Normalized defining polynomial
\( x^{18} - 18x^{16} + 135x^{14} - 546x^{12} + 1287x^{10} - 1782x^{8} + 1386x^{6} - 540x^{4} + 81x^{2} - 3 \)
Invariants
Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[18, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(774455350146061749097070592\) \(\medspace = 2^{18}\cdot 3^{45}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(31.18\) | 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))
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Galois root discriminant: | $2\cdot 3^{5/2}\approx 31.176914536239792$ | ||
Ramified primes: | \(2\), \(3\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
$\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(108=2^{2}\cdot 3^{3}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{108}(1,·)$, $\chi_{108}(71,·)$, $\chi_{108}(73,·)$, $\chi_{108}(11,·)$, $\chi_{108}(13,·)$, $\chi_{108}(83,·)$, $\chi_{108}(85,·)$, $\chi_{108}(23,·)$, $\chi_{108}(25,·)$, $\chi_{108}(95,·)$, $\chi_{108}(97,·)$, $\chi_{108}(35,·)$, $\chi_{108}(37,·)$, $\chi_{108}(107,·)$, $\chi_{108}(47,·)$, $\chi_{108}(49,·)$, $\chi_{108}(59,·)$, $\chi_{108}(61,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $a^{6}-6a^{4}+9a^{2}-1$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{8}-7a^{6}+15a^{4}-10a^{2}+1$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{16}-15a^{14}+91a^{12}-286a^{10}+494a^{8}-455a^{6}+194a^{4}-23a^{2}-1$, $a^{14}-15a^{12}+90a^{10}-275a^{8}+449a^{6}-371a^{4}+125a^{2}-5$, $a^{12}-13a^{10}+65a^{8}-156a^{6}+182a^{4}-91a^{2}+13$, $a^{4}-4a^{2}+2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+139a^{3}-12a-1$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}+a^{3}-36a^{2}-3a+2$, $a^{5}-5a^{3}+5a-1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a-1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+181a^{5}-86a^{3}+8a-1$, $a^{11}-11a^{9}+43a^{7}-70a^{5}+41a^{3}-4a+1$, $a+1$, $a^{17}-17a^{15}+a^{14}+119a^{13}-14a^{12}-441a^{11}+77a^{10}+924a^{9}-209a^{8}-1078a^{7}+286a^{6}+638a^{5}-176a^{4}-154a^{3}+34a^{2}+11a-2$, $a^{7}-7a^{5}+14a^{3}-7a+1$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 43670324.6529 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{0}\cdot 43670324.6529 \cdot 1}{2\cdot\sqrt{774455350146061749097070592}}\cr\approx \mathstrut & 0.205682884710 \end{aligned}\] (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
\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{27})^+\) |
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 | $18$ | $18$ | ${\href{/padicField/11.9.0.1}{9} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | $18$ | $18$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/47.9.0.1}{9} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 | $x^{18} + 18 x^{17} + 198 x^{16} + 1536 x^{15} + 9312 x^{14} + 45696 x^{13} + 187776 x^{12} + 655872 x^{11} + 2010400 x^{10} + 5500224 x^{9} + 14116288 x^{8} + 34058240 x^{7} + 79898624 x^{6} + 169960448 x^{5} + 335809536 x^{4} + 542121984 x^{3} + 798549248 x^{2} + 783239680 x + 807955968$ | $2$ | $9$ | $18$ | $C_{18}$ | $[2]^{9}$ |
\(3\) | Deg $18$ | $18$ | $1$ | $45$ |