Normalized defining polynomial
\( x^{36} + 74 x^{34} + 2516 x^{32} + 52096 x^{30} + 734080 x^{28} + 7450912 x^{26} + 56242368 x^{24} + \cdots + 9699328 \)
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: | \(138892919952333446776057851184385905517238171566853781889085447929331712\) \(\medspace = 2^{54}\cdot 37^{35}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(94.66\) | 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: | $2^{3/2}37^{35/36}\approx 94.66415411791277$ | ||
Ramified primes: | \(2\), \(37\) | 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: | \(296=2^{3}\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{296}(1,·)$, $\chi_{296}(5,·)$, $\chi_{296}(9,·)$, $\chi_{296}(13,·)$, $\chi_{296}(109,·)$, $\chi_{296}(145,·)$, $\chi_{296}(277,·)$, $\chi_{296}(73,·)$, $\chi_{296}(25,·)$, $\chi_{296}(29,·)$, $\chi_{296}(261,·)$, $\chi_{296}(133,·)$, $\chi_{296}(33,·)$, $\chi_{296}(165,·)$, $\chi_{296}(41,·)$, $\chi_{296}(45,·)$, $\chi_{296}(49,·)$, $\chi_{296}(137,·)$, $\chi_{296}(61,·)$, $\chi_{296}(117,·)$, $\chi_{296}(65,·)$, $\chi_{296}(69,·)$, $\chi_{296}(289,·)$, $\chi_{296}(201,·)$, $\chi_{296}(205,·)$, $\chi_{296}(81,·)$, $\chi_{296}(121,·)$, $\chi_{296}(93,·)$, $\chi_{296}(225,·)$, $\chi_{296}(233,·)$, $\chi_{296}(237,·)$, $\chi_{296}(125,·)$, $\chi_{296}(245,·)$, $\chi_{296}(169,·)$, $\chi_{296}(249,·)$, $\chi_{296}(253,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$, $\frac{1}{32768}a^{30}$, $\frac{1}{32768}a^{31}$, $\frac{1}{65536}a^{32}$, $\frac{1}{65536}a^{33}$, $\frac{1}{131072}a^{34}$, $\frac{1}{131072}a^{35}$
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 $C_{36}$ is not computed |
Intermediate fields
\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.3241792.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.46640037043754870505472.1, \(\Q(\zeta_{37})^+\) |
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 | ${\href{/padicField/3.9.0.1}{9} }^{4}$ | $36$ | ${\href{/padicField/7.9.0.1}{9} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $36$ | $36$ | $36$ | ${\href{/padicField/23.12.0.1}{12} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{9}$ | R | $18^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{9}$ | ${\href{/padicField/47.3.0.1}{3} }^{12}$ | $18^{2}$ | $36$ |
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 $36$ | $2$ | $18$ | $54$ | |||
\(37\) | Deg $36$ | $36$ | $1$ | $35$ |