Normalized defining polynomial
\( x^{36} - x^{35} + 22 x^{34} - 27 x^{33} + 67 x^{32} - 207 x^{31} - 1315 x^{30} - 33 x^{29} + \cdots + 97362911 \)
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: |
\(1552561224397970539319305607385300901659347120825982756912708282470703125\)
\(\medspace = 5^{27}\cdot 37^{34}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(101.23\) | 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: | $5^{3/4}37^{17/18}\approx 101.22945747512976$ | ||
| Ramified primes: |
\(5\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(185=5\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{185}(1,·)$, $\chi_{185}(3,·)$, $\chi_{185}(132,·)$, $\chi_{185}(9,·)$, $\chi_{185}(138,·)$, $\chi_{185}(16,·)$, $\chi_{185}(147,·)$, $\chi_{185}(149,·)$, $\chi_{185}(152,·)$, $\chi_{185}(26,·)$, $\chi_{185}(27,·)$, $\chi_{185}(28,·)$, $\chi_{185}(34,·)$, $\chi_{185}(164,·)$, $\chi_{185}(44,·)$, $\chi_{185}(173,·)$, $\chi_{185}(46,·)$, $\chi_{185}(48,·)$, $\chi_{185}(49,·)$, $\chi_{185}(178,·)$, $\chi_{185}(181,·)$, $\chi_{185}(58,·)$, $\chi_{185}(62,·)$, $\chi_{185}(67,·)$, $\chi_{185}(71,·)$, $\chi_{185}(73,·)$, $\chi_{185}(77,·)$, $\chi_{185}(78,·)$, $\chi_{185}(81,·)$, $\chi_{185}(84,·)$, $\chi_{185}(86,·)$, $\chi_{185}(144,·)$, $\chi_{185}(102,·)$, $\chi_{185}(174,·)$, $\chi_{185}(121,·)$, $\chi_{185}(122,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{179}a^{32}+\frac{48}{179}a^{31}+\frac{17}{179}a^{30}+\frac{75}{179}a^{29}+\frac{81}{179}a^{28}-\frac{14}{179}a^{27}-\frac{6}{179}a^{26}-\frac{21}{179}a^{25}-\frac{74}{179}a^{24}-\frac{19}{179}a^{23}-\frac{87}{179}a^{22}-\frac{4}{179}a^{21}-\frac{70}{179}a^{20}+\frac{56}{179}a^{19}+\frac{49}{179}a^{18}+\frac{70}{179}a^{17}+\frac{24}{179}a^{16}-\frac{53}{179}a^{15}+\frac{88}{179}a^{14}+\frac{62}{179}a^{13}+\frac{86}{179}a^{12}-\frac{4}{179}a^{11}+\frac{21}{179}a^{10}-\frac{40}{179}a^{9}+\frac{49}{179}a^{8}-\frac{1}{179}a^{7}+\frac{52}{179}a^{6}+\frac{22}{179}a^{5}+\frac{46}{179}a^{4}+\frac{61}{179}a^{3}-\frac{72}{179}a^{2}+\frac{88}{179}a-\frac{58}{179}$, $\frac{1}{179}a^{33}+\frac{40}{179}a^{31}-\frac{25}{179}a^{30}+\frac{61}{179}a^{29}+\frac{36}{179}a^{28}-\frac{50}{179}a^{27}+\frac{88}{179}a^{26}+\frac{39}{179}a^{25}-\frac{47}{179}a^{24}-\frac{70}{179}a^{23}+\frac{55}{179}a^{22}-\frac{57}{179}a^{21}+\frac{15}{179}a^{20}+\frac{46}{179}a^{19}+\frac{45}{179}a^{18}+\frac{65}{179}a^{17}+\frac{48}{179}a^{16}-\frac{53}{179}a^{15}-\frac{45}{179}a^{14}-\frac{26}{179}a^{13}-\frac{15}{179}a^{12}+\frac{34}{179}a^{11}+\frac{26}{179}a^{10}-\frac{26}{179}a^{8}-\frac{79}{179}a^{7}+\frac{32}{179}a^{6}+\frac{64}{179}a^{5}+\frac{1}{179}a^{4}+\frac{43}{179}a^{3}-\frac{36}{179}a^{2}+\frac{14}{179}a-\frac{80}{179}$, $\frac{1}{25405649}a^{34}+\frac{13867}{25405649}a^{33}+\frac{53151}{25405649}a^{32}+\frac{2203434}{25405649}a^{31}+\frac{7606367}{25405649}a^{30}+\frac{9126680}{25405649}a^{29}-\frac{1217185}{25405649}a^{28}-\frac{863300}{25405649}a^{27}+\frac{3281295}{25405649}a^{26}-\frac{5382326}{25405649}a^{25}-\frac{11368997}{25405649}a^{24}+\frac{8289665}{25405649}a^{23}+\frac{2285251}{25405649}a^{22}-\frac{3461413}{25405649}a^{21}-\frac{10866440}{25405649}a^{20}+\frac{10343080}{25405649}a^{19}+\frac{12270136}{25405649}a^{18}+\frac{7494808}{25405649}a^{17}+\frac{8690138}{25405649}a^{16}+\frac{1391595}{25405649}a^{15}-\frac{1053384}{25405649}a^{14}-\frac{7758092}{25405649}a^{13}+\frac{5195863}{25405649}a^{12}+\frac{349276}{25405649}a^{11}-\frac{4065072}{25405649}a^{10}-\frac{8126694}{25405649}a^{9}+\frac{11340030}{25405649}a^{8}-\frac{9813604}{25405649}a^{7}-\frac{3246475}{25405649}a^{6}+\frac{4390628}{25405649}a^{5}+\frac{12646054}{25405649}a^{4}+\frac{6754611}{25405649}a^{3}-\frac{937584}{25405649}a^{2}+\frac{3130452}{25405649}a+\frac{11766978}{25405649}$, $\frac{1}{86\!\cdots\!61}a^{35}-\frac{21\!\cdots\!31}{86\!\cdots\!61}a^{34}+\frac{64\!\cdots\!35}{86\!\cdots\!61}a^{33}+\frac{31\!\cdots\!89}{86\!\cdots\!61}a^{32}-\frac{82\!\cdots\!10}{86\!\cdots\!61}a^{31}+\frac{14\!\cdots\!74}{86\!\cdots\!61}a^{30}+\frac{35\!\cdots\!87}{86\!\cdots\!61}a^{29}+\frac{81\!\cdots\!26}{86\!\cdots\!61}a^{28}-\frac{31\!\cdots\!46}{86\!\cdots\!61}a^{27}+\frac{17\!\cdots\!94}{86\!\cdots\!61}a^{26}+\frac{66\!\cdots\!52}{86\!\cdots\!61}a^{25}+\frac{73\!\cdots\!88}{86\!\cdots\!61}a^{24}+\frac{53\!\cdots\!94}{86\!\cdots\!61}a^{23}+\frac{19\!\cdots\!96}{86\!\cdots\!61}a^{22}+\frac{19\!\cdots\!92}{86\!\cdots\!61}a^{21}-\frac{13\!\cdots\!72}{86\!\cdots\!61}a^{20}+\frac{30\!\cdots\!09}{86\!\cdots\!61}a^{19}-\frac{40\!\cdots\!38}{86\!\cdots\!61}a^{18}+\frac{37\!\cdots\!79}{86\!\cdots\!61}a^{17}-\frac{31\!\cdots\!12}{86\!\cdots\!61}a^{16}-\frac{89\!\cdots\!23}{86\!\cdots\!61}a^{15}-\frac{27\!\cdots\!90}{86\!\cdots\!61}a^{14}-\frac{26\!\cdots\!05}{86\!\cdots\!61}a^{13}+\frac{26\!\cdots\!31}{86\!\cdots\!61}a^{12}+\frac{25\!\cdots\!39}{86\!\cdots\!61}a^{11}-\frac{40\!\cdots\!88}{86\!\cdots\!61}a^{10}+\frac{15\!\cdots\!44}{86\!\cdots\!61}a^{9}-\frac{21\!\cdots\!10}{86\!\cdots\!61}a^{8}-\frac{22\!\cdots\!72}{86\!\cdots\!61}a^{7}-\frac{49\!\cdots\!88}{86\!\cdots\!61}a^{6}+\frac{35\!\cdots\!50}{86\!\cdots\!61}a^{5}+\frac{22\!\cdots\!67}{86\!\cdots\!61}a^{4}-\frac{18\!\cdots\!78}{86\!\cdots\!61}a^{3}-\frac{14\!\cdots\!40}{86\!\cdots\!61}a^{2}+\frac{24\!\cdots\!35}{86\!\cdots\!61}a+\frac{63\!\cdots\!81}{86\!\cdots\!61}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $17$ | 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: | not computed | 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: | not computed | 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 $
Galois group
| A cyclic group of order 36 |
| The 36 conjugacy class representatives for $C_{36}$ |
| Character table for $C_{36}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 3.3.1369.1, 4.0.171125.1, 6.6.234270125.1, 9.9.3512479453921.1, 12.0.9391766352378611328125.1, 18.18.24096702957455403051316876953125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $36$ | $36$ | R | $36$ | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $36$ | $36$ | ${\href{/padicField/19.9.0.1}{9} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.2.0.1}{2} }^{18}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{9}$ | ${\href{/padicField/47.12.0.1}{12} }^{3}$ | $36$ | ${\href{/padicField/59.9.0.1}{9} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $36$ | $4$ | $9$ | $27$ | |||
|
\(37\)
| Deg $36$ | $18$ | $2$ | $34$ |