Normalized defining polynomial
\( x^{33} - 3x - 5 \)
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: | \(3006077365456454934206852878036196952204160738658522246054409521287734177\) \(\medspace = 3^{33}\cdot 79\cdot 263\cdot 465137788789\cdot 55\!\cdots\!83\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(157.15\) | 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: | not computed | ||
Ramified primes: | \(3\), \(79\), \(263\), \(465137788789\), \(55954\!\cdots\!32583\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{16222\!\cdots\!61697}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
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 $S_{33}$ |
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 | $15^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | $32{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $21{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/19.2.0.1}{2} }$ | $15^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/41.7.0.1}{7} }$ | $33$ | $18{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $29{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ |
3.15.15.39 | $x^{15} - 18 x^{14} + 636 x^{13} - 1623 x^{12} - 378 x^{11} + 83475 x^{10} + 197730 x^{9} + 142074 x^{8} + 49653 x^{7} + 194913 x^{6} + 392931 x^{5} + 317520 x^{4} + 107811 x^{3} + 12393 x^{2} + 486 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
3.15.15.2 | $x^{15} + 6 x^{14} + 201 x^{13} - 750 x^{12} - 5202 x^{11} + 13104 x^{10} + 50148 x^{9} + 206469 x^{8} + 249966 x^{7} + 370359 x^{6} + 133083 x^{5} + 169938 x^{4} - 14418 x^{3} + 17010 x^{2} - 2673 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
\(79\) | $\Q_{79}$ | $x + 76$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
79.2.1.2 | $x^{2} + 79$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
79.7.0.1 | $x^{7} + 4 x + 76$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
79.7.0.1 | $x^{7} + 4 x + 76$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
79.16.0.1 | $x^{16} - 4 x + 34$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
\(263\) | $\Q_{263}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(465137788789\) | $\Q_{465137788789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{465137788789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(559\!\cdots\!583\) | $\Q_{55\!\cdots\!83}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |