Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49)
gp: K = bnfinit(y^18 - 6*y^17 + 19*y^16 - 52*y^15 + 91*y^14 - 60*y^13 - 165*y^12 + 662*y^11 - 1427*y^10 + 3000*y^9 - 5876*y^8 + 7990*y^7 - 6575*y^6 + 1628*y^5 + 3595*y^4 - 4532*y^3 + 2963*y^2 - 686*y + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49)
\( x^{18} - 6 x^{17} + 19 x^{16} - 52 x^{15} + 91 x^{14} - 60 x^{13} - 165 x^{12} + 662 x^{11} - 1427 x^{10} + \cdots + 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(5847152042079256430524760064\)
\(\medspace = 2^{27}\cdot 3^{9}\cdot 19^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.88\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}19^{2/3}\approx 34.88253362095716$ | ||
| Ramified primes: |
\(2\), \(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{6}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}+\frac{1}{3}a^{11}+\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{13}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{14}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a$, $\frac{1}{63}a^{15}+\frac{2}{63}a^{14}-\frac{1}{63}a^{13}+\frac{1}{63}a^{12}+\frac{23}{63}a^{11}-\frac{1}{21}a^{10}+\frac{19}{63}a^{9}-\frac{10}{21}a^{8}-\frac{20}{63}a^{7}+\frac{1}{3}a^{6}+\frac{10}{63}a^{5}+\frac{13}{63}a^{4}+\frac{5}{21}a^{3}+\frac{3}{7}a^{2}+\frac{23}{63}a+\frac{2}{9}$, $\frac{1}{315}a^{16}-\frac{2}{315}a^{15}+\frac{4}{105}a^{14}-\frac{16}{315}a^{13}+\frac{19}{315}a^{12}-\frac{19}{63}a^{11}-\frac{11}{315}a^{10}-\frac{127}{315}a^{9}-\frac{131}{315}a^{8}+\frac{59}{315}a^{7}-\frac{11}{315}a^{6}+\frac{1}{21}a^{5}-\frac{16}{315}a^{4}+\frac{38}{105}a^{3}-\frac{17}{63}a^{2}+\frac{3}{35}a+\frac{13}{45}$, $\frac{1}{66\!\cdots\!75}a^{17}-\frac{43\!\cdots\!89}{66\!\cdots\!75}a^{16}-\frac{33\!\cdots\!94}{66\!\cdots\!75}a^{15}-\frac{12\!\cdots\!38}{89\!\cdots\!37}a^{14}+\frac{28\!\cdots\!66}{66\!\cdots\!75}a^{13}-\frac{27\!\cdots\!71}{22\!\cdots\!25}a^{12}-\frac{14\!\cdots\!11}{66\!\cdots\!75}a^{11}-\frac{39\!\cdots\!21}{26\!\cdots\!11}a^{10}-\frac{34\!\cdots\!53}{74\!\cdots\!75}a^{9}-\frac{27\!\cdots\!84}{66\!\cdots\!75}a^{8}-\frac{10\!\cdots\!43}{22\!\cdots\!25}a^{7}+\frac{15\!\cdots\!58}{74\!\cdots\!75}a^{6}+\frac{47\!\cdots\!24}{66\!\cdots\!75}a^{5}+\frac{17\!\cdots\!11}{66\!\cdots\!75}a^{4}-\frac{32\!\cdots\!74}{95\!\cdots\!25}a^{3}+\frac{10\!\cdots\!54}{22\!\cdots\!25}a^{2}-\frac{53\!\cdots\!08}{66\!\cdots\!75}a+\frac{16\!\cdots\!04}{95\!\cdots\!25}$
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: | $11$ | 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: |
$\frac{17\!\cdots\!66}{66\!\cdots\!75}a^{17}-\frac{15\!\cdots\!87}{95\!\cdots\!25}a^{16}+\frac{33\!\cdots\!16}{66\!\cdots\!75}a^{15}-\frac{17\!\cdots\!04}{13\!\cdots\!55}a^{14}+\frac{15\!\cdots\!41}{66\!\cdots\!75}a^{13}-\frac{28\!\cdots\!41}{22\!\cdots\!25}a^{12}-\frac{33\!\cdots\!01}{66\!\cdots\!75}a^{11}+\frac{37\!\cdots\!79}{21\!\cdots\!85}a^{10}-\frac{24\!\cdots\!62}{66\!\cdots\!75}a^{9}+\frac{55\!\cdots\!74}{74\!\cdots\!75}a^{8}-\frac{99\!\cdots\!54}{66\!\cdots\!75}a^{7}+\frac{18\!\cdots\!91}{95\!\cdots\!25}a^{6}-\frac{30\!\cdots\!97}{22\!\cdots\!25}a^{5}+\frac{33\!\cdots\!87}{22\!\cdots\!25}a^{4}+\frac{10\!\cdots\!96}{95\!\cdots\!25}a^{3}-\frac{99\!\cdots\!19}{95\!\cdots\!25}a^{2}+\frac{35\!\cdots\!03}{74\!\cdots\!75}a-\frac{20\!\cdots\!47}{31\!\cdots\!75}$, $\frac{17\!\cdots\!66}{66\!\cdots\!75}a^{17}-\frac{15\!\cdots\!87}{95\!\cdots\!25}a^{16}+\frac{33\!\cdots\!16}{66\!\cdots\!75}a^{15}-\frac{17\!\cdots\!04}{13\!\cdots\!55}a^{14}+\frac{15\!\cdots\!41}{66\!\cdots\!75}a^{13}-\frac{28\!\cdots\!41}{22\!\cdots\!25}a^{12}-\frac{33\!\cdots\!01}{66\!\cdots\!75}a^{11}+\frac{37\!\cdots\!79}{21\!\cdots\!85}a^{10}-\frac{24\!\cdots\!62}{66\!\cdots\!75}a^{9}+\frac{55\!\cdots\!74}{74\!\cdots\!75}a^{8}-\frac{99\!\cdots\!54}{66\!\cdots\!75}a^{7}+\frac{18\!\cdots\!91}{95\!\cdots\!25}a^{6}-\frac{30\!\cdots\!97}{22\!\cdots\!25}a^{5}+\frac{33\!\cdots\!87}{22\!\cdots\!25}a^{4}+\frac{10\!\cdots\!96}{95\!\cdots\!25}a^{3}-\frac{99\!\cdots\!19}{95\!\cdots\!25}a^{2}+\frac{35\!\cdots\!03}{74\!\cdots\!75}a-\frac{17\!\cdots\!72}{31\!\cdots\!75}$, $\frac{31\!\cdots\!44}{66\!\cdots\!75}a^{17}-\frac{21\!\cdots\!21}{66\!\cdots\!75}a^{16}+\frac{69\!\cdots\!74}{66\!\cdots\!75}a^{15}-\frac{35\!\cdots\!02}{13\!\cdots\!55}a^{14}+\frac{51\!\cdots\!43}{10\!\cdots\!25}a^{13}-\frac{15\!\cdots\!17}{66\!\cdots\!75}a^{12}-\frac{26\!\cdots\!53}{22\!\cdots\!25}a^{11}+\frac{74\!\cdots\!73}{19\!\cdots\!65}a^{10}-\frac{49\!\cdots\!28}{66\!\cdots\!75}a^{9}+\frac{95\!\cdots\!59}{66\!\cdots\!75}a^{8}-\frac{19\!\cdots\!46}{66\!\cdots\!75}a^{7}+\frac{88\!\cdots\!91}{22\!\cdots\!25}a^{6}-\frac{43\!\cdots\!73}{22\!\cdots\!25}a^{5}-\frac{99\!\cdots\!86}{66\!\cdots\!75}a^{4}+\frac{19\!\cdots\!88}{66\!\cdots\!75}a^{3}-\frac{41\!\cdots\!74}{22\!\cdots\!25}a^{2}+\frac{10\!\cdots\!21}{22\!\cdots\!25}a+\frac{91\!\cdots\!36}{95\!\cdots\!25}$, $\frac{25\!\cdots\!52}{66\!\cdots\!75}a^{17}-\frac{14\!\cdots\!08}{66\!\cdots\!75}a^{16}+\frac{46\!\cdots\!58}{74\!\cdots\!75}a^{15}-\frac{74\!\cdots\!84}{44\!\cdots\!85}a^{14}+\frac{17\!\cdots\!12}{66\!\cdots\!75}a^{13}-\frac{55\!\cdots\!46}{66\!\cdots\!75}a^{12}-\frac{15\!\cdots\!99}{22\!\cdots\!25}a^{11}+\frac{32\!\cdots\!19}{14\!\cdots\!95}a^{10}-\frac{28\!\cdots\!19}{66\!\cdots\!75}a^{9}+\frac{68\!\cdots\!43}{74\!\cdots\!75}a^{8}-\frac{12\!\cdots\!03}{66\!\cdots\!75}a^{7}+\frac{14\!\cdots\!99}{66\!\cdots\!75}a^{6}-\frac{86\!\cdots\!02}{66\!\cdots\!75}a^{5}-\frac{89\!\cdots\!23}{66\!\cdots\!75}a^{4}+\frac{81\!\cdots\!94}{66\!\cdots\!75}a^{3}-\frac{66\!\cdots\!26}{66\!\cdots\!75}a^{2}+\frac{35\!\cdots\!99}{66\!\cdots\!75}a-\frac{58\!\cdots\!07}{95\!\cdots\!25}$, $\frac{33\!\cdots\!72}{95\!\cdots\!25}a^{17}+\frac{65\!\cdots\!41}{74\!\cdots\!75}a^{16}-\frac{37\!\cdots\!01}{66\!\cdots\!75}a^{15}+\frac{14\!\cdots\!95}{89\!\cdots\!37}a^{14}-\frac{98\!\cdots\!12}{22\!\cdots\!25}a^{13}+\frac{48\!\cdots\!48}{66\!\cdots\!75}a^{12}-\frac{12\!\cdots\!19}{66\!\cdots\!75}a^{11}-\frac{56\!\cdots\!13}{26\!\cdots\!11}a^{10}+\frac{40\!\cdots\!67}{66\!\cdots\!75}a^{9}-\frac{73\!\cdots\!86}{66\!\cdots\!75}a^{8}+\frac{22\!\cdots\!12}{95\!\cdots\!25}a^{7}-\frac{31\!\cdots\!37}{66\!\cdots\!75}a^{6}+\frac{36\!\cdots\!21}{66\!\cdots\!75}a^{5}-\frac{49\!\cdots\!77}{22\!\cdots\!25}a^{4}-\frac{13\!\cdots\!72}{66\!\cdots\!75}a^{3}+\frac{25\!\cdots\!23}{66\!\cdots\!75}a^{2}-\frac{19\!\cdots\!01}{95\!\cdots\!25}a+\frac{24\!\cdots\!99}{10\!\cdots\!25}$, $\frac{16\!\cdots\!90}{26\!\cdots\!11}a^{17}-\frac{49\!\cdots\!93}{13\!\cdots\!55}a^{16}+\frac{21\!\cdots\!38}{19\!\cdots\!65}a^{15}-\frac{39\!\cdots\!76}{13\!\cdots\!55}a^{14}+\frac{75\!\cdots\!22}{14\!\cdots\!95}a^{13}-\frac{29\!\cdots\!42}{13\!\cdots\!55}a^{12}-\frac{16\!\cdots\!42}{12\!\cdots\!91}a^{11}+\frac{54\!\cdots\!33}{13\!\cdots\!55}a^{10}-\frac{10\!\cdots\!94}{13\!\cdots\!55}a^{9}+\frac{21\!\cdots\!68}{13\!\cdots\!55}a^{8}-\frac{43\!\cdots\!72}{13\!\cdots\!55}a^{7}+\frac{18\!\cdots\!61}{44\!\cdots\!85}a^{6}-\frac{21\!\cdots\!30}{89\!\cdots\!37}a^{5}-\frac{94\!\cdots\!32}{13\!\cdots\!55}a^{4}+\frac{34\!\cdots\!08}{13\!\cdots\!55}a^{3}-\frac{17\!\cdots\!89}{89\!\cdots\!37}a^{2}+\frac{23\!\cdots\!38}{44\!\cdots\!85}a-\frac{82\!\cdots\!14}{19\!\cdots\!65}$, $\frac{15\!\cdots\!28}{66\!\cdots\!75}a^{17}-\frac{94\!\cdots\!88}{74\!\cdots\!75}a^{16}+\frac{26\!\cdots\!68}{66\!\cdots\!75}a^{15}-\frac{29\!\cdots\!35}{26\!\cdots\!11}a^{14}+\frac{12\!\cdots\!48}{66\!\cdots\!75}a^{13}-\frac{27\!\cdots\!63}{22\!\cdots\!25}a^{12}-\frac{73\!\cdots\!11}{22\!\cdots\!25}a^{11}+\frac{36\!\cdots\!49}{26\!\cdots\!11}a^{10}-\frac{20\!\cdots\!81}{66\!\cdots\!75}a^{9}+\frac{61\!\cdots\!89}{95\!\cdots\!25}a^{8}-\frac{82\!\cdots\!62}{66\!\cdots\!75}a^{7}+\frac{11\!\cdots\!91}{66\!\cdots\!75}a^{6}-\frac{99\!\cdots\!53}{66\!\cdots\!75}a^{5}+\frac{47\!\cdots\!94}{95\!\cdots\!25}a^{4}+\frac{48\!\cdots\!96}{66\!\cdots\!75}a^{3}-\frac{66\!\cdots\!14}{66\!\cdots\!75}a^{2}+\frac{51\!\cdots\!26}{66\!\cdots\!75}a-\frac{10\!\cdots\!38}{95\!\cdots\!25}$, $\frac{19\!\cdots\!62}{66\!\cdots\!75}a^{17}-\frac{40\!\cdots\!61}{22\!\cdots\!25}a^{16}+\frac{40\!\cdots\!02}{66\!\cdots\!75}a^{15}-\frac{73\!\cdots\!92}{44\!\cdots\!85}a^{14}+\frac{66\!\cdots\!94}{22\!\cdots\!25}a^{13}-\frac{15\!\cdots\!41}{66\!\cdots\!75}a^{12}-\frac{32\!\cdots\!32}{66\!\cdots\!75}a^{11}+\frac{40\!\cdots\!29}{19\!\cdots\!65}a^{10}-\frac{44\!\cdots\!42}{95\!\cdots\!25}a^{9}+\frac{64\!\cdots\!07}{66\!\cdots\!75}a^{8}-\frac{18\!\cdots\!69}{95\!\cdots\!25}a^{7}+\frac{17\!\cdots\!54}{66\!\cdots\!75}a^{6}-\frac{15\!\cdots\!37}{66\!\cdots\!75}a^{5}+\frac{39\!\cdots\!08}{74\!\cdots\!75}a^{4}+\frac{86\!\cdots\!24}{66\!\cdots\!75}a^{3}-\frac{10\!\cdots\!31}{66\!\cdots\!75}a^{2}+\frac{62\!\cdots\!49}{66\!\cdots\!75}a-\frac{40\!\cdots\!24}{31\!\cdots\!75}$, $\frac{88\!\cdots\!98}{22\!\cdots\!25}a^{17}-\frac{15\!\cdots\!41}{66\!\cdots\!75}a^{16}+\frac{50\!\cdots\!14}{66\!\cdots\!75}a^{15}-\frac{18\!\cdots\!53}{89\!\cdots\!37}a^{14}+\frac{24\!\cdots\!54}{66\!\cdots\!75}a^{13}-\frac{16\!\cdots\!97}{66\!\cdots\!75}a^{12}-\frac{43\!\cdots\!09}{66\!\cdots\!75}a^{11}+\frac{70\!\cdots\!31}{26\!\cdots\!11}a^{10}-\frac{38\!\cdots\!63}{66\!\cdots\!75}a^{9}+\frac{79\!\cdots\!29}{66\!\cdots\!75}a^{8}-\frac{15\!\cdots\!01}{66\!\cdots\!75}a^{7}+\frac{21\!\cdots\!68}{66\!\cdots\!75}a^{6}-\frac{59\!\cdots\!98}{22\!\cdots\!25}a^{5}+\frac{39\!\cdots\!84}{66\!\cdots\!75}a^{4}+\frac{34\!\cdots\!36}{22\!\cdots\!25}a^{3}-\frac{12\!\cdots\!72}{66\!\cdots\!75}a^{2}+\frac{26\!\cdots\!66}{22\!\cdots\!25}a-\frac{15\!\cdots\!24}{95\!\cdots\!25}$, $\frac{81\!\cdots\!14}{31\!\cdots\!75}a^{17}-\frac{10\!\cdots\!51}{66\!\cdots\!75}a^{16}+\frac{10\!\cdots\!03}{22\!\cdots\!25}a^{15}-\frac{17\!\cdots\!79}{13\!\cdots\!55}a^{14}+\frac{48\!\cdots\!38}{22\!\cdots\!25}a^{13}-\frac{29\!\cdots\!29}{22\!\cdots\!25}a^{12}-\frac{28\!\cdots\!84}{66\!\cdots\!75}a^{11}+\frac{22\!\cdots\!07}{13\!\cdots\!55}a^{10}-\frac{33\!\cdots\!49}{95\!\cdots\!25}a^{9}+\frac{48\!\cdots\!89}{66\!\cdots\!75}a^{8}-\frac{94\!\cdots\!16}{66\!\cdots\!75}a^{7}+\frac{12\!\cdots\!03}{66\!\cdots\!75}a^{6}-\frac{99\!\cdots\!94}{66\!\cdots\!75}a^{5}+\frac{16\!\cdots\!94}{66\!\cdots\!75}a^{4}+\frac{21\!\cdots\!56}{22\!\cdots\!25}a^{3}-\frac{71\!\cdots\!47}{66\!\cdots\!75}a^{2}+\frac{42\!\cdots\!28}{66\!\cdots\!75}a-\frac{69\!\cdots\!29}{95\!\cdots\!25}$, $\frac{26\!\cdots\!78}{22\!\cdots\!25}a^{17}-\frac{43\!\cdots\!54}{74\!\cdots\!75}a^{16}+\frac{10\!\cdots\!49}{66\!\cdots\!75}a^{15}-\frac{56\!\cdots\!09}{13\!\cdots\!55}a^{14}+\frac{37\!\cdots\!79}{66\!\cdots\!75}a^{13}+\frac{48\!\cdots\!43}{66\!\cdots\!75}a^{12}-\frac{13\!\cdots\!74}{66\!\cdots\!75}a^{11}+\frac{79\!\cdots\!08}{14\!\cdots\!95}a^{10}-\frac{66\!\cdots\!48}{66\!\cdots\!75}a^{9}+\frac{16\!\cdots\!31}{74\!\cdots\!75}a^{8}-\frac{28\!\cdots\!51}{66\!\cdots\!75}a^{7}+\frac{30\!\cdots\!12}{74\!\cdots\!75}a^{6}-\frac{12\!\cdots\!59}{66\!\cdots\!75}a^{5}-\frac{35\!\cdots\!91}{66\!\cdots\!75}a^{4}+\frac{55\!\cdots\!66}{22\!\cdots\!25}a^{3}-\frac{31\!\cdots\!64}{22\!\cdots\!25}a^{2}+\frac{91\!\cdots\!33}{66\!\cdots\!75}a-\frac{20\!\cdots\!19}{95\!\cdots\!25}$
| 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: | \( 6627856.786636858 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 6627856.786636858 \cdot 1}{2\cdot\sqrt{5847152042079256430524760064}}\cr\approx \mathstrut & 0.170659359263912 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49)
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^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49, 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^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49);
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^18 - 6*x^17 + 19*x^16 - 52*x^15 + 91*x^14 - 60*x^13 - 165*x^12 + 662*x^11 - 1427*x^10 + 3000*x^9 - 5876*x^8 + 7990*x^7 - 6575*x^6 + 1628*x^5 + 3595*x^4 - 4532*x^3 + 2963*x^2 - 686*x + 49);
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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{6}) \), 3.1.2888.1, 3.3.361.1, 6.2.1801557504.7, 6.6.1801557504.1, 9.3.24087491072.1 |
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]
Sibling fields
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.24904730935296.2 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.24904730935296.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{9}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.2.0.1}{2} }^{9}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.9.7 | $x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 19.9.6.2 | $x^{9} + 981 x^{7} + 108 x^{6} + 316911 x^{5} + 20529 x^{4} + 34115982 x^{3} + 10990188 x^{2} + 130942880 x + 566550143$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.152.6t1.c.a | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.456.6t1.b.a | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.6.1801557504.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| 1.152.6t1.c.b | $1$ | $ 2^{3} \cdot 19 $ | 6.0.66724352.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.456.6t1.b.b | $1$ | $ 2^{3} \cdot 3 \cdot 19 $ | 6.6.1801557504.1 | $C_6$ (as 6T1) | $0$ | $1$ |
| * | 2.2888.3t2.b.a | $2$ | $ 2^{3} \cdot 19^{2}$ | 3.1.2888.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.25992.6t3.a.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19^{2}$ | 6.2.1801557504.7 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.1368.12t18.a.a | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19 $ | 18.6.5847152042079256430524760064.2 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.152.6t5.a.a | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1368.12t18.a.b | $2$ | $ 2^{3} \cdot 3^{2} \cdot 19 $ | 18.6.5847152042079256430524760064.2 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.152.6t5.a.b | $2$ | $ 2^{3} \cdot 19 $ | 6.0.184832.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.