Normalized defining polynomial
\( x^{36} - 4 x^{35} - 116 x^{34} + 460 x^{33} + 5979 x^{32} - 23476 x^{31} - 181078 x^{30} + \cdots - 4592149 \)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[36, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(11166706806076050240252573251230095765902100267008000000000000000000000000000\)
\(\medspace = 2^{54}\cdot 5^{27}\cdot 19^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(129.55\) | 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: | $2^{3/2}5^{3/4}19^{8/9}\approx 129.5514756800367$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(760=2^{3}\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{760}(1,·)$, $\chi_{760}(387,·)$, $\chi_{760}(9,·)$, $\chi_{760}(267,·)$, $\chi_{760}(529,·)$, $\chi_{760}(643,·)$, $\chi_{760}(283,·)$, $\chi_{760}(161,·)$, $\chi_{760}(163,·)$, $\chi_{760}(49,·)$, $\chi_{760}(169,·)$, $\chi_{760}(427,·)$, $\chi_{760}(723,·)$, $\chi_{760}(689,·)$, $\chi_{760}(329,·)$, $\chi_{760}(441,·)$, $\chi_{760}(187,·)$, $\chi_{760}(321,·)$, $\chi_{760}(707,·)$, $\chi_{760}(609,·)$, $\chi_{760}(201,·)$, $\chi_{760}(587,·)$, $\chi_{760}(403,·)$, $\chi_{760}(81,·)$, $\chi_{760}(467,·)$, $\chi_{760}(43,·)$, $\chi_{760}(729,·)$, $\chi_{760}(347,·)$, $\chi_{760}(481,·)$, $\chi_{760}(443,·)$, $\chi_{760}(747,·)$, $\chi_{760}(289,·)$, $\chi_{760}(83,·)$, $\chi_{760}(681,·)$, $\chi_{760}(121,·)$, $\chi_{760}(123,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{458}a^{33}-\frac{45}{458}a^{32}-\frac{45}{229}a^{31}-\frac{35}{229}a^{30}-\frac{19}{229}a^{29}-\frac{19}{458}a^{28}-\frac{81}{458}a^{27}-\frac{47}{229}a^{26}+\frac{1}{458}a^{25}+\frac{2}{229}a^{24}-\frac{53}{229}a^{23}+\frac{35}{458}a^{22}-\frac{71}{458}a^{21}+\frac{113}{458}a^{20}+\frac{3}{229}a^{19}+\frac{27}{458}a^{18}-\frac{90}{229}a^{17}+\frac{147}{458}a^{16}-\frac{23}{229}a^{15}-\frac{37}{229}a^{14}+\frac{113}{458}a^{13}+\frac{62}{229}a^{12}+\frac{106}{229}a^{11}+\frac{14}{229}a^{10}+\frac{52}{229}a^{9}-\frac{109}{229}a^{8}-\frac{59}{458}a^{7}-\frac{117}{458}a^{6}+\frac{68}{229}a^{5}+\frac{119}{458}a^{4}+\frac{87}{229}a^{3}+\frac{199}{458}a^{2}+\frac{191}{458}a+\frac{16}{229}$, $\frac{1}{19\!\cdots\!98}a^{34}+\frac{91\!\cdots\!18}{97\!\cdots\!99}a^{33}-\frac{15\!\cdots\!89}{19\!\cdots\!98}a^{32}-\frac{20\!\cdots\!24}{97\!\cdots\!99}a^{31}+\frac{23\!\cdots\!74}{97\!\cdots\!99}a^{30}+\frac{44\!\cdots\!99}{19\!\cdots\!98}a^{29}-\frac{16\!\cdots\!04}{97\!\cdots\!99}a^{28}+\frac{31\!\cdots\!85}{19\!\cdots\!98}a^{27}+\frac{97\!\cdots\!26}{97\!\cdots\!99}a^{26}-\frac{42\!\cdots\!99}{19\!\cdots\!98}a^{25}-\frac{32\!\cdots\!12}{97\!\cdots\!99}a^{24}-\frac{23\!\cdots\!29}{97\!\cdots\!99}a^{23}-\frac{21\!\cdots\!27}{97\!\cdots\!99}a^{22}-\frac{16\!\cdots\!27}{19\!\cdots\!98}a^{21}+\frac{22\!\cdots\!72}{97\!\cdots\!99}a^{20}-\frac{14\!\cdots\!73}{97\!\cdots\!99}a^{19}+\frac{20\!\cdots\!29}{19\!\cdots\!98}a^{18}-\frac{49\!\cdots\!81}{19\!\cdots\!98}a^{17}-\frac{74\!\cdots\!93}{19\!\cdots\!98}a^{16}-\frac{57\!\cdots\!49}{19\!\cdots\!98}a^{15}+\frac{41\!\cdots\!04}{97\!\cdots\!99}a^{14}-\frac{66\!\cdots\!23}{19\!\cdots\!98}a^{13}-\frac{50\!\cdots\!83}{19\!\cdots\!98}a^{12}+\frac{86\!\cdots\!49}{19\!\cdots\!98}a^{11}-\frac{52\!\cdots\!05}{19\!\cdots\!98}a^{10}+\frac{87\!\cdots\!07}{19\!\cdots\!98}a^{9}-\frac{25\!\cdots\!66}{97\!\cdots\!99}a^{8}+\frac{48\!\cdots\!73}{19\!\cdots\!98}a^{7}+\frac{39\!\cdots\!65}{19\!\cdots\!98}a^{6}-\frac{39\!\cdots\!13}{97\!\cdots\!99}a^{5}+\frac{76\!\cdots\!79}{19\!\cdots\!98}a^{4}+\frac{21\!\cdots\!53}{97\!\cdots\!99}a^{3}-\frac{18\!\cdots\!07}{97\!\cdots\!99}a^{2}-\frac{41\!\cdots\!53}{97\!\cdots\!99}a-\frac{37\!\cdots\!73}{19\!\cdots\!98}$, $\frac{1}{43\!\cdots\!22}a^{35}+\frac{86\!\cdots\!25}{43\!\cdots\!22}a^{34}+\frac{31\!\cdots\!50}{21\!\cdots\!11}a^{33}-\frac{25\!\cdots\!15}{21\!\cdots\!11}a^{32}-\frac{10\!\cdots\!45}{21\!\cdots\!11}a^{31}-\frac{12\!\cdots\!19}{43\!\cdots\!22}a^{30}+\frac{15\!\cdots\!25}{21\!\cdots\!11}a^{29}+\frac{22\!\cdots\!91}{43\!\cdots\!22}a^{28}-\frac{88\!\cdots\!73}{43\!\cdots\!22}a^{27}-\frac{65\!\cdots\!09}{43\!\cdots\!22}a^{26}-\frac{43\!\cdots\!05}{21\!\cdots\!11}a^{25}-\frac{45\!\cdots\!87}{43\!\cdots\!22}a^{24}-\frac{50\!\cdots\!21}{43\!\cdots\!22}a^{23}+\frac{96\!\cdots\!89}{43\!\cdots\!22}a^{22}-\frac{42\!\cdots\!61}{21\!\cdots\!11}a^{21}-\frac{32\!\cdots\!76}{21\!\cdots\!11}a^{20}-\frac{68\!\cdots\!20}{21\!\cdots\!11}a^{19}-\frac{14\!\cdots\!03}{21\!\cdots\!11}a^{18}+\frac{10\!\cdots\!87}{43\!\cdots\!22}a^{17}+\frac{33\!\cdots\!65}{21\!\cdots\!11}a^{16}-\frac{47\!\cdots\!77}{21\!\cdots\!11}a^{15}-\frac{43\!\cdots\!47}{21\!\cdots\!11}a^{14}+\frac{70\!\cdots\!32}{21\!\cdots\!11}a^{13}-\frac{60\!\cdots\!09}{21\!\cdots\!11}a^{12}+\frac{74\!\cdots\!61}{43\!\cdots\!22}a^{11}-\frac{14\!\cdots\!51}{43\!\cdots\!22}a^{10}+\frac{67\!\cdots\!38}{21\!\cdots\!11}a^{9}-\frac{77\!\cdots\!45}{21\!\cdots\!11}a^{8}+\frac{21\!\cdots\!65}{43\!\cdots\!22}a^{7}-\frac{67\!\cdots\!34}{21\!\cdots\!11}a^{6}+\frac{28\!\cdots\!74}{21\!\cdots\!11}a^{5}-\frac{18\!\cdots\!85}{43\!\cdots\!22}a^{4}-\frac{60\!\cdots\!91}{21\!\cdots\!11}a^{3}-\frac{68\!\cdots\!73}{21\!\cdots\!11}a^{2}-\frac{47\!\cdots\!31}{43\!\cdots\!22}a-\frac{15\!\cdots\!11}{43\!\cdots\!22}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $35$ | 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
| \(\Q(\sqrt{5}) \), 3.3.361.1, 4.4.8000.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.8695584276992000000000.1, 18.18.563362135874260093126953125.1 |
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 | $36$ | R | ${\href{/padicField/7.12.0.1}{12} }^{3}$ | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $36$ | $36$ | R | $36$ | ${\href{/padicField/29.9.0.1}{9} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | $36$ | $36$ | $36$ | $18^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $36$ | $2$ | $18$ | $54$ | |||
|
\(5\)
| Deg $36$ | $4$ | $9$ | $27$ | |||
|
\(19\)
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |
| 19.18.16.1 | $x^{18} + 162 x^{17} + 11682 x^{16} + 492480 x^{15} + 13390416 x^{14} + 243982368 x^{13} + 2990277024 x^{12} + 23974071552 x^{11} + 116854153056 x^{10} + 292311592166 x^{9} + 233708309190 x^{8} + 95896505088 x^{7} + 23931351696 x^{6} + 4148844336 x^{5} + 4813362864 x^{4} + 52323118080 x^{3} + 400888193472 x^{2} + 1792784840544 x + 3563298115785$ | $9$ | $2$ | $16$ | $C_{18}$ | $[\ ]_{9}^{2}$ |