Normalized defining polynomial
\( x^{36} + 73 x^{34} + 2336 x^{32} + 43289 x^{30} + 517205 x^{28} + 4199763 x^{26} + 23818951 x^{24} + \cdots + 73 \)
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: |
\(11307847636660752290696972110442359259116305023758972734266350296999135281152\)
\(\medspace = 2^{36}\cdot 73^{35}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(129.60\) | 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: | $2\cdot 73^{35/36}\approx 129.59668343190836$ | ||
| Ramified primes: |
\(2\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{73}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(292=2^{2}\cdot 73\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{292}(1,·)$, $\chi_{292}(3,·)$, $\chi_{292}(9,·)$, $\chi_{292}(267,·)$, $\chi_{292}(109,·)$, $\chi_{292}(143,·)$, $\chi_{292}(145,·)$, $\chi_{292}(19,·)$, $\chi_{292}(23,·)$, $\chi_{292}(27,·)$, $\chi_{292}(35,·)$, $\chi_{292}(37,·)$, $\chi_{292}(41,·)$, $\chi_{292}(171,·)$, $\chi_{292}(137,·)$, $\chi_{292}(57,·)$, $\chi_{292}(65,·)$, $\chi_{292}(67,·)$, $\chi_{292}(69,·)$, $\chi_{292}(201,·)$, $\chi_{292}(77,·)$, $\chi_{292}(79,·)$, $\chi_{292}(81,·)$, $\chi_{292}(89,·)$, $\chi_{292}(207,·)$, $\chi_{292}(221,·)$, $\chi_{292}(195,·)$, $\chi_{292}(231,·)$, $\chi_{292}(105,·)$, $\chi_{292}(237,·)$, $\chi_{292}(111,·)$, $\chi_{292}(243,·)$, $\chi_{292}(119,·)$, $\chi_{292}(217,·)$, $\chi_{292}(123,·)$, $\chi_{292}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{3}a^{24}+\frac{1}{3}a^{22}+\frac{1}{3}a^{20}+\frac{1}{3}a^{18}+\frac{1}{3}a^{16}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{3}a^{25}+\frac{1}{3}a^{23}+\frac{1}{3}a^{21}+\frac{1}{3}a^{19}+\frac{1}{3}a^{17}+\frac{1}{3}a^{15}+\frac{1}{3}a^{13}+\frac{1}{3}a^{11}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{26}-\frac{1}{3}$, $\frac{1}{3}a^{27}-\frac{1}{3}a$, $\frac{1}{3}a^{28}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{29}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{30}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{31}-\frac{1}{3}a^{5}$, $\frac{1}{1155009}a^{32}-\frac{72782}{1155009}a^{30}-\frac{12223}{385003}a^{28}-\frac{66695}{1155009}a^{26}+\frac{16311}{385003}a^{24}+\frac{6446}{385003}a^{22}+\frac{154035}{385003}a^{20}-\frac{98490}{385003}a^{18}-\frac{148098}{385003}a^{16}+\frac{10880}{385003}a^{14}+\frac{50416}{385003}a^{12}+\frac{68616}{385003}a^{10}-\frac{26961}{385003}a^{8}-\frac{219523}{1155009}a^{6}-\frac{452599}{1155009}a^{4}+\frac{93050}{385003}a^{2}+\frac{217832}{1155009}$, $\frac{1}{1155009}a^{33}-\frac{72782}{1155009}a^{31}-\frac{12223}{385003}a^{29}-\frac{66695}{1155009}a^{27}+\frac{16311}{385003}a^{25}+\frac{6446}{385003}a^{23}+\frac{154035}{385003}a^{21}-\frac{98490}{385003}a^{19}-\frac{148098}{385003}a^{17}+\frac{10880}{385003}a^{15}+\frac{50416}{385003}a^{13}+\frac{68616}{385003}a^{11}-\frac{26961}{385003}a^{9}-\frac{219523}{1155009}a^{7}-\frac{452599}{1155009}a^{5}+\frac{93050}{385003}a^{3}+\frac{217832}{1155009}a$, $\frac{1}{52\!\cdots\!67}a^{34}+\frac{54\!\cdots\!76}{17\!\cdots\!89}a^{32}+\frac{20\!\cdots\!39}{52\!\cdots\!67}a^{30}+\frac{23\!\cdots\!86}{52\!\cdots\!67}a^{28}+\frac{14\!\cdots\!59}{17\!\cdots\!89}a^{26}-\frac{35\!\cdots\!74}{52\!\cdots\!67}a^{24}+\frac{23\!\cdots\!61}{52\!\cdots\!67}a^{22}-\frac{34\!\cdots\!42}{94\!\cdots\!31}a^{20}+\frac{11\!\cdots\!42}{52\!\cdots\!67}a^{18}-\frac{33\!\cdots\!64}{52\!\cdots\!67}a^{16}-\frac{90\!\cdots\!73}{52\!\cdots\!67}a^{14}-\frac{69\!\cdots\!33}{52\!\cdots\!67}a^{12}+\frac{12\!\cdots\!77}{52\!\cdots\!67}a^{10}-\frac{14\!\cdots\!88}{17\!\cdots\!89}a^{8}-\frac{20\!\cdots\!10}{52\!\cdots\!67}a^{6}+\frac{19\!\cdots\!42}{52\!\cdots\!67}a^{4}-\frac{59\!\cdots\!84}{52\!\cdots\!67}a^{2}+\frac{12\!\cdots\!74}{52\!\cdots\!67}$, $\frac{1}{52\!\cdots\!67}a^{35}+\frac{54\!\cdots\!76}{17\!\cdots\!89}a^{33}+\frac{20\!\cdots\!39}{52\!\cdots\!67}a^{31}+\frac{23\!\cdots\!86}{52\!\cdots\!67}a^{29}+\frac{14\!\cdots\!59}{17\!\cdots\!89}a^{27}-\frac{35\!\cdots\!74}{52\!\cdots\!67}a^{25}+\frac{23\!\cdots\!61}{52\!\cdots\!67}a^{23}-\frac{34\!\cdots\!42}{94\!\cdots\!31}a^{21}+\frac{11\!\cdots\!42}{52\!\cdots\!67}a^{19}-\frac{33\!\cdots\!64}{52\!\cdots\!67}a^{17}-\frac{90\!\cdots\!73}{52\!\cdots\!67}a^{15}-\frac{69\!\cdots\!33}{52\!\cdots\!67}a^{13}+\frac{12\!\cdots\!77}{52\!\cdots\!67}a^{11}-\frac{14\!\cdots\!88}{17\!\cdots\!89}a^{9}-\frac{20\!\cdots\!10}{52\!\cdots\!67}a^{7}+\frac{19\!\cdots\!42}{52\!\cdots\!67}a^{5}-\frac{59\!\cdots\!84}{52\!\cdots\!67}a^{3}+\frac{12\!\cdots\!74}{52\!\cdots\!67}a$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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 $C_{36}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{73}) \), 3.3.5329.1, 4.0.6224272.1, 6.6.2073071593.1, 9.9.806460091894081.1, 12.0.1285024504088001365512192.1, 18.18.47477585226700098686074966922953.1 |
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 | R | ${\href{/padicField/3.3.0.1}{3} }^{12}$ | $36$ | ${\href{/padicField/7.12.0.1}{12} }^{3}$ | $36$ | $36$ | ${\href{/padicField/17.12.0.1}{12} }^{3}$ | ${\href{/padicField/19.9.0.1}{9} }^{4}$ | ${\href{/padicField/23.9.0.1}{9} }^{4}$ | $36$ | $36$ | ${\href{/padicField/37.9.0.1}{9} }^{4}$ | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | ${\href{/padicField/43.12.0.1}{12} }^{3}$ | $36$ | $36$ | $36$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
| 2.18.18.115 | $x^{18} + 18 x^{17} + 162 x^{16} + 960 x^{15} + 4320 x^{14} + 16128 x^{13} + 53696 x^{12} + 165120 x^{11} + 449824 x^{10} + 1006400 x^{9} + 1826368 x^{8} + 2905088 x^{7} + 3317760 x^{6} - 418816 x^{5} - 6684672 x^{4} - 4984832 x^{3} + 2483456 x^{2} + 3566080 x + 1829376$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ | |
|
\(73\)
| Deg $36$ | $36$ | $1$ | $35$ |