Normalized defining polynomial
\( x^{36} - 5 x^{35} - 106 x^{34} + 576 x^{33} + 4748 x^{32} - 28843 x^{31} - 115949 x^{30} + \cdots - 38741779 \)
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: |
\(1009415729163654044784755917747959274289665160574567203886806964874267578125\)
\(\medspace = 5^{27}\cdot 7^{18}\cdot 19^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(121.18\) | 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: | $5^{3/4}7^{1/2}19^{8/9}\approx 121.18430898641847$ | ||
| Ramified primes: |
\(5\), \(7\), \(19\)
| 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: | \(665=5\cdot 7\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{665}(1,·)$, $\chi_{665}(258,·)$, $\chi_{665}(643,·)$, $\chi_{665}(517,·)$, $\chi_{665}(134,·)$, $\chi_{665}(519,·)$, $\chi_{665}(631,·)$, $\chi_{665}(386,·)$, $\chi_{665}(272,·)$, $\chi_{665}(657,·)$, $\chi_{665}(153,·)$, $\chi_{665}(538,·)$, $\chi_{665}(36,·)$, $\chi_{665}(552,·)$, $\chi_{665}(169,·)$, $\chi_{665}(176,·)$, $\chi_{665}(309,·)$, $\chi_{665}(188,·)$, $\chi_{665}(62,·)$, $\chi_{665}(64,·)$, $\chi_{665}(328,·)$, $\chi_{665}(587,·)$, $\chi_{665}(83,·)$, $\chi_{665}(596,·)$, $\chi_{665}(351,·)$, $\chi_{665}(482,·)$, $\chi_{665}(99,·)$, $\chi_{665}(484,·)$, $\chi_{665}(106,·)$, $\chi_{665}(491,·)$, $\chi_{665}(237,·)$, $\chi_{665}(239,·)$, $\chi_{665}(624,·)$, $\chi_{665}(118,·)$, $\chi_{665}(503,·)$, $\chi_{665}(377,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{151}a^{33}-\frac{28}{151}a^{32}-\frac{6}{151}a^{31}+\frac{21}{151}a^{30}-\frac{24}{151}a^{29}-\frac{35}{151}a^{28}+\frac{28}{151}a^{27}-\frac{45}{151}a^{26}-\frac{15}{151}a^{25}-\frac{10}{151}a^{24}-\frac{58}{151}a^{23}+\frac{59}{151}a^{22}+\frac{66}{151}a^{21}+\frac{35}{151}a^{20}+\frac{57}{151}a^{19}-\frac{74}{151}a^{18}-\frac{21}{151}a^{17}-\frac{9}{151}a^{16}-\frac{70}{151}a^{15}+\frac{11}{151}a^{14}+\frac{37}{151}a^{13}+\frac{21}{151}a^{12}+\frac{22}{151}a^{11}+\frac{53}{151}a^{10}-\frac{13}{151}a^{9}+\frac{59}{151}a^{8}-\frac{52}{151}a^{7}-\frac{41}{151}a^{6}-\frac{30}{151}a^{5}-\frac{31}{151}a^{4}+\frac{39}{151}a^{3}+\frac{12}{151}a^{2}+\frac{39}{151}a+\frac{2}{151}$, $\frac{1}{6604589}a^{34}-\frac{11270}{6604589}a^{33}+\frac{497480}{6604589}a^{32}-\frac{1028485}{6604589}a^{31}+\frac{607984}{6604589}a^{30}+\frac{2397967}{6604589}a^{29}-\frac{3292110}{6604589}a^{28}-\frac{2812814}{6604589}a^{27}-\frac{9337}{6604589}a^{26}-\frac{1283396}{6604589}a^{25}-\frac{1429499}{6604589}a^{24}+\frac{728954}{6604589}a^{23}-\frac{2066153}{6604589}a^{22}-\frac{379235}{6604589}a^{21}+\frac{341957}{6604589}a^{20}+\frac{1258108}{6604589}a^{19}+\frac{954197}{6604589}a^{18}+\frac{2559812}{6604589}a^{17}-\frac{1484090}{6604589}a^{16}+\frac{464717}{6604589}a^{15}-\frac{888440}{6604589}a^{14}+\frac{2735437}{6604589}a^{13}+\frac{59749}{6604589}a^{12}-\frac{954253}{6604589}a^{11}+\frac{1751305}{6604589}a^{10}-\frac{2391501}{6604589}a^{9}-\frac{659253}{6604589}a^{8}-\frac{1397483}{6604589}a^{7}-\frac{1923247}{6604589}a^{6}-\frac{564392}{6604589}a^{5}-\frac{2287768}{6604589}a^{4}-\frac{2892176}{6604589}a^{3}+\frac{1977625}{6604589}a^{2}-\frac{626884}{6604589}a+\frac{1000390}{6604589}$, $\frac{1}{17\!\cdots\!49}a^{35}-\frac{49\!\cdots\!78}{17\!\cdots\!49}a^{34}-\frac{13\!\cdots\!53}{17\!\cdots\!49}a^{33}+\frac{26\!\cdots\!75}{17\!\cdots\!49}a^{32}-\frac{72\!\cdots\!25}{17\!\cdots\!49}a^{31}+\frac{41\!\cdots\!73}{17\!\cdots\!49}a^{30}+\frac{59\!\cdots\!11}{17\!\cdots\!49}a^{29}-\frac{13\!\cdots\!26}{17\!\cdots\!49}a^{28}-\frac{59\!\cdots\!41}{17\!\cdots\!49}a^{27}-\frac{62\!\cdots\!27}{17\!\cdots\!49}a^{26}+\frac{74\!\cdots\!34}{17\!\cdots\!49}a^{25}-\frac{58\!\cdots\!82}{17\!\cdots\!49}a^{24}-\frac{51\!\cdots\!85}{17\!\cdots\!49}a^{23}+\frac{39\!\cdots\!47}{17\!\cdots\!49}a^{22}-\frac{12\!\cdots\!75}{17\!\cdots\!49}a^{21}+\frac{67\!\cdots\!75}{17\!\cdots\!49}a^{20}+\frac{34\!\cdots\!49}{17\!\cdots\!49}a^{19}-\frac{16\!\cdots\!25}{17\!\cdots\!49}a^{18}+\frac{33\!\cdots\!34}{17\!\cdots\!49}a^{17}-\frac{21\!\cdots\!38}{17\!\cdots\!49}a^{16}+\frac{13\!\cdots\!78}{17\!\cdots\!49}a^{15}+\frac{76\!\cdots\!44}{17\!\cdots\!49}a^{14}-\frac{27\!\cdots\!87}{17\!\cdots\!49}a^{13}-\frac{70\!\cdots\!70}{17\!\cdots\!49}a^{12}+\frac{25\!\cdots\!27}{17\!\cdots\!49}a^{11}-\frac{24\!\cdots\!71}{17\!\cdots\!49}a^{10}-\frac{50\!\cdots\!58}{17\!\cdots\!49}a^{9}-\frac{13\!\cdots\!46}{17\!\cdots\!49}a^{8}-\frac{24\!\cdots\!78}{17\!\cdots\!49}a^{7}+\frac{10\!\cdots\!21}{17\!\cdots\!49}a^{6}+\frac{29\!\cdots\!20}{17\!\cdots\!49}a^{5}-\frac{44\!\cdots\!69}{17\!\cdots\!49}a^{4}+\frac{71\!\cdots\!64}{17\!\cdots\!49}a^{3}+\frac{86\!\cdots\!09}{17\!\cdots\!49}a^{2}-\frac{41\!\cdots\!51}{17\!\cdots\!49}a-\frac{12\!\cdots\!62}{30\!\cdots\!19}$
| 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, 4.4.6125.1, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.12.3902537516036345703125.1, 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 | $36$ | $36$ | R | R | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $36$ | $36$ | R | $36$ | $18^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | $18^{2}$ | $36$ | $36$ | $36$ | ${\href{/padicField/59.9.0.1}{9} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $36$ | $4$ | $9$ | $27$ | |||
|
\(7\)
| 7.12.6.2 | $x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ |
| 7.12.6.2 | $x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
| 7.12.6.2 | $x^{12} + 49 x^{8} - 1715 x^{6} + 9604 x^{4} - 100842 x^{2} + 352947$ | $2$ | $6$ | $6$ | $C_{12}$ | $[\ ]_{2}^{6}$ | |
|
\(19\)
| 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 | $x^{9} + 19$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |