14\!\cdots\!44}{66\!\cdots\!81}a^{13}+\frac{36\!\cdots\!01}{66\!\cdots\!81}a^{12}-\frac{55\!\cdots\!56}{66\!\cdots\!81}a^{11}-\frac{37\!\cdots\!89}{66\!\cdots\!81}a^{10}+\frac{54\!\cdots\!83}{66\!\cdots\!81}a^{9}+\frac{29\!\cdots\!51}{66\!\cdots\!81}a^{8}-\frac{22\!\cdots\!64}{66\!\cdots\!81}a^{7}-\frac{14\!\cdots\!98}{66\!\cdots\!81}a^{6}+\frac{43\!\cdots\!86}{66\!\cdots\!81}a^{5}+\frac{34\!\cdots\!15}{66\!\cdots\!81}a^{4}-\frac{25\!\cdots\!19}{66\!\cdots\!81}a^{3}-\frac{30\!\cdots\!66}{66\!\cdots\!81}a^{2}+\frac{22\!\cdots\!28}{66\!\cdots\!81}a+\frac{76\!\cdots\!36}{66\!\cdots\!81}$, $\frac{12\!\cdots\!22}{66\!\cdots\!81}a^{32}-\frac{34\!\cdots\!90}{66\!\cdots\!81}a^{31}-\frac{47\!\cdots\!75}{66\!\cdots\!81}a^{30}+\frac{33\!\cdots\!78}{66\!\cdots\!81}a^{29}-\frac{64\!\cdots\!56}{66\!\cdots\!81}a^{28}+\frac{12\!\cdots\!75}{66\!\cdots\!81}a^{27}+\frac{24\!\cdots\!77}{66\!\cdots\!81}a^{26}-\frac{60\!\cdots\!27}{66\!\cdots\!81}a^{25}+\frac{27\!\cdots\!56}{66\!\cdots\!81}a^{24}+\frac{93\!\cdots\!88}{66\!\cdots\!81}a^{23}-\frac{10\!\cdots\!98}{66\!\cdots\!81}a^{22}+\frac{43\!\cdots\!88}{66\!\cdots\!81}a^{21}-\frac{32\!\cdots\!14}{66\!\cdots\!81}a^{20}+\frac{33\!\cdots\!51}{66\!\cdots\!81}a^{19}+\frac{87\!\cdots\!85}{66\!\cdots\!81}a^{18}-\frac{14\!\cdots\!38}{66\!\cdots\!81}a^{17}-\frac{32\!\cdots\!09}{66\!\cdots\!81}a^{16}+\frac{15\!\cdots\!08}{66\!\cdots\!81}a^{15}-\frac{58\!\cdots\!78}{66\!\cdots\!81}a^{14}-\frac{19\!\cdots\!27}{66\!\cdots\!81}a^{13}+\frac{58\!\cdots\!41}{66\!\cdots\!81}a^{12}-\frac{89\!\cdots\!03}{66\!\cdots\!81}a^{11}-\frac{19\!\cdots\!00}{66\!\cdots\!81}a^{10}+\frac{89\!\cdots\!41}{66\!\cdots\!81}a^{9}+\frac{85\!\cdots\!06}{66\!\cdots\!81}a^{8}-\frac{44\!\cdots\!57}{66\!\cdots\!81}a^{7}-\frac{60\!\cdots\!92}{66\!\cdots\!81}a^{6}+\frac{12\!\cdots\!79}{66\!\cdots\!81}a^{5}+\frac{18\!\cdots\!58}{66\!\cdots\!81}a^{4}-\frac{19\!\cdots\!04}{66\!\cdots\!81}a^{3}-\frac{15\!\cdots\!84}{66\!\cdots\!81}a^{2}+\frac{13\!\cdots\!64}{66\!\cdots\!81}a+\frac{81\!\cdots\!73}{66\!\cdots\!81}$, $\frac{30\!\cdots\!29}{66\!\cdots\!81}a^{32}-\frac{75\!\cdots\!81}{66\!\cdots\!81}a^{31}-\frac{12\!\cdots\!23}{66\!\cdots\!81}a^{30}+\frac{76\!\cdots\!86}{66\!\cdots\!81}a^{29}-\frac{14\!\cdots\!77}{66\!\cdots\!81}a^{28}+\frac{21\!\cdots\!04}{66\!\cdots\!81}a^{27}+\frac{53\!\cdots\!17}{66\!\cdots\!81}a^{26}-\frac{12\!\cdots\!03}{66\!\cdots\!81}a^{25}+\frac{50\!\cdots\!68}{66\!\cdots\!81}a^{24}+\frac{19\!\cdots\!54}{66\!\cdots\!81}a^{23}-\frac{16\!\cdots\!22}{66\!\cdots\!81}a^{22}+\frac{10\!\cdots\!81}{66\!\cdots\!81}a^{21}-\frac{81\!\cdots\!81}{66\!\cdots\!81}a^{20}+\frac{67\!\cdots\!36}{66\!\cdots\!81}a^{19}+\frac{19\!\cdots\!92}{66\!\cdots\!81}a^{18}-\frac{26\!\cdots\!56}{66\!\cdots\!81}a^{17}-\frac{97\!\cdots\!45}{66\!\cdots\!81}a^{16}+\frac{25\!\cdots\!73}{66\!\cdots\!81}a^{15}-\frac{65\!\cdots\!76}{66\!\cdots\!81}a^{14}+\frac{28\!\cdots\!86}{66\!\cdots\!81}a^{13}+\frac{88\!\cdots\!96}{66\!\cdots\!81}a^{12}-\frac{16\!\cdots\!51}{66\!\cdots\!81}a^{11}-\frac{69\!\cdots\!01}{66\!\cdots\!81}a^{10}+\frac{14\!\cdots\!49}{66\!\cdots\!81}a^{9}+\frac{59\!\cdots\!20}{66\!\cdots\!81}a^{8}-\frac{56\!\cdots\!97}{66\!\cdots\!81}a^{7}-\frac{29\!\cdots\!96}{66\!\cdots\!81}a^{6}+\frac{99\!\cdots\!32}{66\!\cdots\!81}a^{5}+\frac{64\!\cdots\!71}{66\!\cdots\!81}a^{4}-\frac{36\!\cdots\!75}{66\!\cdots\!81}a^{3}-\frac{55\!\cdots\!60}{66\!\cdots\!81}a^{2}-\frac{12\!\cdots\!30}{66\!\cdots\!81}a+\frac{92\!\cdots\!49}{66\!\cdots\!81}$, $\frac{82\!\cdots\!92}{66\!\cdots\!81}a^{32}-\frac{19\!\cdots\!28}{66\!\cdots\!81}a^{31}-\frac{31\!\cdots\!81}{66\!\cdots\!81}a^{30}+\frac{19\!\cdots\!11}{66\!\cdots\!81}a^{29}-\frac{36\!\cdots\!43}{66\!\cdots\!81}a^{28}+\frac{86\!\cdots\!69}{66\!\cdots\!81}a^{27}+\frac{13\!\cdots\!86}{66\!\cdots\!81}a^{26}-\frac{32\!\cdots\!79}{66\!\cdots\!81}a^{25}+\frac{14\!\cdots\!89}{66\!\cdots\!81}a^{24}+\frac{42\!\cdots\!73}{66\!\cdots\!81}a^{23}-\frac{30\!\cdots\!64}{66\!\cdots\!81}a^{22}+\frac{30\!\cdots\!25}{66\!\cdots\!81}a^{21}-\frac{22\!\cdots\!12}{66\!\cdots\!81}a^{20}+\frac{18\!\cdots\!47}{66\!\cdots\!81}a^{19}+\frac{47\!\cdots\!87}{66\!\cdots\!81}a^{18}-\frac{59\!\cdots\!13}{66\!\cdots\!81}a^{17}-\frac{24\!\cdots\!93}{66\!\cdots\!81}a^{16}+\frac{45\!\cdots\!28}{66\!\cdots\!81}a^{15}-\frac{51\!\cdots\!33}{66\!\cdots\!81}a^{14}+\frac{95\!\cdots\!43}{66\!\cdots\!81}a^{13}+\frac{13\!\cdots\!57}{66\!\cdots\!81}a^{12}-\frac{36\!\cdots\!90}{66\!\cdots\!81}a^{11}-\frac{20\!\cdots\!22}{66\!\cdots\!81}a^{10}+\frac{26\!\cdots\!73}{66\!\cdots\!81}a^{9}+\frac{34\!\cdots\!72}{11\!\cdots\!19}a^{8}-\frac{68\!\cdots\!17}{66\!\cdots\!81}a^{7}-\frac{95\!\cdots\!05}{66\!\cdots\!81}a^{6}-\frac{80\!\cdots\!28}{66\!\cdots\!81}a^{5}+\frac{18\!\cdots\!37}{66\!\cdots\!81}a^{4}+\frac{59\!\cdots\!06}{66\!\cdots\!81}a^{3}-\frac{10\!\cdots\!76}{66\!\cdots\!81}a^{2}-\frac{63\!\cdots\!88}{66\!\cdots\!81}a+\frac{23\!\cdots\!08}{66\!\cdots\!81}$, $\frac{10\!\cdots\!34}{66\!\cdots\!81}a^{32}-\frac{24\!\cdots\!54}{66\!\cdots\!81}a^{31}-\frac{51\!\cdots\!86}{66\!\cdots\!81}a^{30}+\frac{26\!\cdots\!81}{66\!\cdots\!81}a^{29}-\frac{42\!\cdots\!34}{66\!\cdots\!81}a^{28}-\frac{87\!\cdots\!29}{66\!\cdots\!81}a^{27}+\frac{20\!\cdots\!27}{66\!\cdots\!81}a^{26}-\frac{41\!\cdots\!66}{66\!\cdots\!81}a^{25}+\frac{38\!\cdots\!13}{66\!\cdots\!81}a^{24}+\frac{81\!\cdots\!57}{66\!\cdots\!81}a^{23}-\frac{46\!\cdots\!15}{66\!\cdots\!81}a^{22}+\frac{11\!\cdots\!09}{66\!\cdots\!81}a^{21}-\frac{26\!\cdots\!38}{66\!\cdots\!81}a^{20}+\frac{16\!\cdots\!28}{66\!\cdots\!81}a^{19}+\frac{81\!\cdots\!42}{66\!\cdots\!81}a^{18}-\frac{83\!\cdots\!59}{66\!\cdots\!81}a^{17}-\frac{69\!\cdots\!26}{66\!\cdots\!81}a^{16}+\frac{10\!\cdots\!42}{66\!\cdots\!81}a^{15}+\frac{26\!\cdots\!34}{66\!\cdots\!81}a^{14}-\frac{21\!\cdots\!07}{66\!\cdots\!81}a^{13}+\frac{38\!\cdots\!25}{66\!\cdots\!81}a^{12}-\frac{54\!\cdots\!58}{66\!\cdots\!81}a^{11}-\frac{45\!\cdots\!56}{66\!\cdots\!81}a^{10}+\frac{56\!\cdots\!83}{66\!\cdots\!81}a^{9}+\frac{36\!\cdots\!72}{66\!\cdots\!81}a^{8}-\frac{24\!\cdots\!46}{66\!\cdots\!81}a^{7}-\frac{17\!\cdots\!62}{66\!\cdots\!81}a^{6}+\frac{45\!\cdots\!76}{66\!\cdots\!81}a^{5}+\frac{43\!\cdots\!44}{66\!\cdots\!81}a^{4}-\frac{23\!\cdots\!88}{66\!\cdots\!81}a^{3}-\frac{42\!\cdots\!43}{66\!\cdots\!81}a^{2}-\frac{15\!\cdots\!91}{66\!\cdots\!81}a+\frac{12\!\cdots\!94}{66\!\cdots\!81}$
| 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: | \( 255776199.8931762 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{11}\cdot 255776199.8931762 \cdot 1}{2\cdot\sqrt{1635170022196481349560959748587682926364327}}\cr\approx \mathstrut & 0.123411940469866 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1)
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^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1, 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^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1);
