Normalized defining polynomial
\( x^{33} + 3x - 3 \)
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)
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Discriminant: | \(247368086986134255644672374444546725127599586741055504922971230881\) \(\medspace = 3^{33}\cdot 251\cdot 6097463\cdot 19738007\cdot 14\!\cdots\!17\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(95.86\) | 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: | \(3\), \(251\), \(6097463\), \(19738007\), \(14730\!\cdots\!90817\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{13349\!\cdots\!09441}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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)
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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)
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Fundamental units: | $a-1$, $a^{32}+a^{31}+a^{28}+a^{23}-a^{21}+a^{19}+a^{18}-a^{17}+2a^{15}-a^{14}-a^{13}+a^{11}+a^{10}-a^{9}+a^{6}+a^{5}-2a^{4}+a^{3}+3a^{2}-2a+1$, $5a^{32}+6a^{31}+5a^{30}+4a^{29}+4a^{28}+5a^{27}+5a^{26}+4a^{25}+4a^{24}+5a^{23}+5a^{22}+4a^{21}+3a^{20}+4a^{19}+4a^{18}+2a^{17}+a^{16}+2a^{15}+4a^{14}+a^{13}+3a^{10}+2a^{9}+a^{8}+a^{7}+3a^{6}+2a^{5}+a^{4}+5a^{2}+3a+16$, $2a^{32}+4a^{30}+3a^{29}+3a^{27}+4a^{26}+a^{25}+6a^{23}+2a^{20}+4a^{19}-5a^{18}+2a^{17}+2a^{16}-3a^{15}-4a^{14}+4a^{13}-3a^{12}-6a^{11}+2a^{10}+a^{9}-8a^{8}+4a^{6}-5a^{5}-6a^{4}+9a^{3}-4a^{2}-6a+10$, $15a^{32}+16a^{31}+15a^{30}+14a^{29}+11a^{28}+8a^{27}+7a^{26}+5a^{25}+5a^{24}+a^{23}-a^{22}-5a^{21}-5a^{20}-6a^{19}-6a^{18}-8a^{17}-10a^{16}-10a^{15}-11a^{14}-8a^{13}-11a^{12}-9a^{11}-12a^{10}-8a^{9}-7a^{8}-5a^{7}-5a^{6}-7a^{5}-4a^{4}-3a^{3}+4a^{2}+a+49$, $5a^{32}+2a^{31}-3a^{30}-7a^{29}-8a^{28}-11a^{27}-12a^{26}-9a^{25}-8a^{24}-8a^{23}-2a^{22}+2a^{21}+3a^{20}+8a^{19}+12a^{18}+10a^{17}+10a^{16}+10a^{15}+6a^{14}+3a^{13}+a^{12}-4a^{11}-7a^{10}-10a^{9}-12a^{8}-11a^{7}-9a^{6}-8a^{5}-3a^{4}-a^{3}+a^{2}+7a+23$, $a^{32}+a^{31}-a^{30}+a^{29}+a^{28}-2a^{27}+2a^{26}-5a^{24}+3a^{23}+2a^{22}-3a^{21}+2a^{20}+3a^{19}-a^{18}+2a^{17}+2a^{16}-6a^{15}+4a^{14}+3a^{13}-8a^{12}+4a^{11}+5a^{10}-4a^{9}-a^{8}+2a^{7}-7a^{6}+2a^{5}+6a^{4}-11a^{3}+8a^{2}+10a-8$, $23a^{32}+25a^{31}+22a^{30}+21a^{29}+21a^{28}+18a^{27}+17a^{26}+18a^{25}+14a^{24}+18a^{23}+12a^{22}+16a^{21}+11a^{20}+12a^{19}+11a^{18}+12a^{17}+9a^{16}+12a^{15}+8a^{14}+9a^{13}+7a^{12}+8a^{11}+8a^{10}+7a^{9}+7a^{8}+8a^{7}+2a^{6}+8a^{5}+4a^{4}+4a^{3}+10a^{2}-2a+82$, $a^{32}+a^{31}-a^{28}-3a^{27}+a^{26}-3a^{25}-a^{24}-a^{23}-2a^{22}+a^{21}-3a^{20}-a^{19}+2a^{18}-5a^{17}-3a^{15}-a^{14}-3a^{13}-5a^{12}+a^{11}-4a^{10}-4a^{9}+a^{8}-3a^{7}+2a^{6}-a^{5}-a^{4}+8a^{3}-3a^{2}+2a+10$, $8a^{32}+8a^{31}-a^{30}-7a^{29}-4a^{28}+4a^{27}+6a^{26}+3a^{25}-a^{24}-3a^{23}-6a^{22}-6a^{21}+9a^{19}+11a^{18}+4a^{17}-5a^{16}-9a^{15}-6a^{14}+a^{13}+9a^{12}+8a^{11}-2a^{10}-13a^{9}-8a^{8}+4a^{7}+11a^{6}+4a^{5}+3a^{4}+5a^{3}+2a^{2}-17a+4$, $2a^{32}+a^{28}+4a^{27}+3a^{26}-2a^{25}-a^{24}+6a^{23}+5a^{22}-a^{21}-2a^{18}+5a^{17}+6a^{16}-8a^{15}-8a^{14}+7a^{13}+6a^{12}-5a^{11}-3a^{10}+3a^{9}-a^{8}+a^{7}+8a^{6}-2a^{5}-7a^{4}+8a^{3}+7a^{2}-8a+4$, $17a^{32}+5a^{31}-8a^{30}-18a^{29}-21a^{28}-20a^{27}-14a^{26}-2a^{25}+13a^{24}+24a^{23}+26a^{22}+23a^{21}+16a^{20}+4a^{19}-12a^{18}-25a^{17}-27a^{16}-23a^{15}-13a^{14}-3a^{13}+17a^{12}+28a^{11}+34a^{10}+22a^{9}+16a^{8}-13a^{6}-33a^{5}-35a^{4}-26a^{3}-11a^{2}+2a+67$, $a^{32}+3a^{30}-2a^{29}+8a^{26}-a^{25}-2a^{23}-a^{22}+3a^{21}+a^{20}-a^{19}-5a^{18}+10a^{17}-a^{16}+3a^{15}-6a^{14}-a^{13}+7a^{11}-8a^{9}+7a^{8}-2a^{7}+9a^{6}-9a^{5}-2a^{4}-7a^{3}+15a^{2}+a-2$, $a^{32}-11a^{31}-13a^{30}-3a^{29}+8a^{28}+12a^{27}+6a^{26}-5a^{25}-11a^{24}-8a^{23}+6a^{21}+9a^{20}+6a^{19}-2a^{18}-7a^{17}-10a^{16}-7a^{15}+5a^{14}+13a^{13}+13a^{12}+3a^{11}-14a^{10}-21a^{9}-10a^{8}+12a^{7}+26a^{6}+19a^{5}-7a^{4}-33a^{3}-27a^{2}+3a+32$, $9a^{32}+14a^{31}+16a^{30}+17a^{29}+16a^{28}+12a^{27}+7a^{26}+2a^{25}-a^{24}-8a^{23}-14a^{22}-15a^{21}-16a^{20}-15a^{19}-13a^{18}-9a^{17}-6a^{16}+2a^{15}+9a^{14}+11a^{13}+12a^{12}+16a^{11}+18a^{10}+13a^{9}+10a^{8}+5a^{7}-3a^{6}-3a^{5}-9a^{4}-14a^{3}-21a^{2}-15a+17$, $5a^{32}+5a^{31}+5a^{30}+9a^{29}+a^{28}-13a^{27}-9a^{26}-6a^{25}-7a^{24}+3a^{23}+20a^{22}+11a^{21}+5a^{20}+5a^{19}-5a^{18}-23a^{17}-7a^{16}+2a^{15}+a^{14}+9a^{13}+27a^{12}+3a^{11}-9a^{10}-3a^{9}-8a^{8}-24a^{7}+6a^{6}+20a^{5}+3a^{4}+5a^{3}+19a^{2}-16a-17$ (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)]
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Regulator: | \( 64586856090532960000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 64586856090532960000 \cdot 1}{2\cdot\sqrt{247368086986134255644672374444546725127599586741055504922971230881}}\cr\approx \mathstrut & 0.766214212235084 \end{aligned}\] (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 | $15^{2}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | R | $24{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $32{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | $31{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $31{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $30{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $33$ | $33$ | $1$ | $33$ | |||
\(251\) | $\Q_{251}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(6097463\) | $\Q_{6097463}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(19738007\) | $\Q_{19738007}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(147\!\cdots\!817\) | $\Q_{14\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{14\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{14\!\cdots\!17}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |