Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 1)
gp: K = bnfinit(y^36 + 66*y^34 + 1899*y^32 + 31364*y^30 + 329559*y^28 + 2308518*y^26 + 10994901*y^24 + 35728458*y^22 + 78637230*y^20 + 115641708*y^18 + 111976572*y^16 + 70684362*y^14 + 28999138*y^12 + 7688325*y^10 + 1293075*y^8 + 132111*y^6 + 7476*y^4 + 189*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 1)
\( x^{36} + 66 x^{34} + 1899 x^{32} + 31364 x^{30} + 329559 x^{28} + 2308518 x^{26} + 10994901 x^{24} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14361527681487998235176534750111030030248040605937785686475463983104\)
\(\medspace = 2^{36}\cdot 3^{48}\cdot 13^{30}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(73.36\) | 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: | $2\cdot 3^{4/3}13^{5/6}\approx 73.36312046236682$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(468=2^{2}\cdot 3^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{468}(1,·)$, $\chi_{468}(259,·)$, $\chi_{468}(133,·)$, $\chi_{468}(391,·)$, $\chi_{468}(139,·)$, $\chi_{468}(367,·)$, $\chi_{468}(277,·)$, $\chi_{468}(25,·)$, $\chi_{468}(283,·)$, $\chi_{468}(157,·)$, $\chi_{468}(415,·)$, $\chi_{468}(289,·)$, $\chi_{468}(295,·)$, $\chi_{468}(43,·)$, $\chi_{468}(433,·)$, $\chi_{468}(181,·)$, $\chi_{468}(55,·)$, $\chi_{468}(313,·)$, $\chi_{468}(445,·)$, $\chi_{468}(451,·)$, $\chi_{468}(49,·)$, $\chi_{468}(199,·)$, $\chi_{468}(439,·)$, $\chi_{468}(205,·)$, $\chi_{468}(79,·)$, $\chi_{468}(337,·)$, $\chi_{468}(211,·)$, $\chi_{468}(217,·)$, $\chi_{468}(355,·)$, $\chi_{468}(103,·)$, $\chi_{468}(361,·)$, $\chi_{468}(235,·)$, $\chi_{468}(61,·)$, $\chi_{468}(373,·)$, $\chi_{468}(121,·)$, $\chi_{468}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{53}a^{28}+\frac{13}{53}a^{26}-\frac{4}{53}a^{24}+\frac{2}{53}a^{20}+\frac{6}{53}a^{18}-\frac{7}{53}a^{16}+\frac{22}{53}a^{14}+\frac{16}{53}a^{12}+\frac{1}{53}a^{10}-\frac{12}{53}a^{8}+\frac{19}{53}a^{6}-\frac{24}{53}a^{4}-\frac{8}{53}a^{2}-\frac{25}{53}$, $\frac{1}{53}a^{29}+\frac{13}{53}a^{27}-\frac{4}{53}a^{25}+\frac{2}{53}a^{21}+\frac{6}{53}a^{19}-\frac{7}{53}a^{17}+\frac{22}{53}a^{15}+\frac{16}{53}a^{13}+\frac{1}{53}a^{11}-\frac{12}{53}a^{9}+\frac{19}{53}a^{7}-\frac{24}{53}a^{5}-\frac{8}{53}a^{3}-\frac{25}{53}a$, $\frac{1}{53}a^{30}-\frac{14}{53}a^{26}-\frac{1}{53}a^{24}+\frac{2}{53}a^{22}-\frac{20}{53}a^{20}+\frac{21}{53}a^{18}+\frac{7}{53}a^{16}-\frac{5}{53}a^{14}+\frac{5}{53}a^{12}-\frac{25}{53}a^{10}+\frac{16}{53}a^{8}-\frac{6}{53}a^{6}-\frac{14}{53}a^{4}+\frac{26}{53}a^{2}+\frac{7}{53}$, $\frac{1}{53}a^{31}-\frac{14}{53}a^{27}-\frac{1}{53}a^{25}+\frac{2}{53}a^{23}-\frac{20}{53}a^{21}+\frac{21}{53}a^{19}+\frac{7}{53}a^{17}-\frac{5}{53}a^{15}+\frac{5}{53}a^{13}-\frac{25}{53}a^{11}+\frac{16}{53}a^{9}-\frac{6}{53}a^{7}-\frac{14}{53}a^{5}+\frac{26}{53}a^{3}+\frac{7}{53}a$, $\frac{1}{53}a^{32}+\frac{22}{53}a^{26}-\frac{1}{53}a^{24}-\frac{20}{53}a^{22}-\frac{4}{53}a^{20}-\frac{15}{53}a^{18}+\frac{3}{53}a^{16}-\frac{5}{53}a^{14}-\frac{13}{53}a^{12}-\frac{23}{53}a^{10}-\frac{15}{53}a^{8}-\frac{13}{53}a^{6}+\frac{8}{53}a^{4}+\frac{1}{53}a^{2}+\frac{21}{53}$, $\frac{1}{53}a^{33}+\frac{22}{53}a^{27}-\frac{1}{53}a^{25}-\frac{20}{53}a^{23}-\frac{4}{53}a^{21}-\frac{15}{53}a^{19}+\frac{3}{53}a^{17}-\frac{5}{53}a^{15}-\frac{13}{53}a^{13}-\frac{23}{53}a^{11}-\frac{15}{53}a^{9}-\frac{13}{53}a^{7}+\frac{8}{53}a^{5}+\frac{1}{53}a^{3}+\frac{21}{53}a$, $\frac{1}{47\!\cdots\!53}a^{34}+\frac{38\!\cdots\!23}{47\!\cdots\!53}a^{32}-\frac{33\!\cdots\!27}{47\!\cdots\!53}a^{30}-\frac{20\!\cdots\!78}{47\!\cdots\!53}a^{28}+\frac{22\!\cdots\!51}{47\!\cdots\!53}a^{26}-\frac{16\!\cdots\!90}{47\!\cdots\!53}a^{24}+\frac{89\!\cdots\!95}{47\!\cdots\!53}a^{22}+\frac{15\!\cdots\!98}{47\!\cdots\!53}a^{20}+\frac{10\!\cdots\!91}{47\!\cdots\!53}a^{18}-\frac{16\!\cdots\!05}{47\!\cdots\!53}a^{16}-\frac{12\!\cdots\!42}{47\!\cdots\!53}a^{14}-\frac{80\!\cdots\!48}{47\!\cdots\!53}a^{12}+\frac{19\!\cdots\!70}{47\!\cdots\!53}a^{10}-\frac{22\!\cdots\!55}{47\!\cdots\!53}a^{8}-\frac{14\!\cdots\!17}{47\!\cdots\!53}a^{6}+\frac{13\!\cdots\!85}{47\!\cdots\!53}a^{4}+\frac{47\!\cdots\!32}{47\!\cdots\!53}a^{2}-\frac{55\!\cdots\!39}{47\!\cdots\!53}$, $\frac{1}{47\!\cdots\!53}a^{35}+\frac{38\!\cdots\!23}{47\!\cdots\!53}a^{33}-\frac{33\!\cdots\!27}{47\!\cdots\!53}a^{31}-\frac{20\!\cdots\!78}{47\!\cdots\!53}a^{29}+\frac{22\!\cdots\!51}{47\!\cdots\!53}a^{27}-\frac{16\!\cdots\!90}{47\!\cdots\!53}a^{25}+\frac{89\!\cdots\!95}{47\!\cdots\!53}a^{23}+\frac{15\!\cdots\!98}{47\!\cdots\!53}a^{21}+\frac{10\!\cdots\!91}{47\!\cdots\!53}a^{19}-\frac{16\!\cdots\!05}{47\!\cdots\!53}a^{17}-\frac{12\!\cdots\!42}{47\!\cdots\!53}a^{15}-\frac{80\!\cdots\!48}{47\!\cdots\!53}a^{13}+\frac{19\!\cdots\!70}{47\!\cdots\!53}a^{11}-\frac{22\!\cdots\!55}{47\!\cdots\!53}a^{9}-\frac{14\!\cdots\!17}{47\!\cdots\!53}a^{7}+\frac{13\!\cdots\!85}{47\!\cdots\!53}a^{5}+\frac{47\!\cdots\!32}{47\!\cdots\!53}a^{3}-\frac{55\!\cdots\!39}{47\!\cdots\!53}a$
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
$C_{936}\times C_{936}$, which has order $876096$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{749195149491219199743}{630534726674038762237} a^{35} + \frac{49264890649408878897431}{630534726674038762237} a^{33} + \frac{1410742182733092748400454}{630534726674038762237} a^{31} + \frac{23154259172514622529873625}{630534726674038762237} a^{29} + \frac{241256361838291759281330911}{630534726674038762237} a^{27} + \frac{1670546293159551855975576174}{630534726674038762237} a^{25} + \frac{7827576734875697768556950247}{630534726674038762237} a^{23} + \frac{24838854752462209026830995479}{630534726674038762237} a^{21} + \frac{52753326546701508835173786339}{630534726674038762237} a^{19} + \frac{73420312961266711721355136572}{630534726674038762237} a^{17} + \frac{65205404853817539475253905471}{630534726674038762237} a^{15} + \frac{35940153314839751341488004896}{630534726674038762237} a^{13} + \frac{11960879065473313770190126920}{630534726674038762237} a^{11} + \frac{2290566807458779505722607146}{630534726674038762237} a^{9} + \frac{225799277096236679342538870}{630534726674038762237} a^{7} + \frac{7930801479464632737507834}{630534726674038762237} a^{5} - \frac{164570800343422927644771}{630534726674038762237} a^{3} - \frac{9001957169534022418083}{630534726674038762237} a \)
(order $4$)
| 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: |
$\frac{33\!\cdots\!60}{47\!\cdots\!53}a^{35}+\frac{22\!\cdots\!12}{47\!\cdots\!53}a^{33}+\frac{64\!\cdots\!75}{47\!\cdots\!53}a^{31}+\frac{10\!\cdots\!54}{47\!\cdots\!53}a^{29}+\frac{11\!\cdots\!17}{47\!\cdots\!53}a^{27}+\frac{81\!\cdots\!18}{47\!\cdots\!53}a^{25}+\frac{39\!\cdots\!30}{47\!\cdots\!53}a^{23}+\frac{13\!\cdots\!44}{47\!\cdots\!53}a^{21}+\frac{29\!\cdots\!05}{47\!\cdots\!53}a^{19}+\frac{45\!\cdots\!68}{47\!\cdots\!53}a^{17}+\frac{46\!\cdots\!76}{47\!\cdots\!53}a^{15}+\frac{31\!\cdots\!61}{47\!\cdots\!53}a^{13}+\frac{13\!\cdots\!65}{47\!\cdots\!53}a^{11}+\frac{36\!\cdots\!11}{47\!\cdots\!53}a^{9}+\frac{58\!\cdots\!10}{47\!\cdots\!53}a^{7}+\frac{51\!\cdots\!24}{47\!\cdots\!53}a^{5}+\frac{20\!\cdots\!43}{47\!\cdots\!53}a^{3}+\frac{23\!\cdots\!82}{47\!\cdots\!53}a$, $\frac{23\!\cdots\!77}{47\!\cdots\!53}a^{34}+\frac{15\!\cdots\!75}{47\!\cdots\!53}a^{32}+\frac{44\!\cdots\!02}{47\!\cdots\!53}a^{30}+\frac{72\!\cdots\!40}{47\!\cdots\!53}a^{28}+\frac{75\!\cdots\!85}{47\!\cdots\!53}a^{26}+\frac{52\!\cdots\!52}{47\!\cdots\!53}a^{24}+\frac{24\!\cdots\!05}{47\!\cdots\!53}a^{22}+\frac{79\!\cdots\!99}{47\!\cdots\!53}a^{20}+\frac{17\!\cdots\!71}{47\!\cdots\!53}a^{18}+\frac{24\!\cdots\!89}{47\!\cdots\!53}a^{16}+\frac{41\!\cdots\!16}{89\!\cdots\!01}a^{14}+\frac{12\!\cdots\!70}{47\!\cdots\!53}a^{12}+\frac{46\!\cdots\!00}{47\!\cdots\!53}a^{10}+\frac{10\!\cdots\!86}{47\!\cdots\!53}a^{8}+\frac{13\!\cdots\!93}{47\!\cdots\!53}a^{6}+\frac{92\!\cdots\!28}{47\!\cdots\!53}a^{4}+\frac{26\!\cdots\!85}{47\!\cdots\!53}a^{2}+\frac{15\!\cdots\!43}{47\!\cdots\!53}$, $\frac{37\!\cdots\!87}{47\!\cdots\!53}a^{35}+\frac{24\!\cdots\!99}{47\!\cdots\!53}a^{33}+\frac{70\!\cdots\!90}{47\!\cdots\!53}a^{31}+\frac{11\!\cdots\!04}{47\!\cdots\!53}a^{29}+\frac{12\!\cdots\!06}{47\!\cdots\!53}a^{27}+\frac{85\!\cdots\!24}{47\!\cdots\!53}a^{25}+\frac{40\!\cdots\!23}{47\!\cdots\!53}a^{23}+\frac{12\!\cdots\!90}{47\!\cdots\!53}a^{21}+\frac{28\!\cdots\!56}{47\!\cdots\!53}a^{19}+\frac{40\!\cdots\!91}{47\!\cdots\!53}a^{17}+\frac{37\!\cdots\!20}{47\!\cdots\!53}a^{15}+\frac{22\!\cdots\!05}{47\!\cdots\!53}a^{13}+\frac{84\!\cdots\!05}{47\!\cdots\!53}a^{11}+\frac{19\!\cdots\!71}{47\!\cdots\!53}a^{9}+\frac{26\!\cdots\!70}{47\!\cdots\!53}a^{7}+\frac{19\!\cdots\!90}{47\!\cdots\!53}a^{5}+\frac{64\!\cdots\!44}{47\!\cdots\!53}a^{3}+\frac{10\!\cdots\!20}{47\!\cdots\!53}a$, $\frac{23\!\cdots\!77}{47\!\cdots\!53}a^{34}+\frac{15\!\cdots\!75}{47\!\cdots\!53}a^{32}+\frac{44\!\cdots\!02}{47\!\cdots\!53}a^{30}+\frac{72\!\cdots\!40}{47\!\cdots\!53}a^{28}+\frac{75\!\cdots\!85}{47\!\cdots\!53}a^{26}+\frac{52\!\cdots\!52}{47\!\cdots\!53}a^{24}+\frac{24\!\cdots\!05}{47\!\cdots\!53}a^{22}+\frac{79\!\cdots\!99}{47\!\cdots\!53}a^{20}+\frac{17\!\cdots\!71}{47\!\cdots\!53}a^{18}+\frac{24\!\cdots\!89}{47\!\cdots\!53}a^{16}+\frac{41\!\cdots\!16}{89\!\cdots\!01}a^{14}+\frac{12\!\cdots\!70}{47\!\cdots\!53}a^{12}+\frac{46\!\cdots\!00}{47\!\cdots\!53}a^{10}+\frac{10\!\cdots\!86}{47\!\cdots\!53}a^{8}+\frac{13\!\cdots\!93}{47\!\cdots\!53}a^{6}+\frac{92\!\cdots\!28}{47\!\cdots\!53}a^{4}+\frac{26\!\cdots\!85}{47\!\cdots\!53}a^{2}+\frac{16\!\cdots\!96}{47\!\cdots\!53}$, $\frac{19\!\cdots\!58}{47\!\cdots\!53}a^{35}+\frac{12\!\cdots\!43}{47\!\cdots\!53}a^{33}+\frac{36\!\cdots\!83}{47\!\cdots\!53}a^{31}+\frac{59\!\cdots\!39}{47\!\cdots\!53}a^{29}+\frac{61\!\cdots\!72}{47\!\cdots\!53}a^{27}+\frac{43\!\cdots\!80}{47\!\cdots\!53}a^{25}+\frac{20\!\cdots\!01}{47\!\cdots\!53}a^{23}+\frac{64\!\cdots\!31}{47\!\cdots\!53}a^{21}+\frac{13\!\cdots\!64}{47\!\cdots\!53}a^{19}+\frac{19\!\cdots\!24}{47\!\cdots\!53}a^{17}+\frac{18\!\cdots\!59}{47\!\cdots\!53}a^{15}+\frac{10\!\cdots\!69}{47\!\cdots\!53}a^{13}+\frac{37\!\cdots\!29}{47\!\cdots\!53}a^{11}+\frac{83\!\cdots\!95}{47\!\cdots\!53}a^{9}+\frac{10\!\cdots\!30}{47\!\cdots\!53}a^{7}+\frac{74\!\cdots\!56}{47\!\cdots\!53}a^{5}+\frac{21\!\cdots\!08}{47\!\cdots\!53}a^{3}+\frac{10\!\cdots\!13}{47\!\cdots\!53}a$, $\frac{18\!\cdots\!62}{47\!\cdots\!53}a^{34}+\frac{12\!\cdots\!22}{47\!\cdots\!53}a^{32}+\frac{34\!\cdots\!42}{47\!\cdots\!53}a^{30}+\frac{57\!\cdots\!01}{47\!\cdots\!53}a^{28}+\frac{59\!\cdots\!58}{47\!\cdots\!53}a^{26}+\frac{41\!\cdots\!63}{47\!\cdots\!53}a^{24}+\frac{19\!\cdots\!24}{47\!\cdots\!53}a^{22}+\frac{62\!\cdots\!34}{47\!\cdots\!53}a^{20}+\frac{13\!\cdots\!21}{47\!\cdots\!53}a^{18}+\frac{19\!\cdots\!51}{47\!\cdots\!53}a^{16}+\frac{17\!\cdots\!66}{47\!\cdots\!53}a^{14}+\frac{10\!\cdots\!66}{47\!\cdots\!53}a^{12}+\frac{38\!\cdots\!01}{47\!\cdots\!53}a^{10}+\frac{85\!\cdots\!65}{47\!\cdots\!53}a^{8}+\frac{21\!\cdots\!35}{89\!\cdots\!01}a^{6}+\frac{78\!\cdots\!29}{47\!\cdots\!53}a^{4}+\frac{23\!\cdots\!62}{47\!\cdots\!53}a^{2}+\frac{12\!\cdots\!65}{47\!\cdots\!53}$, $\frac{19\!\cdots\!58}{47\!\cdots\!53}a^{34}+\frac{12\!\cdots\!43}{47\!\cdots\!53}a^{32}+\frac{36\!\cdots\!83}{47\!\cdots\!53}a^{30}+\frac{59\!\cdots\!39}{47\!\cdots\!53}a^{28}+\frac{61\!\cdots\!72}{47\!\cdots\!53}a^{26}+\frac{43\!\cdots\!80}{47\!\cdots\!53}a^{24}+\frac{20\!\cdots\!01}{47\!\cdots\!53}a^{22}+\frac{64\!\cdots\!31}{47\!\cdots\!53}a^{20}+\frac{13\!\cdots\!64}{47\!\cdots\!53}a^{18}+\frac{19\!\cdots\!24}{47\!\cdots\!53}a^{16}+\frac{18\!\cdots\!59}{47\!\cdots\!53}a^{14}+\frac{10\!\cdots\!69}{47\!\cdots\!53}a^{12}+\frac{37\!\cdots\!29}{47\!\cdots\!53}a^{10}+\frac{83\!\cdots\!95}{47\!\cdots\!53}a^{8}+\frac{10\!\cdots\!30}{47\!\cdots\!53}a^{6}+\frac{74\!\cdots\!56}{47\!\cdots\!53}a^{4}+\frac{21\!\cdots\!08}{47\!\cdots\!53}a^{2}+\frac{10\!\cdots\!66}{47\!\cdots\!53}$, $\frac{22\!\cdots\!52}{47\!\cdots\!53}a^{34}+\frac{14\!\cdots\!37}{47\!\cdots\!53}a^{32}+\frac{41\!\cdots\!68}{47\!\cdots\!53}a^{30}+\frac{68\!\cdots\!12}{47\!\cdots\!53}a^{28}+\frac{70\!\cdots\!98}{47\!\cdots\!53}a^{26}+\frac{48\!\cdots\!35}{47\!\cdots\!53}a^{24}+\frac{22\!\cdots\!29}{47\!\cdots\!53}a^{22}+\frac{69\!\cdots\!70}{47\!\cdots\!53}a^{20}+\frac{14\!\cdots\!98}{47\!\cdots\!53}a^{18}+\frac{19\!\cdots\!26}{47\!\cdots\!53}a^{16}+\frac{15\!\cdots\!91}{47\!\cdots\!53}a^{14}+\frac{77\!\cdots\!00}{47\!\cdots\!53}a^{12}+\frac{20\!\cdots\!66}{47\!\cdots\!53}a^{10}+\frac{24\!\cdots\!35}{47\!\cdots\!53}a^{8}-\frac{24\!\cdots\!57}{47\!\cdots\!53}a^{6}-\frac{23\!\cdots\!92}{47\!\cdots\!53}a^{4}-\frac{13\!\cdots\!83}{47\!\cdots\!53}a^{2}-\frac{85\!\cdots\!05}{47\!\cdots\!53}$, $\frac{39\!\cdots\!02}{47\!\cdots\!53}a^{35}+\frac{25\!\cdots\!72}{47\!\cdots\!53}a^{33}+\frac{74\!\cdots\!50}{47\!\cdots\!53}a^{31}+\frac{12\!\cdots\!60}{47\!\cdots\!53}a^{29}+\frac{12\!\cdots\!69}{47\!\cdots\!53}a^{27}+\frac{88\!\cdots\!57}{47\!\cdots\!53}a^{25}+\frac{41\!\cdots\!44}{47\!\cdots\!53}a^{23}+\frac{13\!\cdots\!83}{47\!\cdots\!53}a^{21}+\frac{28\!\cdots\!95}{47\!\cdots\!53}a^{19}+\frac{41\!\cdots\!22}{47\!\cdots\!53}a^{17}+\frac{37\!\cdots\!43}{47\!\cdots\!53}a^{15}+\frac{21\!\cdots\!17}{47\!\cdots\!53}a^{13}+\frac{80\!\cdots\!86}{47\!\cdots\!53}a^{11}+\frac{17\!\cdots\!44}{47\!\cdots\!53}a^{9}+\frac{22\!\cdots\!25}{47\!\cdots\!53}a^{7}+\frac{15\!\cdots\!48}{47\!\cdots\!53}a^{5}+\frac{40\!\cdots\!41}{47\!\cdots\!53}a^{3}+\frac{63\!\cdots\!72}{47\!\cdots\!53}a$, $\frac{46\!\cdots\!76}{47\!\cdots\!53}a^{35}+\frac{30\!\cdots\!70}{47\!\cdots\!53}a^{33}+\frac{88\!\cdots\!30}{47\!\cdots\!53}a^{31}+\frac{14\!\cdots\!68}{47\!\cdots\!53}a^{29}+\frac{15\!\cdots\!81}{47\!\cdots\!53}a^{27}+\frac{10\!\cdots\!05}{47\!\cdots\!53}a^{25}+\frac{49\!\cdots\!90}{47\!\cdots\!53}a^{23}+\frac{15\!\cdots\!63}{47\!\cdots\!53}a^{21}+\frac{34\!\cdots\!07}{47\!\cdots\!53}a^{19}+\frac{49\!\cdots\!04}{47\!\cdots\!53}a^{17}+\frac{45\!\cdots\!83}{47\!\cdots\!53}a^{15}+\frac{26\!\cdots\!27}{47\!\cdots\!53}a^{13}+\frac{97\!\cdots\!96}{47\!\cdots\!53}a^{11}+\frac{21\!\cdots\!86}{47\!\cdots\!53}a^{9}+\frac{28\!\cdots\!65}{47\!\cdots\!53}a^{7}+\frac{19\!\cdots\!28}{47\!\cdots\!53}a^{5}+\frac{53\!\cdots\!29}{47\!\cdots\!53}a^{3}+\frac{26\!\cdots\!12}{47\!\cdots\!53}a$, $\frac{12\!\cdots\!23}{47\!\cdots\!53}a^{35}+\frac{80\!\cdots\!17}{47\!\cdots\!53}a^{33}+\frac{23\!\cdots\!49}{47\!\cdots\!53}a^{31}+\frac{38\!\cdots\!89}{47\!\cdots\!53}a^{29}+\frac{40\!\cdots\!58}{47\!\cdots\!53}a^{27}+\frac{27\!\cdots\!16}{47\!\cdots\!53}a^{25}+\frac{13\!\cdots\!39}{47\!\cdots\!53}a^{23}+\frac{42\!\cdots\!13}{47\!\cdots\!53}a^{21}+\frac{92\!\cdots\!13}{47\!\cdots\!53}a^{19}+\frac{13\!\cdots\!90}{47\!\cdots\!53}a^{17}+\frac{12\!\cdots\!19}{47\!\cdots\!53}a^{15}+\frac{74\!\cdots\!24}{47\!\cdots\!53}a^{13}+\frac{27\!\cdots\!23}{47\!\cdots\!53}a^{11}+\frac{64\!\cdots\!76}{47\!\cdots\!53}a^{9}+\frac{87\!\cdots\!67}{47\!\cdots\!53}a^{7}+\frac{62\!\cdots\!01}{47\!\cdots\!53}a^{5}+\frac{19\!\cdots\!50}{47\!\cdots\!53}a^{3}+\frac{17\!\cdots\!06}{47\!\cdots\!53}a$, $\frac{13\!\cdots\!32}{47\!\cdots\!53}a^{35}+\frac{90\!\cdots\!84}{47\!\cdots\!53}a^{33}+\frac{25\!\cdots\!42}{47\!\cdots\!53}a^{31}+\frac{42\!\cdots\!09}{47\!\cdots\!53}a^{29}+\frac{84\!\cdots\!64}{89\!\cdots\!01}a^{27}+\frac{31\!\cdots\!11}{47\!\cdots\!53}a^{25}+\frac{14\!\cdots\!78}{47\!\cdots\!53}a^{23}+\frac{47\!\cdots\!01}{47\!\cdots\!53}a^{21}+\frac{10\!\cdots\!29}{47\!\cdots\!53}a^{19}+\frac{14\!\cdots\!69}{47\!\cdots\!53}a^{17}+\frac{13\!\cdots\!56}{47\!\cdots\!53}a^{15}+\frac{78\!\cdots\!79}{47\!\cdots\!53}a^{13}+\frac{28\!\cdots\!07}{47\!\cdots\!53}a^{11}+\frac{63\!\cdots\!48}{47\!\cdots\!53}a^{9}+\frac{79\!\cdots\!08}{47\!\cdots\!53}a^{7}+\frac{48\!\cdots\!91}{47\!\cdots\!53}a^{5}+\frac{91\!\cdots\!18}{47\!\cdots\!53}a^{3}-\frac{12\!\cdots\!02}{47\!\cdots\!53}a$, $\frac{12\!\cdots\!35}{47\!\cdots\!53}a^{35}+\frac{80\!\cdots\!04}{47\!\cdots\!53}a^{33}+\frac{22\!\cdots\!78}{47\!\cdots\!53}a^{31}+\frac{37\!\cdots\!04}{47\!\cdots\!53}a^{29}+\frac{39\!\cdots\!75}{47\!\cdots\!53}a^{27}+\frac{27\!\cdots\!49}{47\!\cdots\!53}a^{25}+\frac{12\!\cdots\!18}{47\!\cdots\!53}a^{23}+\frac{40\!\cdots\!29}{47\!\cdots\!53}a^{21}+\frac{86\!\cdots\!32}{47\!\cdots\!53}a^{19}+\frac{12\!\cdots\!16}{47\!\cdots\!53}a^{17}+\frac{10\!\cdots\!05}{47\!\cdots\!53}a^{15}+\frac{61\!\cdots\!08}{47\!\cdots\!53}a^{13}+\frac{21\!\cdots\!90}{47\!\cdots\!53}a^{11}+\frac{47\!\cdots\!61}{47\!\cdots\!53}a^{9}+\frac{61\!\cdots\!61}{47\!\cdots\!53}a^{7}+\frac{46\!\cdots\!37}{47\!\cdots\!53}a^{5}+\frac{17\!\cdots\!03}{47\!\cdots\!53}a^{3}+\frac{16\!\cdots\!87}{47\!\cdots\!53}a$, $\frac{14\!\cdots\!59}{47\!\cdots\!53}a^{35}+\frac{92\!\cdots\!71}{47\!\cdots\!53}a^{33}+\frac{26\!\cdots\!57}{47\!\cdots\!53}a^{31}+\frac{43\!\cdots\!59}{47\!\cdots\!53}a^{29}+\frac{45\!\cdots\!81}{47\!\cdots\!53}a^{27}+\frac{31\!\cdots\!17}{47\!\cdots\!53}a^{25}+\frac{14\!\cdots\!71}{47\!\cdots\!53}a^{23}+\frac{47\!\cdots\!47}{47\!\cdots\!53}a^{21}+\frac{10\!\cdots\!80}{47\!\cdots\!53}a^{19}+\frac{14\!\cdots\!92}{47\!\cdots\!53}a^{17}+\frac{12\!\cdots\!00}{47\!\cdots\!53}a^{15}+\frac{70\!\cdots\!23}{47\!\cdots\!53}a^{13}+\frac{44\!\cdots\!99}{89\!\cdots\!01}a^{11}+\frac{46\!\cdots\!08}{47\!\cdots\!53}a^{9}+\frac{47\!\cdots\!68}{47\!\cdots\!53}a^{7}+\frac{16\!\cdots\!57}{47\!\cdots\!53}a^{5}-\frac{49\!\cdots\!81}{47\!\cdots\!53}a^{3}-\frac{25\!\cdots\!64}{47\!\cdots\!53}a$, $\frac{63\!\cdots\!63}{47\!\cdots\!53}a^{35}+\frac{41\!\cdots\!93}{47\!\cdots\!53}a^{33}+\frac{11\!\cdots\!65}{47\!\cdots\!53}a^{31}+\frac{19\!\cdots\!91}{47\!\cdots\!53}a^{29}+\frac{20\!\cdots\!12}{47\!\cdots\!53}a^{27}+\frac{14\!\cdots\!08}{47\!\cdots\!53}a^{25}+\frac{67\!\cdots\!71}{47\!\cdots\!53}a^{23}+\frac{21\!\cdots\!77}{47\!\cdots\!53}a^{21}+\frac{46\!\cdots\!45}{47\!\cdots\!53}a^{19}+\frac{66\!\cdots\!96}{47\!\cdots\!53}a^{17}+\frac{61\!\cdots\!93}{47\!\cdots\!53}a^{15}+\frac{35\!\cdots\!00}{47\!\cdots\!53}a^{13}+\frac{13\!\cdots\!81}{47\!\cdots\!53}a^{11}+\frac{29\!\cdots\!68}{47\!\cdots\!53}a^{9}+\frac{38\!\cdots\!01}{47\!\cdots\!53}a^{7}+\frac{27\!\cdots\!15}{47\!\cdots\!53}a^{5}+\frac{82\!\cdots\!96}{47\!\cdots\!53}a^{3}+\frac{55\!\cdots\!54}{47\!\cdots\!53}a$, $\frac{42\!\cdots\!20}{47\!\cdots\!53}a^{34}+\frac{27\!\cdots\!75}{47\!\cdots\!53}a^{32}+\frac{15\!\cdots\!45}{89\!\cdots\!01}a^{30}+\frac{13\!\cdots\!90}{47\!\cdots\!53}a^{28}+\frac{13\!\cdots\!60}{47\!\cdots\!53}a^{26}+\frac{95\!\cdots\!30}{47\!\cdots\!53}a^{24}+\frac{45\!\cdots\!00}{47\!\cdots\!53}a^{22}+\frac{14\!\cdots\!95}{47\!\cdots\!53}a^{20}+\frac{31\!\cdots\!65}{47\!\cdots\!53}a^{18}+\frac{44\!\cdots\!35}{47\!\cdots\!53}a^{16}+\frac{41\!\cdots\!70}{47\!\cdots\!53}a^{14}+\frac{24\!\cdots\!55}{47\!\cdots\!53}a^{12}+\frac{91\!\cdots\!70}{47\!\cdots\!53}a^{10}+\frac{21\!\cdots\!15}{47\!\cdots\!53}a^{8}+\frac{28\!\cdots\!08}{47\!\cdots\!53}a^{6}+\frac{20\!\cdots\!90}{47\!\cdots\!53}a^{4}+\frac{61\!\cdots\!85}{47\!\cdots\!53}a^{2}+\frac{28\!\cdots\!57}{47\!\cdots\!53}$, $\frac{20\!\cdots\!74}{47\!\cdots\!53}a^{35}+\frac{13\!\cdots\!22}{47\!\cdots\!53}a^{33}+\frac{38\!\cdots\!25}{47\!\cdots\!53}a^{31}+\frac{63\!\cdots\!27}{47\!\cdots\!53}a^{29}+\frac{66\!\cdots\!37}{47\!\cdots\!53}a^{27}+\frac{45\!\cdots\!27}{47\!\cdots\!53}a^{25}+\frac{21\!\cdots\!04}{47\!\cdots\!53}a^{23}+\frac{69\!\cdots\!04}{47\!\cdots\!53}a^{21}+\frac{14\!\cdots\!94}{47\!\cdots\!53}a^{19}+\frac{21\!\cdots\!83}{47\!\cdots\!53}a^{17}+\frac{19\!\cdots\!30}{47\!\cdots\!53}a^{15}+\frac{11\!\cdots\!78}{47\!\cdots\!53}a^{13}+\frac{42\!\cdots\!90}{47\!\cdots\!53}a^{11}+\frac{18\!\cdots\!01}{89\!\cdots\!01}a^{9}+\frac{13\!\cdots\!06}{47\!\cdots\!53}a^{7}+\frac{99\!\cdots\!77}{47\!\cdots\!53}a^{5}+\frac{35\!\cdots\!11}{47\!\cdots\!53}a^{3}+\frac{39\!\cdots\!62}{47\!\cdots\!53}a$
| 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: | \( 20980577392492.816 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 20980577392492.816 \cdot 876096}{4\cdot\sqrt{14361527681487998235176534750111030030248040605937785686475463983104}}\cr\approx \mathstrut & 0.282453169667971 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 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^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 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^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 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^36 + 66*x^34 + 1899*x^32 + 31364*x^30 + 329559*x^28 + 2308518*x^26 + 10994901*x^24 + 35728458*x^22 + 78637230*x^20 + 115641708*x^18 + 111976572*x^16 + 70684362*x^14 + 28999138*x^12 + 7688325*x^10 + 1293075*x^8 + 132111*x^6 + 7476*x^4 + 189*x^2 + 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)
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ is not computed |
Intermediate fields
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 | R | R | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | ${\href{/padicField/7.6.0.1}{6} }^{6}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{12}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.1.0.1}{1} }^{36}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
| 2.12.12.26 | $x^{12} + 12 x^{11} + 98 x^{10} + 542 x^{9} + 2359 x^{8} + 7956 x^{7} + 21831 x^{6} + 47308 x^{5} + 82476 x^{4} + 109442 x^{3} + 112071 x^{2} + 76900 x + 33205$ | $2$ | $6$ | $12$ | $C_6\times C_2$ | $[2]^{6}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
| Deg $18$ | $3$ | $6$ | $24$ | ||||
|
\(13\)
| 13.18.15.1 | $x^{18} + 12 x^{16} + 66 x^{15} + 60 x^{14} + 660 x^{13} + 2014 x^{12} + 2640 x^{11} + 14136 x^{10} + 10450 x^{9} + 46092 x^{8} + 190740 x^{7} + 945517 x^{6} + 280368 x^{5} + 1183620 x^{4} - 1338964 x^{3} + 1184376 x^{2} + 976800 x + 2479736$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |
| 13.18.15.1 | $x^{18} + 12 x^{16} + 66 x^{15} + 60 x^{14} + 660 x^{13} + 2014 x^{12} + 2640 x^{11} + 14136 x^{10} + 10450 x^{9} + 46092 x^{8} + 190740 x^{7} + 945517 x^{6} + 280368 x^{5} + 1183620 x^{4} - 1338964 x^{3} + 1184376 x^{2} + 976800 x + 2479736$ | $6$ | $3$ | $15$ | $C_6 \times C_3$ | $[\ ]_{6}^{3}$ |