Normalized defining polynomial
\( x^{36} + 95 x^{34} + 4085 x^{32} + 105450 x^{30} + 1827800 x^{28} + 22545875 x^{26} + 204644250 x^{24} + \cdots + 705078125 \)
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: | \(15377735736821953341412273192192323957407601152000000000000000000000000000\) \(\medspace = 2^{36}\cdot 5^{27}\cdot 19^{34}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(107.89\) | 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: | $2\cdot 5^{3/4}19^{17/18}\approx 107.88702887996082$ | ||
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: | \(380=2^{2}\cdot 5\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(3,·)$, $\chi_{380}(9,·)$, $\chi_{380}(143,·)$, $\chi_{380}(147,·)$, $\chi_{380}(149,·)$, $\chi_{380}(27,·)$, $\chi_{380}(329,·)$, $\chi_{380}(287,·)$, $\chi_{380}(289,·)$, $\chi_{380}(167,·)$, $\chi_{380}(169,·)$, $\chi_{380}(301,·)$, $\chi_{380}(303,·)$, $\chi_{380}(49,·)$, $\chi_{380}(307,·)$, $\chi_{380}(309,·)$, $\chi_{380}(183,·)$, $\chi_{380}(61,·)$, $\chi_{380}(321,·)$, $\chi_{380}(67,·)$, $\chi_{380}(161,·)$, $\chi_{380}(201,·)$, $\chi_{380}(203,·)$, $\chi_{380}(81,·)$, $\chi_{380}(349,·)$, $\chi_{380}(223,·)$, $\chi_{380}(107,·)$, $\chi_{380}(227,·)$, $\chi_{380}(101,·)$, $\chi_{380}(103,·)$, $\chi_{380}(363,·)$, $\chi_{380}(229,·)$, $\chi_{380}(243,·)$, $\chi_{380}(121,·)$, $\chi_{380}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{625}a^{16}$, $\frac{1}{625}a^{17}$, $\frac{1}{11875}a^{18}$, $\frac{1}{11875}a^{19}$, $\frac{1}{59375}a^{20}$, $\frac{1}{59375}a^{21}$, $\frac{1}{59375}a^{22}$, $\frac{1}{59375}a^{23}$, $\frac{1}{296875}a^{24}$, $\frac{1}{296875}a^{25}$, $\frac{1}{296875}a^{26}$, $\frac{1}{296875}a^{27}$, $\frac{1}{1484375}a^{28}$, $\frac{1}{1484375}a^{29}$, $\frac{1}{1484375}a^{30}$, $\frac{1}{1484375}a^{31}$, $\frac{1}{7421875}a^{32}$, $\frac{1}{7421875}a^{33}$, $\frac{1}{24574651953125}a^{34}-\frac{63628}{1293402734375}a^{32}-\frac{11663}{51736109375}a^{30}+\frac{52747}{258680546875}a^{28}-\frac{14232}{10347221875}a^{26}+\frac{68982}{51736109375}a^{24}+\frac{58252}{10347221875}a^{22}+\frac{25629}{10347221875}a^{20}-\frac{71707}{2069444375}a^{18}-\frac{1616226}{2069444375}a^{16}-\frac{77373}{21783625}a^{14}+\frac{9108}{21783625}a^{12}+\frac{69108}{4356725}a^{10}+\frac{75688}{4356725}a^{8}-\frac{60292}{871345}a^{6}+\frac{30682}{871345}a^{4}-\frac{58138}{174269}a^{2}+\frac{56265}{174269}$, $\frac{1}{24574651953125}a^{35}-\frac{63628}{1293402734375}a^{33}-\frac{11663}{51736109375}a^{31}+\frac{52747}{258680546875}a^{29}-\frac{14232}{10347221875}a^{27}+\frac{68982}{51736109375}a^{25}+\frac{58252}{10347221875}a^{23}+\frac{25629}{10347221875}a^{21}-\frac{71707}{2069444375}a^{19}-\frac{1616226}{2069444375}a^{17}-\frac{77373}{21783625}a^{15}+\frac{9108}{21783625}a^{13}+\frac{69108}{4356725}a^{11}+\frac{75688}{4356725}a^{9}-\frac{60292}{871345}a^{7}+\frac{30682}{871345}a^{5}-\frac{58138}{174269}a^{3}+\frac{56265}{174269}a$
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, 4.0.722000.3, 6.6.16290125.1, \(\Q(\zeta_{19})^+\), 12.0.49048530062408000000000.1, 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 | $36$ | ${\href{/padicField/29.9.0.1}{9} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{12}$ | ${\href{/padicField/37.4.0.1}{4} }^{9}$ | $18^{2}$ | $36$ | $36$ | $36$ | $18^{2}$ |
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 $36$ | $2$ | $18$ | $36$ | |||
\(5\) | Deg $36$ | $4$ | $9$ | $27$ | |||
\(19\) | 19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |
19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |