Normalized defining polynomial
\( x^{33} + 5x - 1 \)
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: | \(170141183460469231731816413755971866755839616029934664535539337183380513\) \(\medspace = 59\cdot 107\cdot 113\cdot 167\cdot 9170594383\cdot 510233763264227\cdot 1295992890923813\cdot 235510051675084013415407\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(144.05\) | 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: | $59^{1/2}107^{1/2}113^{1/2}167^{1/2}9170594383^{1/2}510233763264227^{1/2}1295992890923813^{1/2}235510051675084013415407^{1/2}\approx 4.124817371235595e+35$ | ||
Ramified primes: | \(59\), \(107\), \(113\), \(167\), \(9170594383\), \(510233763264227\), \(1295992890923813\), \(235510051675084013415407\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{17014\!\cdots\!80513}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$
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 | $15^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $26{,}\,{\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $25{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | $30{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $29{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.14.0.1}{14} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | R |
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 |
---|---|---|---|---|---|---|---|
\(59\) | $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
59.2.1.2 | $x^{2} + 59$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
59.8.0.1 | $x^{8} + 16 x^{4} + 32 x^{3} + 2 x^{2} + 50 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
59.10.0.1 | $x^{10} + x^{6} + 28 x^{5} + 25 x^{4} + 4 x^{3} + 39 x^{2} + 15 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
59.12.0.1 | $x^{12} + 39 x^{7} + 25 x^{6} + 51 x^{5} + 21 x^{4} + 38 x^{3} + 8 x^{2} + x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(107\) | 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
107.6.0.1 | $x^{6} + x^{4} + 52 x^{3} + 22 x^{2} + 79 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
107.8.0.1 | $x^{8} + 2 x^{4} + 105 x^{3} + 24 x^{2} + 95 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
107.15.0.1 | $x^{15} - x + 39$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(113\) | $\Q_{113}$ | $x + 110$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
113.2.1.1 | $x^{2} + 113$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
113.2.0.1 | $x^{2} + 101 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
113.5.0.1 | $x^{5} + 7 x + 110$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
113.5.0.1 | $x^{5} + 7 x + 110$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
113.18.0.1 | $x^{18} - x + 6$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(167\) | $\Q_{167}$ | $x + 162$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
167.2.1.1 | $x^{2} + 835$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(9170594383\) | $\Q_{9170594383}$ | $x$ | $1$ | $1$ | $0$ | 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 $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(510233763264227\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(1295992890923813\) | $\Q_{1295992890923813}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(235\!\cdots\!407\) | $\Q_{23\!\cdots\!07}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |