Normalized defining polynomial
\( x^{36} - 4 x^{35} + 64 x^{34} - 220 x^{33} + 2159 x^{32} - 6676 x^{31} + 49642 x^{30} + \cdots + 267925632607951 \)
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: |
\(11166706806076050240252573251230095765902100267008000000000000000000000000000\)
\(\medspace = 2^{54}\cdot 5^{27}\cdot 19^{32}\)
| 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$$\chi_{760}(1,·)$, $\chi_{760}(517,·)$, $\chi_{760}(9,·)$, $\chi_{760}(653,·)$, $\chi_{760}(529,·)$, $\chi_{760}(533,·)$, $\chi_{760}(157,·)$, $\chi_{760}(197,·)$, $\chi_{760}(161,·)$, $\chi_{760}(49,·)$, $\chi_{760}(169,·)$, $\chi_{760}(557,·)$, $\chi_{760}(93,·)$, $\chi_{760}(689,·)$, $\chi_{760}(693,·)$, $\chi_{760}(329,·)$, $\chi_{760}(441,·)$, $\chi_{760}(321,·)$, $\chi_{760}(453,·)$, $\chi_{760}(289,·)$, $\chi_{760}(201,·)$, $\chi_{760}(77,·)$, $\chi_{760}(397,·)$, $\chi_{760}(81,·)$, $\chi_{760}(213,·)$, $\chi_{760}(729,·)$, $\chi_{760}(733,·)$, $\chi_{760}(609,·)$, $\chi_{760}(613,·)$, $\chi_{760}(237,·)$, $\chi_{760}(481,·)$, $\chi_{760}(757,·)$, $\chi_{760}(681,·)$, $\chi_{760}(121,·)$, $\chi_{760}(253,·)$, $\chi_{760}(277,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{114458}a^{33}+\frac{21689}{114458}a^{32}+\frac{11379}{57229}a^{31}-\frac{2503}{114458}a^{30}-\frac{803}{114458}a^{29}-\frac{11712}{57229}a^{28}+\frac{11351}{57229}a^{27}+\frac{11185}{57229}a^{26}+\frac{3447}{114458}a^{25}-\frac{27631}{114458}a^{24}+\frac{120}{57229}a^{23}+\frac{4934}{57229}a^{22}+\frac{7205}{114458}a^{21}-\frac{5253}{114458}a^{20}+\frac{20777}{114458}a^{19}+\frac{4557}{114458}a^{18}-\frac{27295}{114458}a^{17}-\frac{5481}{57229}a^{16}+\frac{30977}{114458}a^{15}-\frac{22025}{57229}a^{14}-\frac{17131}{114458}a^{13}-\frac{49137}{114458}a^{12}-\frac{12933}{114458}a^{11}-\frac{12182}{57229}a^{10}-\frac{4411}{57229}a^{9}+\frac{7231}{114458}a^{8}-\frac{4688}{57229}a^{7}+\frac{51141}{114458}a^{6}-\frac{19795}{114458}a^{5}-\frac{23061}{57229}a^{4}-\frac{16873}{57229}a^{3}+\frac{40289}{114458}a^{2}+\frac{3289}{114458}a+\frac{339}{758}$, $\frac{1}{69\!\cdots\!02}a^{34}-\frac{19\!\cdots\!61}{69\!\cdots\!02}a^{33}-\frac{61\!\cdots\!16}{34\!\cdots\!01}a^{32}+\frac{19\!\cdots\!06}{34\!\cdots\!01}a^{31}-\frac{57\!\cdots\!43}{34\!\cdots\!01}a^{30}-\frac{13\!\cdots\!39}{69\!\cdots\!02}a^{29}+\frac{34\!\cdots\!29}{34\!\cdots\!01}a^{28}-\frac{14\!\cdots\!95}{69\!\cdots\!02}a^{27}+\frac{17\!\cdots\!19}{69\!\cdots\!02}a^{26}+\frac{35\!\cdots\!62}{34\!\cdots\!01}a^{25}+\frac{93\!\cdots\!11}{69\!\cdots\!02}a^{24}+\frac{59\!\cdots\!57}{69\!\cdots\!02}a^{23}+\frac{16\!\cdots\!11}{69\!\cdots\!02}a^{22}-\frac{15\!\cdots\!65}{34\!\cdots\!01}a^{21}+\frac{23\!\cdots\!70}{34\!\cdots\!01}a^{20}+\frac{55\!\cdots\!83}{34\!\cdots\!01}a^{19}-\frac{75\!\cdots\!84}{34\!\cdots\!01}a^{18}-\frac{81\!\cdots\!32}{34\!\cdots\!01}a^{17}+\frac{11\!\cdots\!47}{69\!\cdots\!02}a^{16}+\frac{72\!\cdots\!21}{69\!\cdots\!02}a^{15}+\frac{76\!\cdots\!43}{69\!\cdots\!02}a^{14}+\frac{37\!\cdots\!60}{34\!\cdots\!01}a^{13}-\frac{16\!\cdots\!14}{34\!\cdots\!01}a^{12}+\frac{53\!\cdots\!83}{34\!\cdots\!01}a^{11}+\frac{74\!\cdots\!52}{34\!\cdots\!01}a^{10}-\frac{14\!\cdots\!43}{34\!\cdots\!01}a^{9}+\frac{16\!\cdots\!41}{69\!\cdots\!02}a^{8}+\frac{31\!\cdots\!95}{69\!\cdots\!02}a^{7}+\frac{28\!\cdots\!49}{69\!\cdots\!02}a^{6}+\frac{13\!\cdots\!41}{69\!\cdots\!02}a^{5}+\frac{15\!\cdots\!57}{34\!\cdots\!01}a^{4}-\frac{27\!\cdots\!03}{34\!\cdots\!01}a^{3}+\frac{71\!\cdots\!73}{69\!\cdots\!02}a^{2}+\frac{14\!\cdots\!78}{34\!\cdots\!01}a-\frac{26\!\cdots\!77}{20\!\cdots\!38}$, $\frac{1}{10\!\cdots\!58}a^{35}+\frac{45\!\cdots\!31}{53\!\cdots\!29}a^{34}-\frac{18\!\cdots\!76}{53\!\cdots\!29}a^{33}-\frac{34\!\cdots\!38}{53\!\cdots\!29}a^{32}+\frac{41\!\cdots\!04}{53\!\cdots\!29}a^{31}-\frac{33\!\cdots\!57}{10\!\cdots\!58}a^{30}+\frac{15\!\cdots\!61}{10\!\cdots\!58}a^{29}-\frac{10\!\cdots\!13}{53\!\cdots\!29}a^{28}+\frac{78\!\cdots\!43}{10\!\cdots\!58}a^{27}+\frac{16\!\cdots\!73}{10\!\cdots\!58}a^{26}-\frac{74\!\cdots\!29}{10\!\cdots\!58}a^{25}+\frac{55\!\cdots\!03}{35\!\cdots\!79}a^{24}+\frac{77\!\cdots\!72}{53\!\cdots\!29}a^{23}-\frac{15\!\cdots\!07}{10\!\cdots\!58}a^{22}+\frac{11\!\cdots\!25}{10\!\cdots\!58}a^{21}+\frac{12\!\cdots\!70}{53\!\cdots\!29}a^{20}+\frac{46\!\cdots\!74}{53\!\cdots\!29}a^{19}+\frac{19\!\cdots\!55}{10\!\cdots\!58}a^{18}+\frac{34\!\cdots\!33}{10\!\cdots\!58}a^{17}+\frac{12\!\cdots\!62}{53\!\cdots\!29}a^{16}-\frac{13\!\cdots\!31}{10\!\cdots\!58}a^{15}+\frac{55\!\cdots\!91}{53\!\cdots\!29}a^{14}+\frac{29\!\cdots\!77}{10\!\cdots\!58}a^{13}+\frac{14\!\cdots\!43}{10\!\cdots\!58}a^{12}-\frac{37\!\cdots\!81}{10\!\cdots\!58}a^{11}-\frac{12\!\cdots\!76}{53\!\cdots\!29}a^{10}-\frac{95\!\cdots\!32}{53\!\cdots\!29}a^{9}-\frac{25\!\cdots\!20}{53\!\cdots\!29}a^{8}+\frac{38\!\cdots\!91}{10\!\cdots\!58}a^{7}-\frac{16\!\cdots\!61}{10\!\cdots\!58}a^{6}+\frac{13\!\cdots\!39}{70\!\cdots\!58}a^{5}-\frac{13\!\cdots\!10}{53\!\cdots\!29}a^{4}-\frac{44\!\cdots\!51}{53\!\cdots\!29}a^{3}+\frac{23\!\cdots\!69}{10\!\cdots\!58}a^{2}+\frac{16\!\cdots\!47}{53\!\cdots\!29}a-\frac{14\!\cdots\!11}{33\!\cdots\!82}$
| 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
| \(\Q(\sqrt{5}) \), 3.3.361.1, 4.0.8000.2, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.8695584276992000000000.1, 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 | ${\href{/padicField/7.12.0.1}{12} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | $36$ | $36$ | R | $36$ | ${\href{/padicField/29.9.0.1}{9} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{12}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | $36$ | $36$ | $36$ | ${\href{/padicField/59.9.0.1}{9} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $36$ | $2$ | $18$ | $54$ | |||
|
\(5\)
| Deg $36$ | $4$ | $9$ | $27$ | |||
|
\(19\)
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.8 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |