Normalized defining polynomial
\( x^{18} + 37 x^{16} + 518 x^{14} + 3515 x^{12} + 12284 x^{10} + 22200 x^{8} + 20683 x^{6} + 9287 x^{4} + \cdots + 37 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(-119665574759900159778264689410048\)
\(\medspace = -\,2^{18}\cdot 37^{17}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(60.55\) | 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: | $2\cdot 37^{17/18}\approx 60.54933834364826$ | ||
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: | \(\Q(\sqrt{-37}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(148=2^{2}\cdot 37\) | ||
Dirichlet character group: | $\lbrace$$\chi_{148}(1,·)$, $\chi_{148}(67,·)$, $\chi_{148}(9,·)$, $\chi_{148}(11,·)$, $\chi_{148}(115,·)$, $\chi_{148}(145,·)$, $\chi_{148}(3,·)$, $\chi_{148}(139,·)$, $\chi_{148}(27,·)$, $\chi_{148}(95,·)$, $\chi_{148}(33,·)$, $\chi_{148}(99,·)$, $\chi_{148}(81,·)$, $\chi_{148}(49,·)$, $\chi_{148}(147,·)$, $\chi_{148}(53,·)$, $\chi_{148}(137,·)$, $\chi_{148}(121,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{256}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{43}a^{14}+\frac{14}{43}a^{10}-\frac{13}{43}a^{8}-\frac{10}{43}a^{6}+\frac{11}{43}a^{4}-\frac{11}{43}a^{2}-\frac{21}{43}$, $\frac{1}{43}a^{15}+\frac{14}{43}a^{11}-\frac{13}{43}a^{9}-\frac{10}{43}a^{7}+\frac{11}{43}a^{5}-\frac{11}{43}a^{3}-\frac{21}{43}a$, $\frac{1}{2769316831}a^{16}+\frac{31484785}{2769316831}a^{14}-\frac{1186318945}{2769316831}a^{12}+\frac{28402830}{2769316831}a^{10}+\frac{454901551}{2769316831}a^{8}+\frac{552171309}{2769316831}a^{6}-\frac{1045941801}{2769316831}a^{4}-\frac{852680188}{2769316831}a^{2}-\frac{7266499}{2769316831}$, $\frac{1}{2769316831}a^{17}+\frac{31484785}{2769316831}a^{15}-\frac{1186318945}{2769316831}a^{13}+\frac{28402830}{2769316831}a^{11}+\frac{454901551}{2769316831}a^{9}+\frac{552171309}{2769316831}a^{7}-\frac{1045941801}{2769316831}a^{5}-\frac{852680188}{2769316831}a^{3}-\frac{7266499}{2769316831}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{1526}$, which has order $1526$ (assuming GRH)
Unit group
Rank: | $8$ | 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: |
$\frac{49550997}{2769316831}a^{16}+\frac{1796481397}{2769316831}a^{14}+\frac{24318057492}{2769316831}a^{12}+\frac{155752661387}{2769316831}a^{10}+\frac{490064468150}{2769316831}a^{8}+\frac{726997454480}{2769316831}a^{6}+\frac{477017477491}{2769316831}a^{4}+\frac{111421696335}{2769316831}a^{2}+\frac{4046606249}{2769316831}$, $\frac{20539826}{2769316831}a^{16}+\frac{752755988}{2769316831}a^{14}+\frac{10379690287}{2769316831}a^{12}+\frac{68725581667}{2769316831}a^{10}+\frac{230595822631}{2769316831}a^{8}+\frac{9063228486}{64402717}a^{6}+\frac{325415701697}{2769316831}a^{4}+\frac{108605606320}{2769316831}a^{2}+\frac{2281626251}{2769316831}$, $\frac{92128816}{2769316831}a^{16}+\frac{3316313630}{2769316831}a^{14}+\frac{44415630755}{2769316831}a^{12}+\frac{279956301389}{2769316831}a^{10}+\frac{859148913456}{2769316831}a^{8}+\frac{1229100970102}{2769316831}a^{6}+\frac{788279851211}{2769316831}a^{4}+\frac{181928040922}{2769316831}a^{2}+\frac{1603028794}{2769316831}$, $\frac{77384195}{2769316831}a^{16}+\frac{2785829244}{2769316831}a^{14}+\frac{37283763962}{2769316831}a^{12}+\frac{234174019392}{2769316831}a^{10}+\frac{709184229015}{2769316831}a^{8}+\frac{964011941859}{2769316831}a^{6}+\frac{503666485130}{2769316831}a^{4}+\frac{40043423891}{2769316831}a^{2}-\frac{8897723172}{2769316831}$, $\frac{51050940}{2769316831}a^{16}+\frac{1826221458}{2769316831}a^{14}+\frac{24203912252}{2769316831}a^{12}+\frac{149734341028}{2769316831}a^{10}+\frac{442690226320}{2769316831}a^{8}+\frac{582522198502}{2769316831}a^{6}+\frac{312405557761}{2769316831}a^{4}+\frac{56689381808}{2769316831}a^{2}+\frac{6031397848}{2769316831}$, $\frac{71588990}{2769316831}a^{16}+\frac{2563557642}{2769316831}a^{14}+\frac{34035940468}{2769316831}a^{12}+\frac{211230719722}{2769316831}a^{10}+\frac{628553090825}{2769316831}a^{8}+\frac{839382145204}{2769316831}a^{6}+\frac{462864149514}{2769316831}a^{4}+\frac{73322434602}{2769316831}a^{2}-\frac{3447914288}{2769316831}$, $\frac{2404082}{2769316831}a^{16}+\frac{71226723}{2769316831}a^{14}+\frac{609640170}{2769316831}a^{12}+\frac{9263678}{2769316831}a^{10}-\frac{22644520692}{2769316831}a^{8}-\frac{99722371709}{2769316831}a^{6}-\frac{147574524030}{2769316831}a^{4}-\frac{81840141221}{2769316831}a^{2}-\frac{13353825087}{2769316831}$, $\frac{84231303}{2769316831}a^{16}+\frac{3043887602}{2769316831}a^{14}+\frac{41008869355}{2769316831}a^{12}+\frac{260810318668}{2769316831}a^{10}+\frac{811315960802}{2769316831}a^{8}+\frac{1180613782724}{2769316831}a^{6}+\frac{757033992705}{2769316831}a^{4}+\frac{177803098919}{2769316831}a^{2}+\frac{12685978632}{2769316831}$
| 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: | \( 409151.310213 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 409151.310213 \cdot 1526}{2\cdot\sqrt{119665574759900159778264689410048}}\cr\approx \mathstrut & 0.435554679313 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 18 |
The 18 conjugacy class representatives for $C_{18}$ |
Character table for $C_{18}$ |
Intermediate fields
\(\Q(\sqrt{-37}) \), 3.3.1369.1, 6.0.4438013248.1, 9.9.3512479453921.1 |
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 | $18$ | $18$ | $18$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | $18$ | $18$ | ${\href{/padicField/19.9.0.1}{9} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.1.0.1}{1} }^{18}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{18}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.9.0.1}{9} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| 2.18.18.118 | $x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
\(37\)
| 37.18.17.1 | $x^{18} + 37$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |