Normalized defining polynomial
\( x^{36} - x^{35} + x^{33} - x^{32} + x^{30} - x^{29} + x^{27} - x^{26} + x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{16} + x^{15} - x^{13} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)
Invariants
Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(11636034958735032166924075841251447518799351583251569\) \(\medspace = 3^{18}\cdot 19^{34}\) | 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: | \(27.94\) | 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: | $3^{1/2}19^{17/18}\approx 27.942956945645935$ | ||
Ramified primes: | \(3\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(57=3\cdot 19\) | ||
Dirichlet character group: | $\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(2,·)$, $\chi_{57}(4,·)$, $\chi_{57}(5,·)$, $\chi_{57}(7,·)$, $\chi_{57}(8,·)$, $\chi_{57}(10,·)$, $\chi_{57}(11,·)$, $\chi_{57}(13,·)$, $\chi_{57}(14,·)$, $\chi_{57}(16,·)$, $\chi_{57}(17,·)$, $\chi_{57}(20,·)$, $\chi_{57}(22,·)$, $\chi_{57}(23,·)$, $\chi_{57}(25,·)$, $\chi_{57}(26,·)$, $\chi_{57}(28,·)$, $\chi_{57}(29,·)$, $\chi_{57}(31,·)$, $\chi_{57}(32,·)$, $\chi_{57}(34,·)$, $\chi_{57}(35,·)$, $\chi_{57}(37,·)$, $\chi_{57}(40,·)$, $\chi_{57}(41,·)$, $\chi_{57}(43,·)$, $\chi_{57}(44,·)$, $\chi_{57}(46,·)$, $\chi_{57}(47,·)$, $\chi_{57}(49,·)$, $\chi_{57}(50,·)$, $\chi_{57}(52,·)$, $\chi_{57}(53,·)$, $\chi_{57}(55,·)$, $\chi_{57}(56,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{9}$, which has order $9$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -a \) (order $114$) | 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: | $a^{15}+1$, $a^{30}+1$, $a^{3}+1$, $a^{30}+a^{15}+1$, $a^{29}+a^{20}+a^{10}+a-1$, $a^{35}-a^{27}+a^{23}+a^{20}+a^{16}-a^{12}+a^{4}+a$, $a^{35}-a^{33}+a^{32}-a^{30}+a^{29}-a^{27}+a^{26}-a^{24}+a^{23}-a^{21}+a^{20}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{10}-a^{9}+a^{7}-a^{6}+a^{4}-a^{3}+a$, $a^{26}-a^{18}-a^{15}+a^{7}$, $a-1$, $a^{2}-1$, $a^{4}-1$, $a^{8}-1$, $a^{10}-1$, $a^{5}-1$, $a^{7}-1$, $a^{13}-1$, $a^{34}-a^{33}-a^{30}+a^{29}+a^{25}-a^{24}-a^{21}+a^{20}-a^{14}+a^{13}-a^{12}+a^{10}-a^{5}+a^{4}-a^{3}+a-1$ (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: | \( 719116485989.9799 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 719116485989.9799 \cdot 9}{114\cdot\sqrt{11636034958735032166924075841251447518799351583251569}}\cr\approx \mathstrut & 0.122594793117570 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{18}$ (as 36T2):
An abelian group of order 36 |
The 36 conjugacy class representatives for $C_2\times C_{18}$ |
Character table for $C_2\times C_{18}$ is not computed |
Intermediate fields
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 | $18^{2}$ | R | $18^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{12}$ | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | $18^{2}$ | $18^{2}$ | R | $18^{2}$ | $18^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{18}$ | $18^{2}$ | ${\href{/padicField/43.9.0.1}{9} }^{4}$ | $18^{2}$ | $18^{2}$ | $18^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $36$ | $2$ | $18$ | $18$ | |||
\(19\) | 19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |
19.18.17.14 | $x^{18} + 19$ | $18$ | $1$ | $17$ | $C_{18}$ | $[\ ]_{18}$ |