Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79)
gp: K = bnfinit(y^32 - 5*y^31 + 22*y^30 - 73*y^29 + 220*y^28 - 519*y^27 + 1255*y^26 - 2247*y^25 + 4413*y^24 - 6238*y^23 + 9744*y^22 - 10830*y^21 + 13240*y^20 - 10628*y^19 + 12053*y^18 - 5501*y^17 + 12591*y^16 - 3062*y^15 + 17256*y^14 + 194*y^13 + 18165*y^12 + 5659*y^11 + 13232*y^10 - 2318*y^9 - 319*y^8 - 13445*y^7 - 6940*y^6 - 4990*y^5 + 1466*y^4 + 2074*y^3 + 1626*y^2 + 264*y + 79, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79)
\( x^{32} - 5 x^{31} + 22 x^{30} - 73 x^{29} + 220 x^{28} - 519 x^{27} + 1255 x^{26} - 2247 x^{25} + \cdots + 79 \)
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: |
\(515207889456069254383620937908064472169867121\)
\(\medspace = 7^{16}\cdot 13^{28}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.96\) | 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: | $7^{2/3}13^{7/8}\approx 34.52225265079909$ | ||
| Ramified primes: |
\(7\), \(13\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{3}a^{29}+\frac{1}{3}a^{28}+\frac{1}{3}a^{27}+\frac{1}{3}a^{26}-\frac{1}{3}a^{24}+\frac{1}{3}a^{21}-\frac{1}{3}a^{20}-\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}$, $\frac{1}{387}a^{30}+\frac{47}{387}a^{29}-\frac{190}{387}a^{28}-\frac{109}{387}a^{27}-\frac{38}{387}a^{26}-\frac{145}{387}a^{25}-\frac{4}{9}a^{24}+\frac{29}{129}a^{23}-\frac{179}{387}a^{22}-\frac{11}{43}a^{21}+\frac{140}{387}a^{20}+\frac{104}{387}a^{19}+\frac{139}{387}a^{18}+\frac{59}{129}a^{17}-\frac{14}{387}a^{16}+\frac{56}{387}a^{15}+\frac{55}{387}a^{14}+\frac{19}{129}a^{13}-\frac{85}{387}a^{12}-\frac{44}{387}a^{11}+\frac{149}{387}a^{10}-\frac{148}{387}a^{9}-\frac{112}{387}a^{8}-\frac{119}{387}a^{7}-\frac{59}{387}a^{6}+\frac{13}{387}a^{5}-\frac{103}{387}a^{4}+\frac{2}{43}a^{3}+\frac{13}{129}a^{2}+\frac{25}{387}a-\frac{38}{387}$, $\frac{1}{18\!\cdots\!19}a^{31}+\frac{98\!\cdots\!55}{60\!\cdots\!73}a^{30}+\frac{12\!\cdots\!53}{18\!\cdots\!19}a^{29}+\frac{39\!\cdots\!46}{18\!\cdots\!19}a^{28}-\frac{19\!\cdots\!49}{60\!\cdots\!73}a^{27}+\frac{44\!\cdots\!93}{20\!\cdots\!91}a^{26}-\frac{51\!\cdots\!06}{18\!\cdots\!19}a^{25}-\frac{61\!\cdots\!41}{18\!\cdots\!19}a^{24}+\frac{78\!\cdots\!39}{18\!\cdots\!19}a^{23}+\frac{43\!\cdots\!03}{18\!\cdots\!19}a^{22}+\frac{15\!\cdots\!09}{18\!\cdots\!19}a^{21}+\frac{69\!\cdots\!88}{18\!\cdots\!19}a^{20}-\frac{28\!\cdots\!20}{60\!\cdots\!73}a^{19}+\frac{96\!\cdots\!31}{18\!\cdots\!19}a^{18}-\frac{86\!\cdots\!54}{18\!\cdots\!19}a^{17}-\frac{63\!\cdots\!43}{24\!\cdots\!11}a^{16}+\frac{81\!\cdots\!36}{20\!\cdots\!91}a^{15}+\frac{49\!\cdots\!34}{18\!\cdots\!19}a^{14}+\frac{79\!\cdots\!39}{18\!\cdots\!19}a^{13}-\frac{11\!\cdots\!88}{60\!\cdots\!73}a^{12}-\frac{34\!\cdots\!25}{67\!\cdots\!97}a^{11}-\frac{51\!\cdots\!56}{18\!\cdots\!19}a^{10}+\frac{20\!\cdots\!17}{42\!\cdots\!33}a^{9}+\frac{22\!\cdots\!61}{20\!\cdots\!91}a^{8}-\frac{70\!\cdots\!17}{18\!\cdots\!19}a^{7}-\frac{78\!\cdots\!99}{18\!\cdots\!19}a^{6}+\frac{26\!\cdots\!26}{60\!\cdots\!73}a^{5}+\frac{85\!\cdots\!02}{18\!\cdots\!19}a^{4}-\frac{20\!\cdots\!54}{60\!\cdots\!73}a^{3}+\frac{27\!\cdots\!18}{18\!\cdots\!19}a^{2}+\frac{16\!\cdots\!68}{18\!\cdots\!19}a-\frac{88\!\cdots\!55}{22\!\cdots\!61}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: | $15$ | 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: |
$\frac{13\!\cdots\!12}{84\!\cdots\!43}a^{31}-\frac{94\!\cdots\!21}{94\!\cdots\!27}a^{30}+\frac{38\!\cdots\!27}{84\!\cdots\!43}a^{29}-\frac{13\!\cdots\!59}{84\!\cdots\!43}a^{28}+\frac{15\!\cdots\!66}{31\!\cdots\!09}a^{27}-\frac{34\!\cdots\!50}{28\!\cdots\!81}a^{26}+\frac{25\!\cdots\!83}{84\!\cdots\!43}a^{25}-\frac{49\!\cdots\!14}{84\!\cdots\!43}a^{24}+\frac{92\!\cdots\!69}{84\!\cdots\!43}a^{23}-\frac{14\!\cdots\!44}{84\!\cdots\!43}a^{22}+\frac{21\!\cdots\!75}{84\!\cdots\!43}a^{21}-\frac{26\!\cdots\!63}{84\!\cdots\!43}a^{20}+\frac{99\!\cdots\!75}{28\!\cdots\!81}a^{19}-\frac{26\!\cdots\!14}{84\!\cdots\!43}a^{18}+\frac{23\!\cdots\!66}{84\!\cdots\!43}a^{17}-\frac{47\!\cdots\!35}{28\!\cdots\!81}a^{16}+\frac{51\!\cdots\!54}{28\!\cdots\!81}a^{15}-\frac{13\!\cdots\!89}{84\!\cdots\!43}a^{14}+\frac{23\!\cdots\!67}{84\!\cdots\!43}a^{13}-\frac{20\!\cdots\!91}{94\!\cdots\!27}a^{12}+\frac{69\!\cdots\!24}{28\!\cdots\!81}a^{11}-\frac{88\!\cdots\!09}{84\!\cdots\!43}a^{10}+\frac{54\!\cdots\!53}{84\!\cdots\!43}a^{9}-\frac{40\!\cdots\!99}{28\!\cdots\!81}a^{8}+\frac{26\!\cdots\!61}{84\!\cdots\!43}a^{7}-\frac{11\!\cdots\!44}{84\!\cdots\!43}a^{6}+\frac{23\!\cdots\!11}{28\!\cdots\!81}a^{5}+\frac{50\!\cdots\!65}{84\!\cdots\!43}a^{4}+\frac{10\!\cdots\!66}{28\!\cdots\!81}a^{3}+\frac{87\!\cdots\!98}{84\!\cdots\!43}a^{2}-\frac{13\!\cdots\!74}{84\!\cdots\!43}a-\frac{10\!\cdots\!84}{84\!\cdots\!43}$, $\frac{14\!\cdots\!57}{84\!\cdots\!43}a^{31}-\frac{89\!\cdots\!58}{94\!\cdots\!27}a^{30}+\frac{35\!\cdots\!76}{84\!\cdots\!43}a^{29}-\frac{12\!\cdots\!73}{84\!\cdots\!43}a^{28}+\frac{13\!\cdots\!18}{31\!\cdots\!09}a^{27}-\frac{30\!\cdots\!93}{28\!\cdots\!81}a^{26}+\frac{21\!\cdots\!38}{84\!\cdots\!43}a^{25}-\frac{41\!\cdots\!50}{84\!\cdots\!43}a^{24}+\frac{78\!\cdots\!44}{84\!\cdots\!43}a^{23}-\frac{11\!\cdots\!46}{84\!\cdots\!43}a^{22}+\frac{17\!\cdots\!17}{84\!\cdots\!43}a^{21}-\frac{20\!\cdots\!65}{84\!\cdots\!43}a^{20}+\frac{73\!\cdots\!13}{28\!\cdots\!81}a^{19}-\frac{17\!\cdots\!43}{84\!\cdots\!43}a^{18}+\frac{14\!\cdots\!43}{84\!\cdots\!43}a^{17}-\frac{14\!\cdots\!90}{28\!\cdots\!81}a^{16}+\frac{31\!\cdots\!12}{28\!\cdots\!81}a^{15}-\frac{22\!\cdots\!16}{84\!\cdots\!43}a^{14}+\frac{18\!\cdots\!29}{84\!\cdots\!43}a^{13}-\frac{36\!\cdots\!01}{94\!\cdots\!27}a^{12}+\frac{57\!\cdots\!14}{28\!\cdots\!81}a^{11}+\frac{65\!\cdots\!19}{84\!\cdots\!43}a^{10}+\frac{63\!\cdots\!79}{84\!\cdots\!43}a^{9}-\frac{95\!\cdots\!84}{28\!\cdots\!81}a^{8}-\frac{90\!\cdots\!51}{84\!\cdots\!43}a^{7}-\frac{13\!\cdots\!56}{84\!\cdots\!43}a^{6}-\frac{16\!\cdots\!85}{28\!\cdots\!81}a^{5}+\frac{83\!\cdots\!48}{84\!\cdots\!43}a^{4}-\frac{60\!\cdots\!89}{28\!\cdots\!81}a^{3}+\frac{25\!\cdots\!93}{84\!\cdots\!43}a^{2}+\frac{15\!\cdots\!19}{84\!\cdots\!43}a+\frac{59\!\cdots\!63}{84\!\cdots\!43}$, $\frac{12\!\cdots\!27}{18\!\cdots\!19}a^{31}-\frac{89\!\cdots\!68}{60\!\cdots\!73}a^{30}+\frac{15\!\cdots\!00}{18\!\cdots\!19}a^{29}-\frac{66\!\cdots\!84}{18\!\cdots\!19}a^{28}+\frac{77\!\cdots\!38}{60\!\cdots\!73}a^{27}-\frac{76\!\cdots\!26}{20\!\cdots\!91}a^{26}+\frac{17\!\cdots\!96}{18\!\cdots\!19}a^{25}-\frac{40\!\cdots\!53}{18\!\cdots\!19}a^{24}+\frac{80\!\cdots\!30}{18\!\cdots\!19}a^{23}-\frac{14\!\cdots\!65}{18\!\cdots\!19}a^{22}+\frac{23\!\cdots\!38}{18\!\cdots\!19}a^{21}-\frac{32\!\cdots\!76}{18\!\cdots\!19}a^{20}+\frac{14\!\cdots\!04}{60\!\cdots\!73}a^{19}-\frac{45\!\cdots\!19}{18\!\cdots\!19}a^{18}+\frac{44\!\cdots\!41}{18\!\cdots\!19}a^{17}-\frac{47\!\cdots\!77}{24\!\cdots\!11}a^{16}+\frac{32\!\cdots\!84}{20\!\cdots\!91}a^{15}-\frac{21\!\cdots\!01}{18\!\cdots\!19}a^{14}+\frac{32\!\cdots\!00}{18\!\cdots\!19}a^{13}-\frac{84\!\cdots\!48}{60\!\cdots\!73}a^{12}+\frac{16\!\cdots\!39}{67\!\cdots\!97}a^{11}-\frac{12\!\cdots\!46}{18\!\cdots\!19}a^{10}+\frac{42\!\cdots\!49}{18\!\cdots\!19}a^{9}+\frac{18\!\cdots\!58}{20\!\cdots\!91}a^{8}+\frac{52\!\cdots\!28}{18\!\cdots\!19}a^{7}+\frac{21\!\cdots\!51}{18\!\cdots\!19}a^{6}+\frac{12\!\cdots\!88}{60\!\cdots\!73}a^{5}-\frac{51\!\cdots\!79}{18\!\cdots\!19}a^{4}-\frac{33\!\cdots\!98}{60\!\cdots\!73}a^{3}-\frac{17\!\cdots\!87}{18\!\cdots\!19}a^{2}-\frac{93\!\cdots\!88}{18\!\cdots\!19}a-\frac{48\!\cdots\!13}{22\!\cdots\!61}$, $\frac{12\!\cdots\!11}{42\!\cdots\!33}a^{31}-\frac{92\!\cdots\!47}{60\!\cdots\!73}a^{30}+\frac{12\!\cdots\!24}{18\!\cdots\!19}a^{29}-\frac{41\!\cdots\!35}{18\!\cdots\!19}a^{28}+\frac{41\!\cdots\!21}{60\!\cdots\!73}a^{27}-\frac{33\!\cdots\!51}{20\!\cdots\!91}a^{26}+\frac{73\!\cdots\!45}{18\!\cdots\!19}a^{25}-\frac{31\!\cdots\!88}{42\!\cdots\!33}a^{24}+\frac{26\!\cdots\!96}{18\!\cdots\!19}a^{23}-\frac{38\!\cdots\!82}{18\!\cdots\!19}a^{22}+\frac{60\!\cdots\!21}{18\!\cdots\!19}a^{21}-\frac{69\!\cdots\!32}{18\!\cdots\!19}a^{20}+\frac{28\!\cdots\!72}{60\!\cdots\!73}a^{19}-\frac{73\!\cdots\!26}{18\!\cdots\!19}a^{18}+\frac{81\!\cdots\!61}{18\!\cdots\!19}a^{17}-\frac{17\!\cdots\!41}{74\!\cdots\!33}a^{16}+\frac{89\!\cdots\!28}{20\!\cdots\!91}a^{15}-\frac{21\!\cdots\!80}{18\!\cdots\!19}a^{14}+\frac{10\!\cdots\!37}{18\!\cdots\!19}a^{13}-\frac{17\!\cdots\!03}{60\!\cdots\!73}a^{12}+\frac{44\!\cdots\!23}{67\!\cdots\!97}a^{11}+\frac{29\!\cdots\!61}{18\!\cdots\!19}a^{10}+\frac{89\!\cdots\!12}{18\!\cdots\!19}a^{9}-\frac{54\!\cdots\!24}{20\!\cdots\!91}a^{8}+\frac{10\!\cdots\!88}{18\!\cdots\!19}a^{7}-\frac{74\!\cdots\!34}{18\!\cdots\!19}a^{6}-\frac{11\!\cdots\!55}{60\!\cdots\!73}a^{5}-\frac{34\!\cdots\!88}{18\!\cdots\!19}a^{4}+\frac{16\!\cdots\!24}{60\!\cdots\!73}a^{3}+\frac{10\!\cdots\!58}{18\!\cdots\!19}a^{2}+\frac{11\!\cdots\!30}{18\!\cdots\!19}a+\frac{17\!\cdots\!31}{22\!\cdots\!61}$, $\frac{20\!\cdots\!55}{18\!\cdots\!19}a^{31}-\frac{63\!\cdots\!86}{60\!\cdots\!73}a^{30}+\frac{90\!\cdots\!32}{18\!\cdots\!19}a^{29}-\frac{34\!\cdots\!83}{18\!\cdots\!19}a^{28}+\frac{37\!\cdots\!54}{60\!\cdots\!73}a^{27}-\frac{34\!\cdots\!52}{20\!\cdots\!91}a^{26}+\frac{73\!\cdots\!43}{18\!\cdots\!19}a^{25}-\frac{16\!\cdots\!05}{18\!\cdots\!19}a^{24}+\frac{30\!\cdots\!49}{18\!\cdots\!19}a^{23}-\frac{53\!\cdots\!84}{18\!\cdots\!19}a^{22}+\frac{78\!\cdots\!74}{18\!\cdots\!19}a^{21}-\frac{11\!\cdots\!09}{18\!\cdots\!19}a^{20}+\frac{42\!\cdots\!68}{60\!\cdots\!73}a^{19}-\frac{14\!\cdots\!50}{18\!\cdots\!19}a^{18}+\frac{12\!\cdots\!31}{18\!\cdots\!19}a^{17}-\frac{14\!\cdots\!23}{22\!\cdots\!99}a^{16}+\frac{77\!\cdots\!28}{20\!\cdots\!91}a^{15}-\frac{10\!\cdots\!25}{18\!\cdots\!19}a^{14}+\frac{61\!\cdots\!49}{18\!\cdots\!19}a^{13}-\frac{48\!\cdots\!21}{60\!\cdots\!73}a^{12}+\frac{15\!\cdots\!68}{67\!\cdots\!97}a^{11}-\frac{13\!\cdots\!19}{18\!\cdots\!19}a^{10}-\frac{21\!\cdots\!05}{18\!\cdots\!19}a^{9}-\frac{11\!\cdots\!35}{20\!\cdots\!91}a^{8}+\frac{15\!\cdots\!92}{18\!\cdots\!19}a^{7}-\frac{28\!\cdots\!97}{18\!\cdots\!19}a^{6}+\frac{28\!\cdots\!50}{60\!\cdots\!73}a^{5}+\frac{35\!\cdots\!68}{18\!\cdots\!19}a^{4}+\frac{99\!\cdots\!60}{60\!\cdots\!73}a^{3}-\frac{71\!\cdots\!68}{18\!\cdots\!19}a^{2}-\frac{12\!\cdots\!97}{18\!\cdots\!19}a-\frac{10\!\cdots\!75}{22\!\cdots\!61}$, $\frac{44\!\cdots\!35}{18\!\cdots\!19}a^{31}-\frac{70\!\cdots\!78}{60\!\cdots\!73}a^{30}+\frac{92\!\cdots\!84}{18\!\cdots\!19}a^{29}-\frac{29\!\cdots\!12}{18\!\cdots\!19}a^{28}+\frac{29\!\cdots\!75}{60\!\cdots\!73}a^{27}-\frac{22\!\cdots\!20}{20\!\cdots\!91}a^{26}+\frac{48\!\cdots\!57}{18\!\cdots\!19}a^{25}-\frac{80\!\cdots\!02}{18\!\cdots\!19}a^{24}+\frac{15\!\cdots\!78}{18\!\cdots\!19}a^{23}-\frac{20\!\cdots\!45}{18\!\cdots\!19}a^{22}+\frac{32\!\cdots\!45}{18\!\cdots\!19}a^{21}-\frac{29\!\cdots\!07}{18\!\cdots\!19}a^{20}+\frac{12\!\cdots\!66}{60\!\cdots\!73}a^{19}-\frac{44\!\cdots\!92}{42\!\cdots\!33}a^{18}+\frac{31\!\cdots\!64}{18\!\cdots\!19}a^{17}+\frac{47\!\cdots\!44}{74\!\cdots\!33}a^{16}+\frac{51\!\cdots\!93}{20\!\cdots\!91}a^{15}+\frac{90\!\cdots\!18}{18\!\cdots\!19}a^{14}+\frac{77\!\cdots\!19}{18\!\cdots\!19}a^{13}+\frac{12\!\cdots\!36}{60\!\cdots\!73}a^{12}+\frac{32\!\cdots\!85}{67\!\cdots\!97}a^{11}+\frac{72\!\cdots\!76}{18\!\cdots\!19}a^{10}+\frac{85\!\cdots\!78}{18\!\cdots\!19}a^{9}+\frac{40\!\cdots\!14}{20\!\cdots\!91}a^{8}+\frac{26\!\cdots\!25}{18\!\cdots\!19}a^{7}-\frac{26\!\cdots\!58}{18\!\cdots\!19}a^{6}-\frac{94\!\cdots\!11}{60\!\cdots\!73}a^{5}-\frac{23\!\cdots\!95}{18\!\cdots\!19}a^{4}-\frac{18\!\cdots\!03}{60\!\cdots\!73}a^{3}-\frac{20\!\cdots\!00}{18\!\cdots\!19}a^{2}+\frac{17\!\cdots\!95}{18\!\cdots\!19}a-\frac{41\!\cdots\!93}{22\!\cdots\!61}$, $\frac{61\!\cdots\!78}{18\!\cdots\!19}a^{31}-\frac{10\!\cdots\!93}{60\!\cdots\!73}a^{30}+\frac{13\!\cdots\!43}{18\!\cdots\!19}a^{29}-\frac{44\!\cdots\!42}{18\!\cdots\!19}a^{28}+\frac{45\!\cdots\!54}{60\!\cdots\!73}a^{27}-\frac{36\!\cdots\!63}{20\!\cdots\!91}a^{26}+\frac{80\!\cdots\!28}{18\!\cdots\!19}a^{25}-\frac{14\!\cdots\!12}{18\!\cdots\!19}a^{24}+\frac{29\!\cdots\!43}{18\!\cdots\!19}a^{23}-\frac{42\!\cdots\!42}{18\!\cdots\!19}a^{22}+\frac{69\!\cdots\!69}{18\!\cdots\!19}a^{21}-\frac{79\!\cdots\!91}{18\!\cdots\!19}a^{20}+\frac{34\!\cdots\!68}{60\!\cdots\!73}a^{19}-\frac{90\!\cdots\!96}{18\!\cdots\!19}a^{18}+\frac{10\!\cdots\!39}{18\!\cdots\!19}a^{17}-\frac{24\!\cdots\!47}{74\!\cdots\!33}a^{16}+\frac{12\!\cdots\!00}{20\!\cdots\!91}a^{15}-\frac{30\!\cdots\!80}{18\!\cdots\!19}a^{14}+\frac{14\!\cdots\!71}{18\!\cdots\!19}a^{13}+\frac{81\!\cdots\!66}{60\!\cdots\!73}a^{12}+\frac{61\!\cdots\!67}{67\!\cdots\!97}a^{11}+\frac{50\!\cdots\!20}{18\!\cdots\!19}a^{10}+\frac{14\!\cdots\!68}{18\!\cdots\!19}a^{9}+\frac{22\!\cdots\!08}{20\!\cdots\!91}a^{8}+\frac{51\!\cdots\!00}{18\!\cdots\!19}a^{7}-\frac{69\!\cdots\!16}{18\!\cdots\!19}a^{6}-\frac{10\!\cdots\!89}{60\!\cdots\!73}a^{5}-\frac{55\!\cdots\!43}{18\!\cdots\!19}a^{4}-\frac{20\!\cdots\!33}{60\!\cdots\!73}a^{3}-\frac{26\!\cdots\!75}{18\!\cdots\!19}a^{2}+\frac{71\!\cdots\!74}{18\!\cdots\!19}a+\frac{94\!\cdots\!13}{22\!\cdots\!61}$, $\frac{81\!\cdots\!62}{60\!\cdots\!73}a^{31}-\frac{13\!\cdots\!52}{20\!\cdots\!91}a^{30}+\frac{18\!\cdots\!61}{60\!\cdots\!73}a^{29}-\frac{59\!\cdots\!73}{60\!\cdots\!73}a^{28}+\frac{60\!\cdots\!53}{20\!\cdots\!91}a^{27}-\frac{47\!\cdots\!34}{67\!\cdots\!97}a^{26}+\frac{10\!\cdots\!97}{60\!\cdots\!73}a^{25}-\frac{18\!\cdots\!95}{60\!\cdots\!73}a^{24}+\frac{35\!\cdots\!16}{60\!\cdots\!73}a^{23}-\frac{50\!\cdots\!95}{60\!\cdots\!73}a^{22}+\frac{77\!\cdots\!97}{60\!\cdots\!73}a^{21}-\frac{85\!\cdots\!24}{60\!\cdots\!73}a^{20}+\frac{33\!\cdots\!48}{20\!\cdots\!91}a^{19}-\frac{78\!\cdots\!27}{60\!\cdots\!73}a^{18}+\frac{86\!\cdots\!63}{60\!\cdots\!73}a^{17}-\frac{12\!\cdots\!93}{22\!\cdots\!99}a^{16}+\frac{97\!\cdots\!32}{67\!\cdots\!97}a^{15}-\frac{18\!\cdots\!48}{60\!\cdots\!73}a^{14}+\frac{12\!\cdots\!30}{60\!\cdots\!73}a^{13}+\frac{74\!\cdots\!63}{20\!\cdots\!91}a^{12}+\frac{47\!\cdots\!31}{22\!\cdots\!99}a^{11}+\frac{42\!\cdots\!24}{60\!\cdots\!73}a^{10}+\frac{83\!\cdots\!32}{60\!\cdots\!73}a^{9}-\frac{29\!\cdots\!62}{67\!\cdots\!97}a^{8}-\frac{21\!\cdots\!27}{60\!\cdots\!73}a^{7}-\frac{11\!\cdots\!65}{60\!\cdots\!73}a^{6}-\frac{19\!\cdots\!83}{20\!\cdots\!91}a^{5}-\frac{28\!\cdots\!05}{60\!\cdots\!73}a^{4}+\frac{53\!\cdots\!60}{20\!\cdots\!91}a^{3}+\frac{23\!\cdots\!46}{60\!\cdots\!73}a^{2}+\frac{12\!\cdots\!29}{60\!\cdots\!73}a+\frac{24\!\cdots\!84}{76\!\cdots\!87}$, $\frac{72\!\cdots\!39}{18\!\cdots\!19}a^{31}-\frac{10\!\cdots\!35}{60\!\cdots\!73}a^{30}+\frac{13\!\cdots\!77}{18\!\cdots\!19}a^{29}-\frac{41\!\cdots\!29}{18\!\cdots\!19}a^{28}+\frac{40\!\cdots\!04}{60\!\cdots\!73}a^{27}-\frac{28\!\cdots\!44}{20\!\cdots\!91}a^{26}+\frac{63\!\cdots\!54}{18\!\cdots\!19}a^{25}-\frac{96\!\cdots\!37}{18\!\cdots\!19}a^{24}+\frac{20\!\cdots\!27}{18\!\cdots\!19}a^{23}-\frac{21\!\cdots\!81}{18\!\cdots\!19}a^{22}+\frac{37\!\cdots\!84}{18\!\cdots\!19}a^{21}-\frac{26\!\cdots\!33}{18\!\cdots\!19}a^{20}+\frac{12\!\cdots\!92}{60\!\cdots\!73}a^{19}-\frac{58\!\cdots\!93}{18\!\cdots\!19}a^{18}+\frac{30\!\cdots\!78}{18\!\cdots\!19}a^{17}+\frac{32\!\cdots\!02}{22\!\cdots\!99}a^{16}+\frac{68\!\cdots\!70}{20\!\cdots\!91}a^{15}+\frac{47\!\cdots\!60}{18\!\cdots\!19}a^{14}+\frac{10\!\cdots\!08}{18\!\cdots\!19}a^{13}+\frac{32\!\cdots\!64}{60\!\cdots\!73}a^{12}+\frac{49\!\cdots\!97}{67\!\cdots\!97}a^{11}+\frac{14\!\cdots\!69}{18\!\cdots\!19}a^{10}+\frac{13\!\cdots\!51}{18\!\cdots\!19}a^{9}+\frac{65\!\cdots\!34}{20\!\cdots\!91}a^{8}-\frac{10\!\cdots\!15}{18\!\cdots\!19}a^{7}-\frac{92\!\cdots\!38}{18\!\cdots\!19}a^{6}-\frac{38\!\cdots\!47}{60\!\cdots\!73}a^{5}-\frac{65\!\cdots\!09}{18\!\cdots\!19}a^{4}-\frac{31\!\cdots\!74}{60\!\cdots\!73}a^{3}+\frac{25\!\cdots\!90}{18\!\cdots\!19}a^{2}+\frac{20\!\cdots\!20}{18\!\cdots\!19}a+\frac{88\!\cdots\!57}{22\!\cdots\!61}$, $\frac{59\!\cdots\!26}{18\!\cdots\!19}a^{31}-\frac{10\!\cdots\!52}{60\!\cdots\!73}a^{30}+\frac{14\!\cdots\!35}{18\!\cdots\!19}a^{29}-\frac{47\!\cdots\!88}{18\!\cdots\!19}a^{28}+\frac{47\!\cdots\!62}{60\!\cdots\!73}a^{27}-\frac{38\!\cdots\!37}{20\!\cdots\!91}a^{26}+\frac{82\!\cdots\!54}{18\!\cdots\!19}a^{25}-\frac{15\!\cdots\!64}{18\!\cdots\!19}a^{24}+\frac{29\!\cdots\!57}{18\!\cdots\!19}a^{23}-\frac{42\!\cdots\!78}{18\!\cdots\!19}a^{22}+\frac{64\!\cdots\!34}{18\!\cdots\!19}a^{21}-\frac{74\!\cdots\!42}{18\!\cdots\!19}a^{20}+\frac{67\!\cdots\!24}{14\!\cdots\!11}a^{19}-\frac{73\!\cdots\!01}{18\!\cdots\!19}a^{18}+\frac{76\!\cdots\!39}{18\!\cdots\!19}a^{17}-\frac{53\!\cdots\!11}{22\!\cdots\!99}a^{16}+\frac{19\!\cdots\!40}{46\!\cdots\!37}a^{15}-\frac{37\!\cdots\!19}{18\!\cdots\!19}a^{14}+\frac{10\!\cdots\!42}{18\!\cdots\!19}a^{13}-\frac{81\!\cdots\!19}{60\!\cdots\!73}a^{12}+\frac{34\!\cdots\!19}{67\!\cdots\!97}a^{11}+\frac{93\!\cdots\!83}{18\!\cdots\!19}a^{10}+\frac{59\!\cdots\!21}{18\!\cdots\!19}a^{9}-\frac{35\!\cdots\!64}{20\!\cdots\!91}a^{8}+\frac{81\!\cdots\!82}{18\!\cdots\!19}a^{7}-\frac{59\!\cdots\!28}{18\!\cdots\!19}a^{6}-\frac{19\!\cdots\!27}{60\!\cdots\!73}a^{5}-\frac{39\!\cdots\!64}{18\!\cdots\!19}a^{4}+\frac{59\!\cdots\!19}{60\!\cdots\!73}a^{3}+\frac{66\!\cdots\!27}{18\!\cdots\!19}a^{2}+\frac{13\!\cdots\!30}{18\!\cdots\!19}a-\frac{33\!\cdots\!38}{22\!\cdots\!61}$, $\frac{23\!\cdots\!39}{18\!\cdots\!19}a^{31}+\frac{80\!\cdots\!91}{60\!\cdots\!73}a^{30}-\frac{10\!\cdots\!12}{18\!\cdots\!19}a^{29}+\frac{48\!\cdots\!89}{18\!\cdots\!19}a^{28}-\frac{51\!\cdots\!08}{60\!\cdots\!73}a^{27}+\frac{52\!\cdots\!13}{20\!\cdots\!91}a^{26}-\frac{98\!\cdots\!52}{18\!\cdots\!19}a^{25}+\frac{25\!\cdots\!81}{18\!\cdots\!19}a^{24}-\frac{36\!\cdots\!43}{18\!\cdots\!19}a^{23}+\frac{80\!\cdots\!91}{18\!\cdots\!19}a^{22}-\frac{83\!\cdots\!64}{18\!\cdots\!19}a^{21}+\frac{14\!\cdots\!85}{18\!\cdots\!19}a^{20}-\frac{34\!\cdots\!50}{60\!\cdots\!73}a^{19}+\frac{14\!\cdots\!63}{18\!\cdots\!19}a^{18}-\frac{24\!\cdots\!52}{18\!\cdots\!19}a^{17}+\frac{56\!\cdots\!67}{74\!\cdots\!33}a^{16}+\frac{94\!\cdots\!56}{20\!\cdots\!91}a^{15}+\frac{65\!\cdots\!26}{42\!\cdots\!33}a^{14}+\frac{16\!\cdots\!91}{18\!\cdots\!19}a^{13}+\frac{14\!\cdots\!22}{60\!\cdots\!73}a^{12}+\frac{13\!\cdots\!22}{67\!\cdots\!97}a^{11}+\frac{49\!\cdots\!26}{18\!\cdots\!19}a^{10}+\frac{44\!\cdots\!96}{18\!\cdots\!19}a^{9}+\frac{44\!\cdots\!35}{20\!\cdots\!91}a^{8}-\frac{16\!\cdots\!83}{18\!\cdots\!19}a^{7}-\frac{29\!\cdots\!67}{18\!\cdots\!19}a^{6}-\frac{16\!\cdots\!90}{60\!\cdots\!73}a^{5}-\frac{44\!\cdots\!04}{18\!\cdots\!19}a^{4}-\frac{46\!\cdots\!05}{60\!\cdots\!73}a^{3}+\frac{13\!\cdots\!54}{18\!\cdots\!19}a^{2}+\frac{13\!\cdots\!57}{18\!\cdots\!19}a+\frac{55\!\cdots\!27}{22\!\cdots\!61}$, $\frac{21\!\cdots\!94}{60\!\cdots\!73}a^{31}-\frac{36\!\cdots\!06}{20\!\cdots\!91}a^{30}+\frac{47\!\cdots\!50}{60\!\cdots\!73}a^{29}-\frac{15\!\cdots\!94}{60\!\cdots\!73}a^{28}+\frac{15\!\cdots\!84}{20\!\cdots\!91}a^{27}-\frac{12\!\cdots\!08}{67\!\cdots\!97}a^{26}+\frac{27\!\cdots\!36}{60\!\cdots\!73}a^{25}-\frac{48\!\cdots\!00}{60\!\cdots\!73}a^{24}+\frac{95\!\cdots\!24}{60\!\cdots\!73}a^{23}-\frac{13\!\cdots\!56}{60\!\cdots\!73}a^{22}+\frac{20\!\cdots\!76}{60\!\cdots\!73}a^{21}-\frac{23\!\cdots\!80}{60\!\cdots\!73}a^{20}+\frac{91\!\cdots\!81}{20\!\cdots\!91}a^{19}-\frac{21\!\cdots\!19}{60\!\cdots\!73}a^{18}+\frac{23\!\cdots\!06}{60\!\cdots\!73}a^{17}-\frac{12\!\cdots\!49}{74\!\cdots\!33}a^{16}+\frac{27\!\cdots\!74}{67\!\cdots\!97}a^{15}-\frac{57\!\cdots\!86}{60\!\cdots\!73}a^{14}+\frac{34\!\cdots\!96}{60\!\cdots\!73}a^{13}+\frac{33\!\cdots\!14}{20\!\cdots\!91}a^{12}+\frac{12\!\cdots\!59}{22\!\cdots\!99}a^{11}+\frac{11\!\cdots\!25}{60\!\cdots\!73}a^{10}+\frac{22\!\cdots\!67}{60\!\cdots\!73}a^{9}-\frac{60\!\cdots\!25}{67\!\cdots\!97}a^{8}-\frac{47\!\cdots\!90}{60\!\cdots\!73}a^{7}-\frac{26\!\cdots\!84}{60\!\cdots\!73}a^{6}-\frac{46\!\cdots\!98}{20\!\cdots\!91}a^{5}-\frac{66\!\cdots\!36}{60\!\cdots\!73}a^{4}+\frac{17\!\cdots\!46}{20\!\cdots\!91}a^{3}+\frac{50\!\cdots\!67}{60\!\cdots\!73}a^{2}+\frac{21\!\cdots\!85}{60\!\cdots\!73}a+\frac{47\!\cdots\!73}{76\!\cdots\!87}$, $\frac{75\!\cdots\!62}{18\!\cdots\!19}a^{31}-\frac{13\!\cdots\!68}{60\!\cdots\!73}a^{30}+\frac{18\!\cdots\!18}{18\!\cdots\!19}a^{29}-\frac{61\!\cdots\!42}{18\!\cdots\!19}a^{28}+\frac{62\!\cdots\!52}{60\!\cdots\!73}a^{27}-\frac{50\!\cdots\!76}{20\!\cdots\!91}a^{26}+\frac{10\!\cdots\!63}{18\!\cdots\!19}a^{25}-\frac{20\!\cdots\!82}{18\!\cdots\!19}a^{24}+\frac{38\!\cdots\!59}{18\!\cdots\!19}a^{23}-\frac{57\!\cdots\!44}{18\!\cdots\!19}a^{22}+\frac{85\!\cdots\!24}{18\!\cdots\!19}a^{21}-\frac{10\!\cdots\!12}{18\!\cdots\!19}a^{20}+\frac{38\!\cdots\!77}{60\!\cdots\!73}a^{19}-\frac{99\!\cdots\!90}{18\!\cdots\!19}a^{18}+\frac{99\!\cdots\!95}{18\!\cdots\!19}a^{17}-\frac{68\!\cdots\!66}{22\!\cdots\!99}a^{16}+\frac{99\!\cdots\!94}{20\!\cdots\!91}a^{15}-\frac{45\!\cdots\!34}{18\!\cdots\!19}a^{14}+\frac{11\!\cdots\!19}{18\!\cdots\!19}a^{13}-\frac{11\!\cdots\!42}{60\!\cdots\!73}a^{12}+\frac{41\!\cdots\!39}{67\!\cdots\!97}a^{11}+\frac{67\!\cdots\!50}{42\!\cdots\!33}a^{10}+\frac{62\!\cdots\!81}{18\!\cdots\!19}a^{9}-\frac{56\!\cdots\!76}{20\!\cdots\!91}a^{8}-\frac{46\!\cdots\!19}{18\!\cdots\!19}a^{7}-\frac{86\!\cdots\!27}{18\!\cdots\!19}a^{6}-\frac{59\!\cdots\!93}{60\!\cdots\!73}a^{5}-\frac{29\!\cdots\!79}{18\!\cdots\!19}a^{4}+\frac{71\!\cdots\!88}{60\!\cdots\!73}a^{3}+\frac{10\!\cdots\!75}{18\!\cdots\!19}a^{2}+\frac{45\!\cdots\!00}{18\!\cdots\!19}a+\frac{11\!\cdots\!03}{22\!\cdots\!61}$, $\frac{15\!\cdots\!49}{60\!\cdots\!73}a^{31}-\frac{25\!\cdots\!51}{20\!\cdots\!91}a^{30}+\frac{34\!\cdots\!60}{60\!\cdots\!73}a^{29}-\frac{11\!\cdots\!43}{60\!\cdots\!73}a^{28}+\frac{11\!\cdots\!94}{20\!\cdots\!91}a^{27}-\frac{89\!\cdots\!07}{67\!\cdots\!97}a^{26}+\frac{19\!\cdots\!91}{60\!\cdots\!73}a^{25}-\frac{34\!\cdots\!95}{60\!\cdots\!73}a^{24}+\frac{68\!\cdots\!28}{60\!\cdots\!73}a^{23}-\frac{95\!\cdots\!28}{60\!\cdots\!73}a^{22}+\frac{15\!\cdots\!60}{60\!\cdots\!73}a^{21}-\frac{16\!\cdots\!42}{60\!\cdots\!73}a^{20}+\frac{69\!\cdots\!99}{20\!\cdots\!91}a^{19}-\frac{16\!\cdots\!32}{60\!\cdots\!73}a^{18}+\frac{19\!\cdots\!51}{60\!\cdots\!73}a^{17}-\frac{31\!\cdots\!74}{24\!\cdots\!11}a^{16}+\frac{22\!\cdots\!66}{67\!\cdots\!97}a^{15}-\frac{30\!\cdots\!68}{60\!\cdots\!73}a^{14}+\frac{29\!\cdots\!68}{60\!\cdots\!73}a^{13}+\frac{11\!\cdots\!11}{20\!\cdots\!91}a^{12}+\frac{12\!\cdots\!66}{22\!\cdots\!99}a^{11}+\frac{13\!\cdots\!89}{60\!\cdots\!73}a^{10}+\frac{65\!\cdots\!79}{14\!\cdots\!11}a^{9}+\frac{40\!\cdots\!47}{67\!\cdots\!97}a^{8}+\frac{69\!\cdots\!95}{60\!\cdots\!73}a^{7}-\frac{14\!\cdots\!43}{60\!\cdots\!73}a^{6}-\frac{27\!\cdots\!99}{20\!\cdots\!91}a^{5}-\frac{81\!\cdots\!71}{60\!\cdots\!73}a^{4}-\frac{37\!\cdots\!74}{20\!\cdots\!91}a^{3}-\frac{35\!\cdots\!05}{60\!\cdots\!73}a^{2}+\frac{51\!\cdots\!77}{60\!\cdots\!73}a-\frac{33\!\cdots\!79}{76\!\cdots\!87}$, $\frac{27\!\cdots\!46}{18\!\cdots\!19}a^{31}-\frac{69\!\cdots\!75}{60\!\cdots\!73}a^{30}+\frac{10\!\cdots\!13}{18\!\cdots\!19}a^{29}-\frac{37\!\cdots\!46}{18\!\cdots\!19}a^{28}+\frac{40\!\cdots\!26}{60\!\cdots\!73}a^{27}-\frac{36\!\cdots\!66}{20\!\cdots\!91}a^{26}+\frac{79\!\cdots\!08}{18\!\cdots\!19}a^{25}-\frac{17\!\cdots\!90}{18\!\cdots\!19}a^{24}+\frac{32\!\cdots\!03}{18\!\cdots\!19}a^{23}-\frac{56\!\cdots\!35}{18\!\cdots\!19}a^{22}+\frac{86\!\cdots\!57}{18\!\cdots\!19}a^{21}-\frac{28\!\cdots\!39}{42\!\cdots\!33}a^{20}+\frac{48\!\cdots\!91}{60\!\cdots\!73}a^{19}-\frac{15\!\cdots\!37}{18\!\cdots\!19}a^{18}+\frac{14\!\cdots\!20}{18\!\cdots\!19}a^{17}-\frac{15\!\cdots\!97}{22\!\cdots\!99}a^{16}+\frac{11\!\cdots\!22}{20\!\cdots\!91}a^{15}-\frac{10\!\cdots\!43}{18\!\cdots\!19}a^{14}+\frac{95\!\cdots\!79}{18\!\cdots\!19}a^{13}-\frac{43\!\cdots\!24}{60\!\cdots\!73}a^{12}+\frac{32\!\cdots\!54}{67\!\cdots\!97}a^{11}-\frac{11\!\cdots\!43}{18\!\cdots\!19}a^{10}+\frac{27\!\cdots\!93}{18\!\cdots\!19}a^{9}-\frac{10\!\cdots\!59}{20\!\cdots\!91}a^{8}+\frac{30\!\cdots\!96}{18\!\cdots\!19}a^{7}-\frac{58\!\cdots\!19}{18\!\cdots\!19}a^{6}+\frac{20\!\cdots\!27}{60\!\cdots\!73}a^{5}+\frac{31\!\cdots\!74}{18\!\cdots\!19}a^{4}+\frac{77\!\cdots\!60}{60\!\cdots\!73}a^{3}+\frac{15\!\cdots\!25}{18\!\cdots\!19}a^{2}+\frac{92\!\cdots\!34}{18\!\cdots\!19}a-\frac{54\!\cdots\!94}{22\!\cdots\!61}$
| 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: | \( 1483068056.9314811 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1483068056.9314811 \cdot 1}{2\cdot\sqrt{515207889456069254383620937908064472169867121}}\cr\approx \mathstrut & 0.192760359061108 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79)
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 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79, 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 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79);
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 - 5*x^31 + 22*x^30 - 73*x^29 + 220*x^28 - 519*x^27 + 1255*x^26 - 2247*x^25 + 4413*x^24 - 6238*x^23 + 9744*x^22 - 10830*x^21 + 13240*x^20 - 10628*x^19 + 12053*x^18 - 5501*x^17 + 12591*x^16 - 3062*x^15 + 17256*x^14 + 194*x^13 + 18165*x^12 + 5659*x^11 + 13232*x^10 - 2318*x^9 - 319*x^8 - 13445*x^7 - 6940*x^6 - 4990*x^5 + 1466*x^4 + 2074*x^3 + 1626*x^2 + 264*x + 79);
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
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 96 |
| The 28 conjugacy class representatives for $C_8.A_4$ |
| Character table for $C_8.A_4$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.0.2197.1, 4.0.8281.1, 8.0.68574961.1, 16.0.134308824412163591281.1 |
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 | $24{,}\,{\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{4}$ | $24{,}\,{\href{/padicField/5.8.0.1}{8} }$ | R | $24{,}\,{\href{/padicField/11.8.0.1}{8} }$ | R | ${\href{/padicField/17.12.0.1}{12} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | $24{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.12.0.1}{12} }^{2}{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | $24{,}\,{\href{/padicField/31.8.0.1}{8} }$ | $24{,}\,{\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{16}$ | $24{,}\,{\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.6.0.1}{6} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | $24{,}\,{\href{/padicField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.8.0.1 | $x^{8} + 4 x^{3} + 6 x^{2} + 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| Deg $24$ | $3$ | $8$ | $16$ | ||||
|
\(13\)
| 13.16.14.1 | $x^{16} - 1508 x^{8} - 6084$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |
| 13.16.14.1 | $x^{16} - 1508 x^{8} - 6084$ | $8$ | $2$ | $14$ | $C_8: C_2$ | $[\ ]_{8}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| 1.91.6t1.j.a | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.91.6t1.j.b | $1$ | $ 7 \cdot 13 $ | 6.6.5274997.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 1.13.4t1.a.a | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| * | 1.13.4t1.a.b | $1$ | $ 13 $ | 4.0.2197.1 | $C_4$ (as 4T1) | $0$ | $-1$ |
| 1.91.12t1.a.a | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.b | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.c | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 1.91.12t1.a.d | $1$ | $ 7 \cdot 13 $ | 12.0.61132828589969773.1 | $C_{12}$ (as 12T1) | $0$ | $-1$ | |
| 2.8281.48.a.a | $2$ | $ 7^{2} \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.8281.48.a.b | $2$ | $ 7^{2} \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.8281.48.a.c | $2$ | $ 7^{2} \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| 2.8281.48.a.d | $2$ | $ 7^{2} \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ | |
| * | 2.1183.32t402.a.a | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.b | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.c | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.d | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.e | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.f | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.g | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 2.1183.32t402.a.h | $2$ | $ 7 \cdot 13^{2}$ | 32.0.515207889456069254383620937908064472169867121.1 | $C_8.A_4$ (as 32T402) | $0$ | $0$ |
| * | 3.8281.4t4.b.a | $3$ | $ 7^{2} \cdot 13^{2}$ | 4.0.8281.1 | $A_4$ (as 4T4) | $1$ | $-1$ |
| * | 3.637.6t6.a.a | $3$ | $ 7^{2} \cdot 13 $ | 6.2.31213.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ |
| * | 3.107653.12t29.a.a | $3$ | $ 7^{2} \cdot 13^{3}$ | 12.8.61132828589969773.1 | $C_4\times A_4$ (as 12T29) | $0$ | $1$ |
| * | 3.107653.12t29.a.b | $3$ | $ 7^{2} \cdot 13^{3}$ | 12.8.61132828589969773.1 | $C_4\times A_4$ (as 12T29) | $0$ | $1$ |
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 *.