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^33 - 3*x^32 - 3*x^31 + 28*x^30 - 58*x^29 + 22*x^28 + 191*x^27 - 526*x^26 + 330*x^25 + 728*x^24 - 1012*x^23 + 413*x^22 - 2524*x^21 + 3437*x^20 + 6311*x^19 - 13555*x^18 - 573*x^17 + 14501*x^16 - 6805*x^15 - 2479*x^14 + 5007*x^13 - 7594*x^12 - 344*x^11 + 8602*x^10 - 626*x^9 - 5041*x^8 + 15*x^7 + 1789*x^6 + 121*x^5 - 352*x^4 - 33*x^3 + 32*x^2 + 2*x - 1);
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
$S_3\times C_{11}$ (as 33T2):
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 66 |
| The 33 conjugacy class representatives for $S_3\times C_{11}$ |
| Character table for $S_3\times C_{11}$ is not computed |
Intermediate fields
| 3.1.23.1, \(\Q(\zeta_{23})^+\) |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $33$ | $33$ | $22{,}\,{\href{/padicField/5.11.0.1}{11} }$ | $22{,}\,{\href{/padicField/7.11.0.1}{11} }$ | $22{,}\,{\href{/padicField/11.11.0.1}{11} }$ | $33$ | $22{,}\,{\href{/padicField/17.11.0.1}{11} }$ | $22{,}\,{\href{/padicField/19.11.0.1}{11} }$ | R | $33$ | $33$ | $22{,}\,{\href{/padicField/37.11.0.1}{11} }$ | $33$ | $22{,}\,{\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.3.0.1}{3} }^{11}$ | $22{,}\,{\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.11.10.10 | $x^{11} + 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.22.21.17 | $x^{22} + 23$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.23.2t1.a.a | $1$ | $ 23 $ | \(\Q(\sqrt{-23}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.23.11t1.a.a | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| 1.23.22t1.a.a | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.b | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| * | 1.23.11t1.a.b | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.c | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.d | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.e | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.f | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| 1.23.22t1.a.c | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.d | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.e | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.f | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.g | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| * | 1.23.11t1.a.g | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| 1.23.22t1.a.h | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.i | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| 1.23.22t1.a.j | $1$ | $ 23 $ | \(\Q(\zeta_{23})\) | $C_{22}$ (as 22T1) | $0$ | $-1$ | |
| * | 1.23.11t1.a.h | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.i | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 1.23.11t1.a.j | $1$ | $ 23 $ | \(\Q(\zeta_{23})^+\) | $C_{11}$ (as 11T1) | $0$ | $1$ |
| * | 2.23.3t2.b.a | $2$ | $ 23 $ | 3.1.23.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.529.33t2.a.a | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.b | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.c | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.d | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.e | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.f | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.g | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.h | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.i | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
| * | 2.529.33t2.a.j | $2$ | $ 23^{2}$ | 33.11.1635170022196481349560959748587682926364327.1 | $S_3\times C_{11}$ (as 33T2) | $0$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.