Normalized defining polynomial
\( x^{33} + 5x - 2 \)
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: | \(42535295865255938782862371575079650527441685299218120731034014514675712\) \(\medspace = 2^{30}\cdot 41\cdot 179\cdot 1579\cdot 2236133\cdot 39308148498265324319\cdot 38891111153367554795380843499\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(138.12\) | 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: | $2^{30/31}41^{1/2}179^{1/2}1579^{1/2}2236133^{1/2}39308148498265324319^{1/2}38891111153367554795380843499^{1/2}\approx 1.2309605779465077e+31$ | ||
Ramified primes: | \(2\), \(41\), \(179\), \(1579\), \(2236133\), \(39308148498265324319\), \(38891111153367554795380843499\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{39614\!\cdots\!80513}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{31}-\frac{1}{2}a^{30}-\frac{1}{2}a^{29}-\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-\frac{1}{2}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
not computed
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: | 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 non-solvable group of order 8683317618811886495518194401280000000 |
The 10143 conjugacy class representatives for $S_{33}$ are not computed |
Character table for $S_{33}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $26{,}\,{\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $30{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | R | $32{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $31$ | $31$ | $1$ | $30$ | ||||
\(41\) | $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
41.2.1.1 | $x^{2} + 41$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
41.3.0.1 | $x^{3} + x + 35$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(179\) | $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
179.2.1.1 | $x^{2} + 358$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
179.4.0.1 | $x^{4} + x^{2} + 109 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
179.11.0.1 | $x^{11} + 28 x + 177$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
179.15.0.1 | $x^{15} - x + 95$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(1579\) | $\Q_{1579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(2236133\) | $\Q_{2236133}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(39308148498265324319\) | $\Q_{39308148498265324319}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(388\!\cdots\!499\) | $\Q_{38\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{38\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |