Normalized defining polynomial
\( x^{33} + x - 1 \)
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: | \(130571541725091930757819714767380818558993115923489\) \(\medspace = 17\cdot 43\cdot 1579\cdot 2609\cdot 13153817\cdot 216865282965193\cdot 15199633537674804809\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.01\) | 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: | $17^{1/2}43^{1/2}1579^{1/2}2609^{1/2}13153817^{1/2}216865282965193^{1/2}15199633537674804809^{1/2}\approx 1.1426790526000376e+25$ | ||
Ramified primes: | \(17\), \(43\), \(1579\), \(2609\), \(13153817\), \(216865282965193\), \(15199633537674804809\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{13057\!\cdots\!23489}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a$, $a^{32}+a^{31}-a^{9}+1$, $a^{25}+a^{24}+a^{23}$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}-a^{11}+1$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{26}+a^{25}+1$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{12}-a^{4}+1$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{23}+a^{21}+1$, $a^{30}-a^{25}+a^{20}-a^{15}+a^{10}+a^{7}+a^{3}-a^{2}$, $a^{32}+a^{31}+a^{30}+a^{29}+a^{28}+a^{26}+a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+1$, $a^{32}+a^{31}+a^{25}+a^{24}+a^{18}-a^{13}+a^{8}-a^{3}+1$, $a^{31}+a^{30}+a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{23}+a^{21}-a^{14}-a^{9}-a^{4}$, $a^{32}+a^{31}+a^{30}+a^{25}-a^{22}-a^{21}+a^{18}-a^{16}-a^{15}-a^{14}+a^{12}-a^{9}-a^{8}+a^{5}-a+1$, $a^{30}-a^{27}-a^{26}-a^{25}-a^{20}+a^{17}+a^{16}+a^{15}+a^{10}+a^{8}-a^{7}-a^{5}+a^{2}$, $a^{31}+2a^{30}+2a^{29}+2a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}+a^{23}-a^{20}-a^{19}-a^{18}-a^{17}-a^{16}-a^{15}+a^{12}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}-a^{2}-1$, $a^{32}+a^{31}+a^{25}+a^{19}+a^{13}+1$, $a^{32}+a^{31}+2a^{30}-a^{28}-2a^{25}-2a^{24}-a^{23}-2a^{22}-3a^{21}-2a^{20}-a^{19}-2a^{18}-2a^{17}-a^{16}-a^{14}-a^{13}+a^{12}+a^{11}-a^{10}+2a^{8}-a^{6}+a^{5}+a^{4}-a^{2}+a+2$ (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: | \( 794597544869.8735 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 794597544869.8735 \cdot 1}{2\cdot\sqrt{130571541725091930757819714767380818558993115923489}}\cr\approx \mathstrut & 0.410299366611747 \end{aligned}\] (assuming GRH)
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 | $15^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | $27{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $33$ | $30{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $30{,}\,{\href{/padicField/13.3.0.1}{3} }$ | R | $22{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.14.0.1}{14} }$ | $24{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.5.0.1 | $x^{5} + x + 14$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
17.7.0.1 | $x^{7} + 12 x + 14$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
17.19.0.1 | $x^{19} + 11 x + 14$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | |
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.2.1.2 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.3.0.1 | $x^{3} + x + 40$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
43.5.0.1 | $x^{5} + 8 x + 40$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
43.10.0.1 | $x^{10} + 3 x^{6} + 26 x^{5} + 36 x^{4} + 5 x^{3} + 27 x^{2} + 24 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
43.10.0.1 | $x^{10} + 3 x^{6} + 26 x^{5} + 36 x^{4} + 5 x^{3} + 27 x^{2} + 24 x + 3$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(1579\) | $\Q_{1579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(2609\) | $\Q_{2609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(13153817\) | $\Q_{13153817}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13153817}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(216865282965193\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(15199633537674804809\) | $\Q_{15199633537674804809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ |