Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1)
gp: K = bnfinit(y^27 - 6*y^26 + 31*y^25 - 80*y^24 + 117*y^23 - 51*y^22 - 243*y^21 + 807*y^20 - 1350*y^19 + 966*y^18 + 1935*y^17 - 9345*y^16 + 23103*y^15 - 43548*y^14 + 68751*y^13 - 94575*y^12 + 115269*y^11 - 125268*y^10 + 121785*y^9 - 105567*y^8 + 81027*y^7 - 54357*y^6 + 31122*y^5 - 14664*y^4 + 5393*y^3 - 1350*y^2 + 179*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*x - 1)
\( x^{27} - 6 x^{26} + 31 x^{25} - 80 x^{24} + 117 x^{23} - 51 x^{22} - 243 x^{21} + 807 x^{20} - 1350 x^{19} + 966 x^{18} + 1935 x^{17} - 9345 x^{16} + 23103 x^{15} - 43548 x^{14} + 68751 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $27$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1543319746516623033280478216838436483146079\)
\(\medspace = -\,1759^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(36.52\) | 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))
| |
| Ramified primes: |
\(1759\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1759}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{5}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{6}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{7}$, $\frac{1}{9}a^{16}+\frac{1}{9}a^{8}-\frac{2}{9}$, $\frac{1}{9}a^{17}+\frac{1}{9}a^{9}-\frac{2}{9}a$, $\frac{1}{9}a^{18}+\frac{1}{9}a^{10}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{19}+\frac{1}{9}a^{11}-\frac{2}{9}a^{3}$, $\frac{1}{9}a^{20}+\frac{1}{9}a^{12}-\frac{2}{9}a^{4}$, $\frac{1}{9}a^{21}+\frac{1}{9}a^{13}-\frac{2}{9}a^{5}$, $\frac{1}{27}a^{22}+\frac{1}{27}a^{21}-\frac{1}{27}a^{20}-\frac{1}{27}a^{18}-\frac{1}{27}a^{17}+\frac{1}{27}a^{16}+\frac{1}{27}a^{14}+\frac{1}{27}a^{13}-\frac{1}{27}a^{12}-\frac{1}{27}a^{10}-\frac{1}{27}a^{9}+\frac{1}{27}a^{8}-\frac{2}{27}a^{6}-\frac{2}{27}a^{5}+\frac{2}{27}a^{4}+\frac{2}{27}a^{2}+\frac{2}{27}a-\frac{2}{27}$, $\frac{1}{27}a^{23}+\frac{1}{27}a^{21}+\frac{1}{27}a^{20}-\frac{1}{27}a^{19}-\frac{1}{27}a^{17}-\frac{1}{27}a^{16}+\frac{1}{27}a^{15}+\frac{1}{27}a^{13}+\frac{1}{27}a^{12}-\frac{1}{27}a^{11}-\frac{1}{27}a^{9}-\frac{1}{27}a^{8}-\frac{2}{27}a^{7}-\frac{2}{27}a^{5}-\frac{2}{27}a^{4}+\frac{2}{27}a^{3}+\frac{2}{27}a+\frac{2}{27}$, $\frac{1}{27}a^{24}-\frac{1}{9}a^{8}+\frac{2}{27}$, $\frac{1}{202797}a^{25}+\frac{254}{22533}a^{24}+\frac{220}{67599}a^{23}+\frac{442}{28971}a^{22}-\frac{1472}{28971}a^{21}+\frac{7313}{202797}a^{20}+\frac{3403}{67599}a^{19}+\frac{10271}{202797}a^{18}+\frac{9236}{202797}a^{17}+\frac{235}{202797}a^{16}+\frac{3856}{67599}a^{15}+\frac{11230}{202797}a^{14}+\frac{766}{5481}a^{13}-\frac{30865}{202797}a^{12}+\frac{6691}{67599}a^{11}-\frac{8143}{202797}a^{10}+\frac{19178}{202797}a^{9}+\frac{31636}{202797}a^{8}+\frac{32230}{67599}a^{7}-\frac{2675}{28971}a^{6}-\frac{20306}{202797}a^{5}-\frac{72379}{202797}a^{4}-\frac{20714}{67599}a^{3}+\frac{5501}{28971}a^{2}+\frac{14401}{28971}a+\frac{17494}{202797}$, $\frac{1}{34\!\cdots\!29}a^{26}+\frac{43\!\cdots\!75}{34\!\cdots\!29}a^{25}-\frac{58\!\cdots\!47}{34\!\cdots\!29}a^{24}+\frac{69\!\cdots\!05}{11\!\cdots\!43}a^{23}-\frac{60\!\cdots\!85}{16\!\cdots\!49}a^{22}-\frac{46\!\cdots\!71}{17\!\cdots\!99}a^{21}+\frac{10\!\cdots\!71}{37\!\cdots\!81}a^{20}-\frac{44\!\cdots\!24}{11\!\cdots\!43}a^{19}+\frac{45\!\cdots\!51}{16\!\cdots\!49}a^{18}-\frac{20\!\cdots\!54}{37\!\cdots\!81}a^{17}+\frac{97\!\cdots\!88}{37\!\cdots\!81}a^{16}+\frac{98\!\cdots\!24}{75\!\cdots\!93}a^{15}+\frac{44\!\cdots\!85}{75\!\cdots\!93}a^{14}-\frac{11\!\cdots\!87}{36\!\cdots\!27}a^{13}-\frac{38\!\cdots\!66}{53\!\cdots\!83}a^{12}-\frac{29\!\cdots\!86}{11\!\cdots\!43}a^{11}+\frac{13\!\cdots\!37}{11\!\cdots\!43}a^{10}-\frac{15\!\cdots\!67}{11\!\cdots\!43}a^{9}+\frac{86\!\cdots\!57}{11\!\cdots\!43}a^{8}-\frac{20\!\cdots\!90}{11\!\cdots\!43}a^{7}-\frac{34\!\cdots\!05}{11\!\cdots\!43}a^{6}-\frac{17\!\cdots\!36}{37\!\cdots\!81}a^{5}+\frac{37\!\cdots\!58}{12\!\cdots\!27}a^{4}+\frac{22\!\cdots\!72}{11\!\cdots\!43}a^{3}+\frac{91\!\cdots\!65}{48\!\cdots\!47}a^{2}+\frac{22\!\cdots\!67}{34\!\cdots\!29}a+\frac{89\!\cdots\!32}{34\!\cdots\!29}$
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: | $13$ | 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: |
$\frac{16\!\cdots\!70}{34\!\cdots\!29}a^{26}+\frac{60\!\cdots\!38}{34\!\cdots\!29}a^{25}+\frac{35\!\cdots\!07}{34\!\cdots\!29}a^{24}-\frac{11\!\cdots\!54}{11\!\cdots\!43}a^{23}+\frac{38\!\cdots\!62}{16\!\cdots\!49}a^{22}-\frac{48\!\cdots\!47}{51\!\cdots\!97}a^{21}-\frac{31\!\cdots\!41}{37\!\cdots\!81}a^{20}+\frac{12\!\cdots\!80}{16\!\cdots\!49}a^{19}-\frac{18\!\cdots\!11}{11\!\cdots\!43}a^{18}+\frac{25\!\cdots\!33}{12\!\cdots\!27}a^{17}+\frac{40\!\cdots\!74}{37\!\cdots\!81}a^{16}-\frac{48\!\cdots\!38}{75\!\cdots\!93}a^{15}+\frac{50\!\cdots\!87}{28\!\cdots\!27}a^{14}-\frac{11\!\cdots\!32}{36\!\cdots\!27}a^{13}+\frac{18\!\cdots\!73}{37\!\cdots\!81}a^{12}-\frac{91\!\cdots\!71}{16\!\cdots\!49}a^{11}+\frac{64\!\cdots\!97}{11\!\cdots\!43}a^{10}-\frac{48\!\cdots\!28}{11\!\cdots\!43}a^{9}+\frac{16\!\cdots\!64}{11\!\cdots\!43}a^{8}+\frac{19\!\cdots\!43}{11\!\cdots\!43}a^{7}-\frac{40\!\cdots\!02}{11\!\cdots\!43}a^{6}+\frac{66\!\cdots\!61}{12\!\cdots\!27}a^{5}-\frac{57\!\cdots\!22}{12\!\cdots\!27}a^{4}+\frac{41\!\cdots\!42}{11\!\cdots\!43}a^{3}-\frac{87\!\cdots\!31}{48\!\cdots\!47}a^{2}+\frac{24\!\cdots\!87}{34\!\cdots\!29}a-\frac{97\!\cdots\!70}{91\!\cdots\!17}$, $\frac{14\!\cdots\!23}{34\!\cdots\!29}a^{26}-\frac{84\!\cdots\!62}{34\!\cdots\!29}a^{25}+\frac{43\!\cdots\!28}{34\!\cdots\!29}a^{24}-\frac{35\!\cdots\!97}{11\!\cdots\!43}a^{23}+\frac{66\!\cdots\!70}{16\!\cdots\!49}a^{22}-\frac{42\!\cdots\!54}{51\!\cdots\!97}a^{21}-\frac{41\!\cdots\!74}{37\!\cdots\!81}a^{20}+\frac{36\!\cdots\!30}{11\!\cdots\!43}a^{19}-\frac{19\!\cdots\!84}{39\!\cdots\!67}a^{18}+\frac{13\!\cdots\!82}{53\!\cdots\!83}a^{17}+\frac{35\!\cdots\!19}{37\!\cdots\!81}a^{16}-\frac{28\!\cdots\!71}{75\!\cdots\!93}a^{15}+\frac{66\!\cdots\!03}{75\!\cdots\!93}a^{14}-\frac{19\!\cdots\!98}{12\!\cdots\!09}a^{13}+\frac{31\!\cdots\!20}{12\!\cdots\!27}a^{12}-\frac{37\!\cdots\!99}{11\!\cdots\!43}a^{11}+\frac{44\!\cdots\!16}{11\!\cdots\!43}a^{10}-\frac{46\!\cdots\!20}{11\!\cdots\!43}a^{9}+\frac{44\!\cdots\!27}{11\!\cdots\!43}a^{8}-\frac{36\!\cdots\!27}{11\!\cdots\!43}a^{7}+\frac{26\!\cdots\!86}{11\!\cdots\!43}a^{6}-\frac{56\!\cdots\!01}{37\!\cdots\!81}a^{5}+\frac{30\!\cdots\!89}{37\!\cdots\!81}a^{4}-\frac{37\!\cdots\!62}{11\!\cdots\!43}a^{3}+\frac{49\!\cdots\!97}{48\!\cdots\!47}a^{2}-\frac{53\!\cdots\!72}{34\!\cdots\!29}a-\frac{42\!\cdots\!09}{34\!\cdots\!29}$, $\frac{19\!\cdots\!14}{34\!\cdots\!29}a^{26}-\frac{10\!\cdots\!39}{34\!\cdots\!29}a^{25}+\frac{51\!\cdots\!24}{34\!\cdots\!29}a^{24}-\frac{36\!\cdots\!76}{11\!\cdots\!43}a^{23}+\frac{43\!\cdots\!80}{16\!\cdots\!49}a^{22}+\frac{17\!\cdots\!91}{51\!\cdots\!97}a^{21}-\frac{70\!\cdots\!41}{41\!\cdots\!09}a^{20}+\frac{10\!\cdots\!38}{30\!\cdots\!39}a^{19}-\frac{57\!\cdots\!35}{16\!\cdots\!49}a^{18}-\frac{18\!\cdots\!52}{12\!\cdots\!27}a^{17}+\frac{20\!\cdots\!99}{12\!\cdots\!27}a^{16}-\frac{33\!\cdots\!11}{75\!\cdots\!93}a^{15}+\frac{64\!\cdots\!82}{75\!\cdots\!93}a^{14}-\frac{49\!\cdots\!55}{36\!\cdots\!27}a^{13}+\frac{98\!\cdots\!65}{53\!\cdots\!83}a^{12}-\frac{24\!\cdots\!39}{11\!\cdots\!43}a^{11}+\frac{24\!\cdots\!06}{11\!\cdots\!43}a^{10}-\frac{21\!\cdots\!76}{11\!\cdots\!43}a^{9}+\frac{16\!\cdots\!27}{11\!\cdots\!43}a^{8}-\frac{97\!\cdots\!70}{11\!\cdots\!43}a^{7}+\frac{41\!\cdots\!50}{11\!\cdots\!43}a^{6}-\frac{47\!\cdots\!12}{12\!\cdots\!27}a^{5}-\frac{38\!\cdots\!35}{37\!\cdots\!81}a^{4}+\frac{11\!\cdots\!48}{11\!\cdots\!43}a^{3}-\frac{21\!\cdots\!82}{48\!\cdots\!47}a^{2}+\frac{24\!\cdots\!76}{34\!\cdots\!29}a+\frac{38\!\cdots\!44}{34\!\cdots\!29}$, $\frac{33\!\cdots\!06}{34\!\cdots\!29}a^{26}-\frac{27\!\cdots\!72}{48\!\cdots\!47}a^{25}+\frac{14\!\cdots\!12}{48\!\cdots\!47}a^{24}-\frac{81\!\cdots\!12}{11\!\cdots\!43}a^{23}+\frac{14\!\cdots\!80}{16\!\cdots\!49}a^{22}-\frac{26\!\cdots\!05}{51\!\cdots\!97}a^{21}-\frac{10\!\cdots\!64}{37\!\cdots\!81}a^{20}+\frac{85\!\cdots\!05}{11\!\cdots\!43}a^{19}-\frac{12\!\cdots\!85}{11\!\cdots\!43}a^{18}+\frac{17\!\cdots\!54}{37\!\cdots\!81}a^{17}+\frac{92\!\cdots\!20}{37\!\cdots\!81}a^{16}-\frac{96\!\cdots\!42}{10\!\cdots\!99}a^{15}+\frac{15\!\cdots\!19}{75\!\cdots\!93}a^{14}-\frac{43\!\cdots\!71}{12\!\cdots\!09}a^{13}+\frac{67\!\cdots\!34}{12\!\cdots\!27}a^{12}-\frac{79\!\cdots\!42}{11\!\cdots\!43}a^{11}+\frac{91\!\cdots\!41}{11\!\cdots\!43}a^{10}-\frac{93\!\cdots\!20}{11\!\cdots\!43}a^{9}+\frac{84\!\cdots\!29}{11\!\cdots\!43}a^{8}-\frac{67\!\cdots\!63}{11\!\cdots\!43}a^{7}+\frac{47\!\cdots\!17}{11\!\cdots\!43}a^{6}-\frac{92\!\cdots\!96}{37\!\cdots\!81}a^{5}+\frac{63\!\cdots\!79}{53\!\cdots\!83}a^{4}-\frac{47\!\cdots\!97}{11\!\cdots\!43}a^{3}+\frac{46\!\cdots\!95}{48\!\cdots\!47}a^{2}-\frac{30\!\cdots\!36}{34\!\cdots\!29}a-\frac{37\!\cdots\!90}{34\!\cdots\!29}$, $\frac{34\!\cdots\!99}{34\!\cdots\!29}a^{26}-\frac{19\!\cdots\!28}{34\!\cdots\!29}a^{25}+\frac{10\!\cdots\!50}{34\!\cdots\!29}a^{24}-\frac{85\!\cdots\!72}{11\!\cdots\!43}a^{23}+\frac{58\!\cdots\!38}{55\!\cdots\!81}a^{22}-\frac{16\!\cdots\!09}{51\!\cdots\!97}a^{21}-\frac{15\!\cdots\!03}{59\!\cdots\!87}a^{20}+\frac{87\!\cdots\!20}{11\!\cdots\!43}a^{19}-\frac{13\!\cdots\!16}{11\!\cdots\!43}a^{18}+\frac{28\!\cdots\!60}{37\!\cdots\!81}a^{17}+\frac{26\!\cdots\!65}{12\!\cdots\!27}a^{16}-\frac{68\!\cdots\!04}{75\!\cdots\!93}a^{15}+\frac{16\!\cdots\!80}{75\!\cdots\!93}a^{14}-\frac{77\!\cdots\!58}{19\!\cdots\!43}a^{13}+\frac{23\!\cdots\!62}{37\!\cdots\!81}a^{12}-\frac{94\!\cdots\!16}{11\!\cdots\!43}a^{11}+\frac{11\!\cdots\!99}{11\!\cdots\!43}a^{10}-\frac{17\!\cdots\!98}{16\!\cdots\!49}a^{9}+\frac{16\!\cdots\!24}{16\!\cdots\!49}a^{8}-\frac{97\!\cdots\!28}{11\!\cdots\!43}a^{7}+\frac{72\!\cdots\!39}{11\!\cdots\!43}a^{6}-\frac{15\!\cdots\!37}{37\!\cdots\!81}a^{5}+\frac{29\!\cdots\!94}{13\!\cdots\!89}a^{4}-\frac{11\!\cdots\!44}{11\!\cdots\!43}a^{3}+\frac{15\!\cdots\!50}{48\!\cdots\!47}a^{2}-\frac{20\!\cdots\!10}{34\!\cdots\!29}a+\frac{50\!\cdots\!50}{34\!\cdots\!29}$, $\frac{49\!\cdots\!98}{46\!\cdots\!73}a^{26}-\frac{28\!\cdots\!42}{46\!\cdots\!73}a^{25}+\frac{14\!\cdots\!52}{46\!\cdots\!73}a^{24}-\frac{12\!\cdots\!71}{15\!\cdots\!91}a^{23}+\frac{24\!\cdots\!39}{22\!\cdots\!13}a^{22}-\frac{16\!\cdots\!79}{51\!\cdots\!97}a^{21}-\frac{13\!\cdots\!28}{51\!\cdots\!97}a^{20}+\frac{18\!\cdots\!09}{22\!\cdots\!13}a^{19}-\frac{19\!\cdots\!50}{15\!\cdots\!91}a^{18}+\frac{13\!\cdots\!72}{17\!\cdots\!99}a^{17}+\frac{11\!\cdots\!83}{51\!\cdots\!97}a^{16}-\frac{98\!\cdots\!08}{10\!\cdots\!41}a^{15}+\frac{33\!\cdots\!87}{14\!\cdots\!63}a^{14}-\frac{21\!\cdots\!51}{50\!\cdots\!99}a^{13}+\frac{33\!\cdots\!18}{51\!\cdots\!97}a^{12}-\frac{19\!\cdots\!13}{22\!\cdots\!13}a^{11}+\frac{16\!\cdots\!50}{15\!\cdots\!91}a^{10}-\frac{17\!\cdots\!30}{15\!\cdots\!91}a^{9}+\frac{16\!\cdots\!64}{15\!\cdots\!91}a^{8}-\frac{13\!\cdots\!42}{15\!\cdots\!91}a^{7}+\frac{10\!\cdots\!35}{15\!\cdots\!91}a^{6}-\frac{74\!\cdots\!60}{17\!\cdots\!99}a^{5}+\frac{40\!\cdots\!57}{17\!\cdots\!99}a^{4}-\frac{15\!\cdots\!58}{15\!\cdots\!91}a^{3}+\frac{22\!\cdots\!96}{66\!\cdots\!39}a^{2}-\frac{28\!\cdots\!50}{46\!\cdots\!73}a+\frac{73\!\cdots\!98}{46\!\cdots\!73}$, $\frac{21\!\cdots\!24}{34\!\cdots\!29}a^{26}-\frac{28\!\cdots\!32}{91\!\cdots\!17}a^{25}+\frac{55\!\cdots\!33}{34\!\cdots\!29}a^{24}-\frac{37\!\cdots\!04}{11\!\cdots\!43}a^{23}+\frac{58\!\cdots\!32}{16\!\cdots\!49}a^{22}+\frac{15\!\cdots\!46}{17\!\cdots\!99}a^{21}-\frac{54\!\cdots\!00}{37\!\cdots\!81}a^{20}+\frac{39\!\cdots\!49}{11\!\cdots\!43}a^{19}-\frac{52\!\cdots\!16}{11\!\cdots\!43}a^{18}+\frac{51\!\cdots\!18}{53\!\cdots\!83}a^{17}+\frac{49\!\cdots\!60}{37\!\cdots\!81}a^{16}-\frac{33\!\cdots\!68}{75\!\cdots\!93}a^{15}+\frac{72\!\cdots\!86}{75\!\cdots\!93}a^{14}-\frac{62\!\cdots\!54}{36\!\cdots\!27}a^{13}+\frac{85\!\cdots\!97}{34\!\cdots\!71}a^{12}-\frac{37\!\cdots\!13}{11\!\cdots\!43}a^{11}+\frac{43\!\cdots\!07}{11\!\cdots\!43}a^{10}-\frac{44\!\cdots\!73}{11\!\cdots\!43}a^{9}+\frac{41\!\cdots\!29}{11\!\cdots\!43}a^{8}-\frac{33\!\cdots\!64}{11\!\cdots\!43}a^{7}+\frac{24\!\cdots\!21}{11\!\cdots\!43}a^{6}-\frac{50\!\cdots\!65}{37\!\cdots\!81}a^{5}+\frac{25\!\cdots\!68}{37\!\cdots\!81}a^{4}-\frac{31\!\cdots\!78}{11\!\cdots\!43}a^{3}+\frac{38\!\cdots\!70}{48\!\cdots\!47}a^{2}-\frac{28\!\cdots\!81}{34\!\cdots\!29}a-\frac{10\!\cdots\!91}{11\!\cdots\!01}$, $\frac{38\!\cdots\!67}{34\!\cdots\!29}a^{26}-\frac{21\!\cdots\!40}{34\!\cdots\!29}a^{25}+\frac{10\!\cdots\!95}{34\!\cdots\!29}a^{24}-\frac{84\!\cdots\!47}{11\!\cdots\!43}a^{23}+\frac{15\!\cdots\!22}{16\!\cdots\!49}a^{22}-\frac{86\!\cdots\!91}{17\!\cdots\!99}a^{21}-\frac{46\!\cdots\!66}{16\!\cdots\!51}a^{20}+\frac{88\!\cdots\!39}{11\!\cdots\!43}a^{19}-\frac{12\!\cdots\!13}{11\!\cdots\!43}a^{18}+\frac{17\!\cdots\!97}{37\!\cdots\!81}a^{17}+\frac{32\!\cdots\!25}{12\!\cdots\!27}a^{16}-\frac{70\!\cdots\!51}{75\!\cdots\!93}a^{15}+\frac{16\!\cdots\!67}{75\!\cdots\!93}a^{14}-\frac{20\!\cdots\!22}{52\!\cdots\!61}a^{13}+\frac{21\!\cdots\!25}{37\!\cdots\!81}a^{12}-\frac{85\!\cdots\!00}{11\!\cdots\!43}a^{11}+\frac{10\!\cdots\!13}{11\!\cdots\!43}a^{10}-\frac{14\!\cdots\!68}{16\!\cdots\!49}a^{9}+\frac{13\!\cdots\!36}{16\!\cdots\!49}a^{8}-\frac{78\!\cdots\!30}{11\!\cdots\!43}a^{7}+\frac{19\!\cdots\!96}{39\!\cdots\!67}a^{6}-\frac{11\!\cdots\!83}{37\!\cdots\!81}a^{5}+\frac{58\!\cdots\!14}{37\!\cdots\!81}a^{4}-\frac{69\!\cdots\!41}{11\!\cdots\!43}a^{3}+\frac{81\!\cdots\!30}{48\!\cdots\!47}a^{2}-\frac{14\!\cdots\!44}{91\!\cdots\!17}a-\frac{38\!\cdots\!68}{34\!\cdots\!29}$, $\frac{23\!\cdots\!13}{46\!\cdots\!73}a^{26}-\frac{13\!\cdots\!37}{46\!\cdots\!73}a^{25}+\frac{66\!\cdots\!59}{46\!\cdots\!73}a^{24}-\frac{76\!\cdots\!16}{22\!\cdots\!13}a^{23}+\frac{99\!\cdots\!84}{22\!\cdots\!13}a^{22}-\frac{31\!\cdots\!68}{57\!\cdots\!33}a^{21}-\frac{24\!\cdots\!16}{19\!\cdots\!11}a^{20}+\frac{55\!\cdots\!23}{15\!\cdots\!91}a^{19}-\frac{82\!\cdots\!01}{15\!\cdots\!91}a^{18}+\frac{13\!\cdots\!04}{51\!\cdots\!97}a^{17}+\frac{19\!\cdots\!54}{17\!\cdots\!99}a^{16}-\frac{44\!\cdots\!86}{10\!\cdots\!41}a^{15}+\frac{10\!\cdots\!66}{10\!\cdots\!41}a^{14}-\frac{89\!\cdots\!51}{50\!\cdots\!99}a^{13}+\frac{13\!\cdots\!71}{51\!\cdots\!97}a^{12}-\frac{15\!\cdots\!25}{41\!\cdots\!43}a^{11}+\frac{65\!\cdots\!36}{15\!\cdots\!91}a^{10}-\frac{68\!\cdots\!82}{15\!\cdots\!91}a^{9}+\frac{64\!\cdots\!77}{15\!\cdots\!91}a^{8}-\frac{76\!\cdots\!81}{22\!\cdots\!13}a^{7}+\frac{39\!\cdots\!45}{15\!\cdots\!91}a^{6}-\frac{83\!\cdots\!93}{51\!\cdots\!97}a^{5}+\frac{44\!\cdots\!55}{51\!\cdots\!97}a^{4}-\frac{82\!\cdots\!30}{22\!\cdots\!13}a^{3}+\frac{80\!\cdots\!74}{66\!\cdots\!39}a^{2}-\frac{10\!\cdots\!41}{46\!\cdots\!73}a+\frac{10\!\cdots\!64}{66\!\cdots\!39}$, $\frac{10\!\cdots\!69}{34\!\cdots\!29}a^{26}-\frac{57\!\cdots\!60}{34\!\cdots\!29}a^{25}+\frac{29\!\cdots\!26}{34\!\cdots\!29}a^{24}-\frac{23\!\cdots\!83}{11\!\cdots\!43}a^{23}+\frac{41\!\cdots\!18}{16\!\cdots\!49}a^{22}-\frac{15\!\cdots\!59}{51\!\cdots\!97}a^{21}-\frac{28\!\cdots\!71}{37\!\cdots\!81}a^{20}+\frac{23\!\cdots\!92}{11\!\cdots\!43}a^{19}-\frac{35\!\cdots\!37}{11\!\cdots\!43}a^{18}+\frac{53\!\cdots\!91}{37\!\cdots\!81}a^{17}+\frac{35\!\cdots\!55}{53\!\cdots\!83}a^{16}-\frac{19\!\cdots\!32}{75\!\cdots\!93}a^{15}+\frac{43\!\cdots\!67}{75\!\cdots\!93}a^{14}-\frac{12\!\cdots\!70}{12\!\cdots\!09}a^{13}+\frac{67\!\cdots\!09}{41\!\cdots\!09}a^{12}-\frac{83\!\cdots\!11}{39\!\cdots\!67}a^{11}+\frac{41\!\cdots\!39}{16\!\cdots\!49}a^{10}-\frac{30\!\cdots\!86}{11\!\cdots\!43}a^{9}+\frac{28\!\cdots\!07}{11\!\cdots\!43}a^{8}-\frac{24\!\cdots\!07}{11\!\cdots\!43}a^{7}+\frac{18\!\cdots\!85}{11\!\cdots\!43}a^{6}-\frac{55\!\cdots\!95}{53\!\cdots\!83}a^{5}+\frac{21\!\cdots\!85}{37\!\cdots\!81}a^{4}-\frac{29\!\cdots\!54}{11\!\cdots\!43}a^{3}+\frac{44\!\cdots\!88}{48\!\cdots\!47}a^{2}-\frac{69\!\cdots\!66}{34\!\cdots\!29}a+\frac{98\!\cdots\!04}{34\!\cdots\!29}$, $\frac{10\!\cdots\!67}{32\!\cdots\!67}a^{26}-\frac{43\!\cdots\!09}{22\!\cdots\!69}a^{25}+\frac{22\!\cdots\!50}{22\!\cdots\!69}a^{24}-\frac{18\!\cdots\!23}{75\!\cdots\!23}a^{23}+\frac{37\!\cdots\!90}{10\!\cdots\!89}a^{22}-\frac{589848865894903}{490267873736631}a^{21}-\frac{20\!\cdots\!55}{25\!\cdots\!41}a^{20}+\frac{19\!\cdots\!21}{75\!\cdots\!23}a^{19}-\frac{30\!\cdots\!50}{75\!\cdots\!23}a^{18}+\frac{72\!\cdots\!63}{27\!\cdots\!49}a^{17}+\frac{16\!\cdots\!93}{25\!\cdots\!41}a^{16}-\frac{22\!\cdots\!90}{75\!\cdots\!23}a^{15}+\frac{53\!\cdots\!21}{75\!\cdots\!23}a^{14}-\frac{33\!\cdots\!99}{25\!\cdots\!41}a^{13}+\frac{51\!\cdots\!50}{25\!\cdots\!41}a^{12}-\frac{56\!\cdots\!24}{20\!\cdots\!79}a^{11}+\frac{25\!\cdots\!58}{75\!\cdots\!23}a^{10}-\frac{26\!\cdots\!98}{75\!\cdots\!23}a^{9}+\frac{25\!\cdots\!96}{75\!\cdots\!23}a^{8}-\frac{21\!\cdots\!39}{75\!\cdots\!23}a^{7}+\frac{79\!\cdots\!22}{37\!\cdots\!41}a^{6}-\frac{38\!\cdots\!12}{27\!\cdots\!49}a^{5}+\frac{61\!\cdots\!70}{83\!\cdots\!47}a^{4}-\frac{23\!\cdots\!63}{75\!\cdots\!23}a^{3}+\frac{30\!\cdots\!04}{32\!\cdots\!67}a^{2}-\frac{48\!\cdots\!86}{32\!\cdots\!67}a+\frac{39\!\cdots\!52}{22\!\cdots\!69}$, $a$, $\frac{62\!\cdots\!01}{34\!\cdots\!29}a^{26}-\frac{37\!\cdots\!57}{34\!\cdots\!29}a^{25}+\frac{19\!\cdots\!42}{34\!\cdots\!29}a^{24}-\frac{16\!\cdots\!66}{11\!\cdots\!43}a^{23}+\frac{30\!\cdots\!73}{16\!\cdots\!49}a^{22}-\frac{14\!\cdots\!76}{51\!\cdots\!97}a^{21}-\frac{64\!\cdots\!83}{12\!\cdots\!27}a^{20}+\frac{16\!\cdots\!94}{11\!\cdots\!43}a^{19}-\frac{35\!\cdots\!75}{16\!\cdots\!49}a^{18}+\frac{13\!\cdots\!09}{12\!\cdots\!27}a^{17}+\frac{55\!\cdots\!96}{12\!\cdots\!27}a^{16}-\frac{12\!\cdots\!76}{75\!\cdots\!93}a^{15}+\frac{29\!\cdots\!85}{75\!\cdots\!93}a^{14}-\frac{26\!\cdots\!04}{36\!\cdots\!27}a^{13}+\frac{58\!\cdots\!03}{53\!\cdots\!83}a^{12}-\frac{16\!\cdots\!00}{11\!\cdots\!43}a^{11}+\frac{19\!\cdots\!32}{11\!\cdots\!43}a^{10}-\frac{20\!\cdots\!35}{11\!\cdots\!43}a^{9}+\frac{18\!\cdots\!03}{11\!\cdots\!43}a^{8}-\frac{15\!\cdots\!68}{11\!\cdots\!43}a^{7}+\frac{10\!\cdots\!00}{11\!\cdots\!43}a^{6}-\frac{74\!\cdots\!64}{12\!\cdots\!27}a^{5}+\frac{11\!\cdots\!41}{37\!\cdots\!81}a^{4}-\frac{13\!\cdots\!89}{11\!\cdots\!43}a^{3}+\frac{15\!\cdots\!95}{48\!\cdots\!47}a^{2}-\frac{14\!\cdots\!67}{34\!\cdots\!29}a+\frac{17\!\cdots\!86}{34\!\cdots\!29}$
| 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: | \( 71844122831.24568 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{13}\cdot 71844122831.24568 \cdot 1}{2\cdot\sqrt{1543319746516623033280478216838436483146079}}\cr\approx \mathstrut & 1.37562981139276 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*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^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*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^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*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^27 - 6*x^26 + 31*x^25 - 80*x^24 + 117*x^23 - 51*x^22 - 243*x^21 + 807*x^20 - 1350*x^19 + 966*x^18 + 1935*x^17 - 9345*x^16 + 23103*x^15 - 43548*x^14 + 68751*x^13 - 94575*x^12 + 115269*x^11 - 125268*x^10 + 121785*x^9 - 105567*x^8 + 81027*x^7 - 54357*x^6 + 31122*x^5 - 14664*x^4 + 5393*x^3 - 1350*x^2 + 179*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
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 54 |
| The 15 conjugacy class representatives for $D_{27}$ |
| Character table for $D_{27}$ |
Intermediate fields
| 3.1.1759.1, 9.1.9573337234561.1 |
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 | ${\href{/padicField/2.9.0.1}{9} }^{3}$ | ${\href{/padicField/3.2.0.1}{2} }^{13}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $27$ | ${\href{/padicField/7.2.0.1}{2} }^{13}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $27$ | $27$ | $27$ | ${\href{/padicField/19.2.0.1}{2} }^{13}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $27$ | ${\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.9.0.1}{9} }^{3}$ | ${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $27$ | $27$ | $27$ | $27$ | ${\href{/padicField/59.2.0.1}{2} }^{13}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1759\)
| $\Q_{1759}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
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 *.