Normalized defining polynomial
\( x^{33} - 2 \)
Invariants
Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(554523399762182584434496381766741196872482924136058058702848\) \(\medspace = 2^{32}\cdot 3^{33}\cdot 11^{33}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(64.63\) | 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: | $2^{32/33}3^{7/6}11^{119/110}\approx 94.4380214379661$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{33}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $16$ | 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: | $a^{11}-1$, $a^{3}-1$, $a^{2}+a+1$, $a^{18}+a^{3}+1$, $a^{17}-a-1$, $a^{27}+a^{21}+a^{15}+a^{9}+a^{3}+1$, $a^{24}+a^{21}+a^{18}+a^{15}-a^{9}-a^{6}-2a^{3}-1$, $a^{21}+a^{12}-a^{9}-a^{6}-1$, $a^{30}+a^{29}+a^{26}+a^{25}+a^{22}+a^{21}+a^{18}-a^{16}-a^{13}-a^{12}-a^{9}-a^{8}-a^{5}-a^{4}+1$, $a^{31}+a^{30}-a^{28}-a^{27}+a^{25}+a^{24}-a^{22}-a^{21}+a^{19}+a^{18}-a^{16}-a^{15}+a^{13}+a^{12}-a^{10}-a^{9}+a^{7}+a^{6}-a^{4}-a^{3}-1$, $a^{31}+a^{29}+a^{27}+a^{26}+a^{24}+a^{22}-a^{20}-a^{19}-a^{18}-2a^{17}-a^{16}-2a^{15}-a^{14}-a^{13}-2a^{12}-a^{10}+a^{9}-a^{8}+a^{6}+a^{4}+a^{3}+a^{2}+2a+1$, $2a^{32}+2a^{31}+2a^{30}+a^{29}+3a^{28}+a^{27}-a^{26}+2a^{25}-a^{23}-a^{22}-3a^{21}-a^{20}-2a^{19}-5a^{18}-2a^{17}-2a^{16}-4a^{15}-3a^{14}-4a^{13}-2a^{12}-4a^{10}-a^{9}+2a^{8}+2a^{6}+a^{5}+2a^{4}+6a^{3}+2a^{2}+3a+7$, $2a^{32}+2a^{31}-a^{30}+a^{29}-2a^{27}+a^{26}-a^{24}+a^{23}-2a^{21}+3a^{20}+a^{19}-2a^{18}-a^{16}-3a^{15}+2a^{14}+3a^{13}+3a^{11}-2a^{10}-5a^{9}-a^{8}+a^{7}+5a^{5}+2a^{4}-4a^{3}-3a-3$, $3a^{32}-a^{31}+4a^{30}-3a^{29}+4a^{28}-a^{27}+2a^{26}-4a^{25}+a^{24}-3a^{22}-4a^{20}+4a^{19}-3a^{18}+3a^{17}-4a^{16}+7a^{15}+a^{13}+a^{12}+a^{11}+3a^{10}-4a^{9}+2a^{8}-6a^{7}+5a^{6}-7a^{5}-2a^{3}+2a^{2}-a-1$, $19a^{32}+6a^{31}+7a^{30}-10a^{29}-10a^{28}-19a^{27}-9a^{26}-14a^{25}+6a^{24}+8a^{23}+20a^{22}+16a^{21}+13a^{20}+4a^{19}-6a^{18}-18a^{17}-21a^{16}-18a^{15}-9a^{14}+4a^{13}+9a^{12}+28a^{11}+18a^{10}+21a^{9}-2a^{8}-5a^{7}-25a^{6}-19a^{5}-32a^{4}-3a^{3}-4a^{2}+27a+15$, $5a^{32}+5a^{31}+5a^{30}-5a^{29}-3a^{28}-6a^{27}+a^{26}+8a^{25}-a^{24}+11a^{23}-13a^{22}-8a^{20}+5a^{19}+9a^{18}+2a^{17}+a^{16}-9a^{15}-4a^{14}-4a^{13}+9a^{12}+3a^{11}+11a^{10}-13a^{9}-2a^{8}-12a^{7}+6a^{6}+11a^{5}+10a^{3}-18a^{2}-11$ (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)]
| |
Regulator: | \( 95573511152749460 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 95573511152749460 \cdot 1}{2\cdot\sqrt{554523399762182584434496381766741196872482924136058058702848}}\cr\approx \mathstrut & 0.757278272494686 \end{aligned}\] (assuming GRH)
Galois group
$S_3\times F_{11}$ (as 33T11):
A solvable group of order 660 |
The 33 conjugacy class representatives for $S_3\times F_{11}$ |
Character table for $S_3\times F_{11}$ is not computed |
Intermediate fields
3.1.108.1, 11.1.292159150705664.1 |
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 | R | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/7.3.0.1}{3} }$ | R | $30{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.10.0.1}{10} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/19.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.10.0.1}{10} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $15^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{15}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $33$ | $33$ | $1$ | $32$ | |||
\(3\) | 3.3.3.2 | $x^{3} + 3 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.15.15.49 | $x^{15} - 18 x^{14} + 87 x^{13} + 663 x^{12} + 7002 x^{11} + 59229 x^{10} + 24984 x^{9} + 28998 x^{8} + 14823 x^{7} + 8208 x^{6} + 729 x^{5} - 3969 x^{4} - 2997 x^{3} + 972 x^{2} + 1215 x + 243$ | $3$ | $5$ | $15$ | $S_3 \times C_5$ | $[3/2]_{2}^{5}$ | |
3.15.15.49 | $x^{15} - 18 x^{14} + 87 x^{13} + 663 x^{12} + 7002 x^{11} + 59229 x^{10} + 24984 x^{9} + 28998 x^{8} + 14823 x^{7} + 8208 x^{6} + 729 x^{5} - 3969 x^{4} - 2997 x^{3} + 972 x^{2} + 1215 x + 243$ | $3$ | $5$ | $15$ | $S_3 \times C_5$ | $[3/2]_{2}^{5}$ | |
\(11\) | 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
Deg $22$ | $11$ | $2$ | $22$ |