Normalized defining polynomial
\( x^{36} + 111 x^{34} + 5661 x^{32} + 175824 x^{30} + 3716280 x^{28} + 56580363 x^{26} + \cdots + 14334558093 \)
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: | \(205268718591960451664107537071777407379898873740754655345797836302054326272\) \(\medspace = 2^{36}\cdot 3^{18}\cdot 37^{35}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(115.94\) | 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: | $2\cdot 3^{1/2}37^{35/36}\approx 115.93943726053665$ | ||
Ramified primes: | \(2\), \(3\), \(37\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
$\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(444=2^{2}\cdot 3\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{444}(1,·)$, $\chi_{444}(131,·)$, $\chi_{444}(397,·)$, $\chi_{444}(143,·)$, $\chi_{444}(145,·)$, $\chi_{444}(23,·)$, $\chi_{444}(25,·)$, $\chi_{444}(157,·)$, $\chi_{444}(289,·)$, $\chi_{444}(35,·)$, $\chi_{444}(167,·)$, $\chi_{444}(169,·)$, $\chi_{444}(431,·)$, $\chi_{444}(433,·)$, $\chi_{444}(179,·)$, $\chi_{444}(181,·)$, $\chi_{444}(311,·)$, $\chi_{444}(59,·)$, $\chi_{444}(191,·)$, $\chi_{444}(49,·)$, $\chi_{444}(73,·)$, $\chi_{444}(203,·)$, $\chi_{444}(335,·)$, $\chi_{444}(337,·)$, $\chi_{444}(85,·)$, $\chi_{444}(347,·)$, $\chi_{444}(349,·)$, $\chi_{444}(227,·)$, $\chi_{444}(229,·)$, $\chi_{444}(361,·)$, $\chi_{444}(239,·)$, $\chi_{444}(373,·)$, $\chi_{444}(119,·)$, $\chi_{444}(121,·)$, $\chi_{444}(251,·)$, $\chi_{444}(383,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{531441}a^{24}$, $\frac{1}{531441}a^{25}$, $\frac{1}{1594323}a^{26}$, $\frac{1}{1594323}a^{27}$, $\frac{1}{4782969}a^{28}$, $\frac{1}{4782969}a^{29}$, $\frac{1}{14348907}a^{30}$, $\frac{1}{14348907}a^{31}$, $\frac{1}{43046721}a^{32}$, $\frac{1}{43046721}a^{33}$, $\frac{1}{129140163}a^{34}$, $\frac{1}{129140163}a^{35}$
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
\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.7294032.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.531259171951520321851392.1, \(\Q(\zeta_{37})^+\) |
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 | R | $36$ | $18^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | $36$ | $36$ | $36$ | ${\href{/padicField/23.12.0.1}{12} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{9}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{9}$ | ${\href{/padicField/47.3.0.1}{3} }^{12}$ | $18^{2}$ | $36$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $36$ | $2$ | $18$ | $36$ | |||
\(3\) | Deg $18$ | $2$ | $9$ | $9$ | |||
Deg $18$ | $2$ | $9$ | $9$ | ||||
\(37\) | Deg $36$ | $36$ | $1$ | $35$ |