Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831)
gp: K = bnfinit(y^18 - 3*y^17 - 42*y^15 - 110*y^14 + 332*y^13 + 219*y^12 + 1276*y^11 - 3494*y^10 - 12539*y^9 + 24077*y^8 + 22738*y^7 + 12556*y^6 - 9361*y^5 - 277423*y^4 + 8295*y^3 + 548514*y^2 + 236088*y - 60831, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831)
\( x^{18} - 3 x^{17} - 42 x^{15} - 110 x^{14} + 332 x^{13} + 219 x^{12} + 1276 x^{11} - 3494 x^{10} + \cdots - 60831 \)
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: |
\(128607482636630365539205392633856\)
\(\medspace = 2^{12}\cdot 7^{12}\cdot 197^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.79\) | 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: | $2^{2/3}7^{2/3}197^{1/2}\approx 81.53019296306911$ | ||
| Ramified primes: |
\(2\), \(7\), \(197\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{15}a^{11}+\frac{1}{15}a^{10}-\frac{2}{15}a^{9}-\frac{2}{15}a^{8}+\frac{2}{15}a^{7}+\frac{7}{15}a^{6}+\frac{2}{15}a^{5}+\frac{7}{15}a^{4}+\frac{1}{15}a^{3}+\frac{2}{5}a^{2}+\frac{4}{15}a-\frac{2}{5}$, $\frac{1}{15}a^{12}+\frac{2}{15}a^{10}-\frac{1}{15}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{4}{15}a^{4}-\frac{1}{3}a^{3}+\frac{1}{5}a^{2}-\frac{1}{3}a+\frac{2}{5}$, $\frac{1}{15}a^{13}-\frac{2}{15}a^{10}-\frac{2}{15}a^{9}-\frac{1}{15}a^{8}+\frac{1}{15}a^{7}-\frac{4}{15}a^{6}-\frac{4}{15}a^{4}+\frac{1}{15}a^{3}-\frac{2}{15}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{45}a^{14}-\frac{1}{45}a^{13}+\frac{1}{45}a^{11}+\frac{1}{15}a^{10}-\frac{1}{9}a^{9}-\frac{4}{45}a^{8}-\frac{14}{45}a^{7}+\frac{2}{9}a^{6}-\frac{13}{45}a^{5}-\frac{19}{45}a^{4}+\frac{8}{45}a^{2}+\frac{2}{15}a$, $\frac{1}{45}a^{15}-\frac{1}{45}a^{13}+\frac{1}{45}a^{12}+\frac{1}{45}a^{11}-\frac{1}{9}a^{10}-\frac{1}{15}a^{9}+\frac{1}{15}a^{8}+\frac{4}{9}a^{7}-\frac{1}{5}a^{6}-\frac{8}{45}a^{5}+\frac{4}{9}a^{4}-\frac{2}{9}a^{3}+\frac{11}{45}a^{2}-\frac{7}{15}a+\frac{2}{5}$, $\frac{1}{135}a^{16}+\frac{1}{135}a^{15}+\frac{1}{135}a^{14}+\frac{4}{135}a^{13}+\frac{2}{135}a^{12}+\frac{4}{135}a^{11}+\frac{22}{135}a^{10}+\frac{11}{135}a^{9}+\frac{4}{45}a^{8}-\frac{59}{135}a^{7}-\frac{2}{5}a^{6}-\frac{62}{135}a^{5}-\frac{8}{27}a^{4}-\frac{2}{135}a^{3}+\frac{4}{9}a^{2}-\frac{1}{15}a+\frac{2}{5}$, $\frac{1}{24\!\cdots\!25}a^{17}-\frac{37\!\cdots\!01}{55\!\cdots\!25}a^{16}+\frac{55\!\cdots\!87}{55\!\cdots\!25}a^{15}-\frac{20\!\cdots\!74}{82\!\cdots\!75}a^{14}-\frac{21\!\cdots\!36}{24\!\cdots\!25}a^{13}-\frac{78\!\cdots\!06}{24\!\cdots\!25}a^{12}+\frac{19\!\cdots\!73}{91\!\cdots\!75}a^{11}+\frac{37\!\cdots\!69}{24\!\cdots\!25}a^{10}-\frac{25\!\cdots\!92}{24\!\cdots\!25}a^{9}-\frac{14\!\cdots\!29}{99\!\cdots\!85}a^{8}-\frac{11\!\cdots\!23}{24\!\cdots\!25}a^{7}-\frac{29\!\cdots\!71}{24\!\cdots\!25}a^{6}-\frac{79\!\cdots\!37}{24\!\cdots\!25}a^{5}-\frac{77\!\cdots\!32}{24\!\cdots\!25}a^{4}+\frac{10\!\cdots\!96}{24\!\cdots\!25}a^{3}-\frac{47\!\cdots\!29}{82\!\cdots\!75}a^{2}+\frac{59\!\cdots\!52}{27\!\cdots\!25}a+\frac{34\!\cdots\!91}{91\!\cdots\!75}$
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
$C_{12}$, which has order $12$ (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: |
$\frac{33\!\cdots\!58}{45\!\cdots\!25}a^{17}-\frac{36\!\cdots\!53}{10\!\cdots\!25}a^{16}+\frac{39\!\cdots\!76}{10\!\cdots\!25}a^{15}-\frac{49\!\cdots\!42}{15\!\cdots\!75}a^{14}-\frac{74\!\cdots\!38}{45\!\cdots\!25}a^{13}+\frac{18\!\cdots\!02}{45\!\cdots\!25}a^{12}-\frac{77\!\cdots\!98}{50\!\cdots\!25}a^{11}+\frac{43\!\cdots\!52}{45\!\cdots\!25}a^{10}-\frac{21\!\cdots\!36}{45\!\cdots\!25}a^{9}-\frac{96\!\cdots\!21}{18\!\cdots\!65}a^{8}+\frac{14\!\cdots\!66}{45\!\cdots\!25}a^{7}-\frac{51\!\cdots\!68}{45\!\cdots\!25}a^{6}+\frac{12\!\cdots\!54}{45\!\cdots\!25}a^{5}-\frac{24\!\cdots\!06}{45\!\cdots\!25}a^{4}-\frac{11\!\cdots\!82}{45\!\cdots\!25}a^{3}+\frac{41\!\cdots\!18}{15\!\cdots\!75}a^{2}+\frac{14\!\cdots\!66}{50\!\cdots\!25}a-\frac{18\!\cdots\!22}{16\!\cdots\!75}$, $\frac{12\!\cdots\!87}{27\!\cdots\!25}a^{17}-\frac{10\!\cdots\!93}{55\!\cdots\!25}a^{16}+\frac{12\!\cdots\!06}{55\!\cdots\!25}a^{15}-\frac{60\!\cdots\!14}{27\!\cdots\!25}a^{14}-\frac{59\!\cdots\!82}{27\!\cdots\!25}a^{13}+\frac{43\!\cdots\!78}{27\!\cdots\!25}a^{12}-\frac{24\!\cdots\!98}{27\!\cdots\!25}a^{11}+\frac{19\!\cdots\!03}{27\!\cdots\!25}a^{10}-\frac{25\!\cdots\!27}{10\!\cdots\!75}a^{9}-\frac{25\!\cdots\!71}{11\!\cdots\!65}a^{8}+\frac{11\!\cdots\!33}{91\!\cdots\!75}a^{7}-\frac{89\!\cdots\!02}{27\!\cdots\!25}a^{6}+\frac{33\!\cdots\!81}{27\!\cdots\!25}a^{5}-\frac{58\!\cdots\!59}{27\!\cdots\!25}a^{4}-\frac{80\!\cdots\!16}{91\!\cdots\!75}a^{3}+\frac{29\!\cdots\!09}{30\!\cdots\!25}a^{2}+\frac{40\!\cdots\!99}{30\!\cdots\!25}a-\frac{90\!\cdots\!83}{10\!\cdots\!75}$, $\frac{13\!\cdots\!26}{27\!\cdots\!25}a^{17}-\frac{10\!\cdots\!99}{55\!\cdots\!25}a^{16}+\frac{10\!\cdots\!98}{55\!\cdots\!25}a^{15}-\frac{61\!\cdots\!97}{27\!\cdots\!25}a^{14}-\frac{29\!\cdots\!62}{91\!\cdots\!75}a^{13}+\frac{17\!\cdots\!73}{91\!\cdots\!75}a^{12}-\frac{19\!\cdots\!54}{27\!\cdots\!25}a^{11}+\frac{21\!\cdots\!16}{30\!\cdots\!25}a^{10}-\frac{65\!\cdots\!92}{27\!\cdots\!25}a^{9}-\frac{43\!\cdots\!54}{11\!\cdots\!65}a^{8}+\frac{42\!\cdots\!27}{27\!\cdots\!25}a^{7}-\frac{82\!\cdots\!71}{27\!\cdots\!25}a^{6}+\frac{94\!\cdots\!21}{91\!\cdots\!75}a^{5}-\frac{12\!\cdots\!69}{91\!\cdots\!75}a^{4}-\frac{35\!\cdots\!04}{27\!\cdots\!25}a^{3}+\frac{10\!\cdots\!96}{91\!\cdots\!75}a^{2}+\frac{50\!\cdots\!52}{30\!\cdots\!25}a-\frac{33\!\cdots\!84}{10\!\cdots\!75}$, $\frac{71\!\cdots\!43}{27\!\cdots\!25}a^{17}-\frac{61\!\cdots\!07}{55\!\cdots\!25}a^{16}+\frac{79\!\cdots\!89}{55\!\cdots\!25}a^{15}-\frac{35\!\cdots\!71}{27\!\cdots\!25}a^{14}-\frac{32\!\cdots\!48}{27\!\cdots\!25}a^{13}+\frac{28\!\cdots\!67}{27\!\cdots\!25}a^{12}-\frac{19\!\cdots\!72}{27\!\cdots\!25}a^{11}+\frac{11\!\cdots\!17}{27\!\cdots\!25}a^{10}-\frac{13\!\cdots\!52}{91\!\cdots\!75}a^{9}-\frac{15\!\cdots\!07}{11\!\cdots\!65}a^{8}+\frac{24\!\cdots\!04}{30\!\cdots\!25}a^{7}-\frac{10\!\cdots\!03}{27\!\cdots\!25}a^{6}+\frac{22\!\cdots\!34}{27\!\cdots\!25}a^{5}-\frac{39\!\cdots\!76}{27\!\cdots\!25}a^{4}-\frac{47\!\cdots\!99}{91\!\cdots\!75}a^{3}+\frac{58\!\cdots\!03}{91\!\cdots\!75}a^{2}+\frac{65\!\cdots\!37}{10\!\cdots\!75}a-\frac{16\!\cdots\!62}{10\!\cdots\!75}$, $\frac{29\!\cdots\!94}{11\!\cdots\!65}a^{17}-\frac{61\!\cdots\!91}{11\!\cdots\!65}a^{16}-\frac{31\!\cdots\!69}{11\!\cdots\!65}a^{15}-\frac{91\!\cdots\!92}{11\!\cdots\!65}a^{14}-\frac{70\!\cdots\!87}{11\!\cdots\!65}a^{13}+\frac{46\!\cdots\!21}{11\!\cdots\!65}a^{12}+\frac{35\!\cdots\!64}{11\!\cdots\!65}a^{11}+\frac{51\!\cdots\!48}{22\!\cdots\!53}a^{10}+\frac{16\!\cdots\!56}{81\!\cdots\!39}a^{9}-\frac{82\!\cdots\!19}{11\!\cdots\!65}a^{8}+\frac{71\!\cdots\!73}{36\!\cdots\!55}a^{7}+\frac{30\!\cdots\!37}{11\!\cdots\!65}a^{6}-\frac{26\!\cdots\!53}{11\!\cdots\!65}a^{5}+\frac{20\!\cdots\!54}{11\!\cdots\!65}a^{4}-\frac{47\!\cdots\!28}{36\!\cdots\!55}a^{3}-\frac{76\!\cdots\!76}{73\!\cdots\!51}a^{2}+\frac{35\!\cdots\!09}{12\!\cdots\!85}a+\frac{17\!\cdots\!00}{81\!\cdots\!39}$, $\frac{37\!\cdots\!32}{27\!\cdots\!25}a^{17}-\frac{29\!\cdots\!58}{55\!\cdots\!25}a^{16}+\frac{39\!\cdots\!96}{55\!\cdots\!25}a^{15}-\frac{20\!\cdots\!54}{27\!\cdots\!25}a^{14}-\frac{19\!\cdots\!02}{27\!\cdots\!25}a^{13}+\frac{10\!\cdots\!08}{27\!\cdots\!25}a^{12}-\frac{41\!\cdots\!78}{27\!\cdots\!25}a^{11}+\frac{67\!\cdots\!33}{27\!\cdots\!25}a^{10}-\frac{64\!\cdots\!73}{91\!\cdots\!75}a^{9}-\frac{62\!\cdots\!04}{11\!\cdots\!65}a^{8}+\frac{81\!\cdots\!71}{30\!\cdots\!25}a^{7}-\frac{89\!\cdots\!72}{27\!\cdots\!25}a^{6}+\frac{10\!\cdots\!91}{27\!\cdots\!25}a^{5}-\frac{17\!\cdots\!99}{27\!\cdots\!25}a^{4}-\frac{22\!\cdots\!76}{91\!\cdots\!75}a^{3}+\frac{16\!\cdots\!47}{91\!\cdots\!75}a^{2}+\frac{12\!\cdots\!14}{30\!\cdots\!25}a+\frac{13\!\cdots\!37}{10\!\cdots\!75}$, $\frac{23\!\cdots\!23}{22\!\cdots\!53}a^{17}-\frac{96\!\cdots\!03}{22\!\cdots\!53}a^{16}+\frac{12\!\cdots\!95}{22\!\cdots\!53}a^{15}-\frac{12\!\cdots\!05}{22\!\cdots\!53}a^{14}-\frac{39\!\cdots\!53}{73\!\cdots\!51}a^{13}+\frac{26\!\cdots\!33}{73\!\cdots\!51}a^{12}-\frac{42\!\cdots\!06}{22\!\cdots\!53}a^{11}+\frac{15\!\cdots\!13}{73\!\cdots\!51}a^{10}-\frac{13\!\cdots\!53}{22\!\cdots\!53}a^{9}-\frac{11\!\cdots\!78}{22\!\cdots\!53}a^{8}+\frac{54\!\cdots\!49}{22\!\cdots\!53}a^{7}-\frac{19\!\cdots\!97}{22\!\cdots\!53}a^{6}+\frac{42\!\cdots\!12}{73\!\cdots\!51}a^{5}-\frac{15\!\cdots\!45}{24\!\cdots\!17}a^{4}-\frac{40\!\cdots\!42}{22\!\cdots\!53}a^{3}+\frac{90\!\cdots\!17}{81\!\cdots\!39}a^{2}+\frac{99\!\cdots\!12}{81\!\cdots\!39}a-\frac{30\!\cdots\!48}{81\!\cdots\!39}$, $\frac{98\!\cdots\!17}{11\!\cdots\!65}a^{17}-\frac{42\!\cdots\!24}{11\!\cdots\!65}a^{16}+\frac{64\!\cdots\!48}{11\!\cdots\!65}a^{15}-\frac{52\!\cdots\!28}{11\!\cdots\!65}a^{14}-\frac{13\!\cdots\!77}{36\!\cdots\!55}a^{13}+\frac{21\!\cdots\!74}{73\!\cdots\!51}a^{12}-\frac{32\!\cdots\!53}{11\!\cdots\!65}a^{11}+\frac{58\!\cdots\!59}{36\!\cdots\!55}a^{10}-\frac{11\!\cdots\!08}{22\!\cdots\!53}a^{9}-\frac{27\!\cdots\!72}{11\!\cdots\!65}a^{8}+\frac{24\!\cdots\!32}{11\!\cdots\!65}a^{7}-\frac{18\!\cdots\!47}{11\!\cdots\!65}a^{6}+\frac{41\!\cdots\!99}{12\!\cdots\!85}a^{5}-\frac{48\!\cdots\!38}{73\!\cdots\!51}a^{4}-\frac{13\!\cdots\!97}{11\!\cdots\!65}a^{3}+\frac{71\!\cdots\!31}{36\!\cdots\!55}a^{2}+\frac{23\!\cdots\!02}{12\!\cdots\!85}a-\frac{24\!\cdots\!77}{40\!\cdots\!95}$, $\frac{40\!\cdots\!37}{50\!\cdots\!25}a^{17}-\frac{23\!\cdots\!18}{10\!\cdots\!25}a^{16}-\frac{31\!\cdots\!84}{10\!\cdots\!25}a^{15}-\frac{17\!\cdots\!89}{50\!\cdots\!25}a^{14}-\frac{15\!\cdots\!19}{16\!\cdots\!75}a^{13}+\frac{43\!\cdots\!76}{16\!\cdots\!75}a^{12}+\frac{10\!\cdots\!27}{50\!\cdots\!25}a^{11}+\frac{18\!\cdots\!76}{16\!\cdots\!75}a^{10}-\frac{13\!\cdots\!04}{50\!\cdots\!25}a^{9}-\frac{42\!\cdots\!67}{40\!\cdots\!57}a^{8}+\frac{91\!\cdots\!99}{50\!\cdots\!25}a^{7}+\frac{10\!\cdots\!48}{50\!\cdots\!25}a^{6}+\frac{23\!\cdots\!77}{16\!\cdots\!75}a^{5}-\frac{38\!\cdots\!76}{55\!\cdots\!25}a^{4}-\frac{11\!\cdots\!73}{50\!\cdots\!25}a^{3}-\frac{41\!\cdots\!23}{16\!\cdots\!75}a^{2}+\frac{24\!\cdots\!49}{55\!\cdots\!25}a+\frac{48\!\cdots\!42}{18\!\cdots\!75}$, $\frac{80\!\cdots\!89}{81\!\cdots\!39}a^{17}-\frac{17\!\cdots\!69}{12\!\cdots\!85}a^{16}-\frac{36\!\cdots\!33}{12\!\cdots\!85}a^{15}-\frac{19\!\cdots\!83}{40\!\cdots\!95}a^{14}-\frac{42\!\cdots\!25}{24\!\cdots\!17}a^{13}+\frac{10\!\cdots\!49}{12\!\cdots\!85}a^{12}+\frac{69\!\cdots\!88}{12\!\cdots\!85}a^{11}+\frac{91\!\cdots\!26}{40\!\cdots\!95}a^{10}-\frac{53\!\cdots\!97}{12\!\cdots\!85}a^{9}-\frac{58\!\cdots\!18}{40\!\cdots\!95}a^{8}-\frac{11\!\cdots\!66}{12\!\cdots\!85}a^{7}+\frac{87\!\cdots\!48}{24\!\cdots\!17}a^{6}+\frac{28\!\cdots\!22}{40\!\cdots\!95}a^{5}+\frac{20\!\cdots\!69}{40\!\cdots\!95}a^{4}-\frac{97\!\cdots\!84}{40\!\cdots\!95}a^{3}-\frac{49\!\cdots\!92}{12\!\cdots\!85}a^{2}-\frac{60\!\cdots\!19}{40\!\cdots\!95}a+\frac{16\!\cdots\!98}{40\!\cdots\!95}$, $\frac{12\!\cdots\!62}{82\!\cdots\!75}a^{17}-\frac{26\!\cdots\!01}{55\!\cdots\!25}a^{16}+\frac{10\!\cdots\!87}{55\!\cdots\!25}a^{15}-\frac{15\!\cdots\!63}{27\!\cdots\!25}a^{14}-\frac{14\!\cdots\!57}{82\!\cdots\!75}a^{13}+\frac{49\!\cdots\!03}{82\!\cdots\!75}a^{12}+\frac{22\!\cdots\!59}{27\!\cdots\!25}a^{11}+\frac{14\!\cdots\!78}{82\!\cdots\!75}a^{10}-\frac{38\!\cdots\!04}{82\!\cdots\!75}a^{9}-\frac{68\!\cdots\!56}{33\!\cdots\!95}a^{8}+\frac{42\!\cdots\!49}{82\!\cdots\!75}a^{7}+\frac{23\!\cdots\!98}{82\!\cdots\!75}a^{6}+\frac{13\!\cdots\!31}{82\!\cdots\!75}a^{5}+\frac{23\!\cdots\!41}{82\!\cdots\!75}a^{4}-\frac{39\!\cdots\!23}{82\!\cdots\!75}a^{3}+\frac{70\!\cdots\!77}{27\!\cdots\!25}a^{2}+\frac{54\!\cdots\!49}{91\!\cdots\!75}a-\frac{39\!\cdots\!58}{30\!\cdots\!25}$
| 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: | \( 1033559423.06 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 1033559423.06 \cdot 12}{2\cdot\sqrt{128607482636630365539205392633856}}\cr\approx \mathstrut & 2.15334078737 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831)
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 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831, 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 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831);
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 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831);
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 72 |
| The 9 conjugacy class representatives for $C_3:S_4$ |
| Character table for $C_3:S_4$ |
Intermediate fields
| 3.3.9653.1, 3.3.788.1, 3.3.38612.2, 3.3.38612.1, 6.2.93180409.3, 9.9.11340523913674816.1 |
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
| Degree 12 siblings: | data not computed |
| Degree 18 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 12.0.8936332843975755008.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.2.0.1}{2} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.4.0.1}{4} }^{3}{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.12.8.1 | $x^{12} + 11 x^{9} + 3 x^{8} - 9 x^{6} - 90 x^{5} + 3 x^{4} - 27 x^{3} + 135 x^{2} + 27 x + 55$ | $3$ | $4$ | $8$ | $C_3 : C_4$ | $[\ ]_{3}^{4}$ | |
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(197\)
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{197}$ | $x + 195$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.2.1.1 | $x^{2} + 197$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 197.4.2.1 | $x^{4} + 66970 x^{3} + 1127637879 x^{2} + 214058019190 x + 3496206875$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |