Normalized defining polynomial
\( x^{33} + 3x - 2 \)
Invariants
Degree: | $33$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8125130643442315862419837587656427951186617662466440556546359296\) \(\medspace = 2^{32}\cdot 3^{34}\cdot 5273\cdot 13873073\cdot 79403245997\cdot 19528960698348533\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(86.43\) | 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: | \(2\), \(3\), \(5273\), \(13873073\), \(79403245997\), \(19528960698348533\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{11343\!\cdots\!73529}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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: | $a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a+1$, $a^{23}-2a^{12}+a^{11}+a^{2}+a-1$, $13a^{32}+8a^{31}+4a^{30}+a^{29}-a^{28}-a^{27}+a^{26}+3a^{25}+4a^{24}+4a^{23}+2a^{22}-a^{21}-3a^{20}-4a^{19}-3a^{18}+3a^{16}+4a^{15}+3a^{14}+a^{13}-2a^{12}-5a^{11}-5a^{10}-2a^{9}+a^{8}+4a^{7}+6a^{6}+4a^{5}+a^{4}-2a^{3}-5a^{2}-5a+37$, $a^{31}-a^{30}+2a^{29}-2a^{28}+3a^{27}-3a^{26}+4a^{25}-4a^{24}+5a^{23}-5a^{22}+5a^{21}-4a^{20}+3a^{19}-2a^{18}+a^{17}-a^{15}+2a^{14}-3a^{13}+4a^{12}-5a^{11}+5a^{10}-5a^{9}+4a^{8}-4a^{7}+3a^{6}-3a^{5}+2a^{4}-2a^{3}+a^{2}-a+1$, $4a^{32}-5a^{31}+3a^{30}-3a^{29}-a^{28}+3a^{27}-4a^{26}+5a^{25}-2a^{24}+a^{23}+a^{22}-a^{21}-a^{20}+3a^{19}-5a^{18}+5a^{17}-3a^{16}-2a^{15}+6a^{14}-11a^{13}+12a^{12}-10a^{11}+7a^{10}+a^{9}-3a^{8}+6a^{7}-4a^{6}-2a^{5}+2a^{4}-5a^{3}-2a^{2}+9a-3$, $7a^{32}+4a^{31}+3a^{30}+4a^{29}+3a^{28}+2a^{27}+a^{26}-2a^{25}-a^{23}+2a^{22}+a^{21}+a^{20}-2a^{19}-2a^{18}-3a^{17}+a^{16}+5a^{14}-a^{13}+a^{12}-3a^{11}-2a^{10}+a^{9}+3a^{8}+2a^{7}+5a^{6}-5a^{5}-4a^{3}-2a^{2}+5a+21$, $2a^{32}+a^{31}+a^{30}+a^{29}-a^{27}+a^{26}+a^{25}-a^{23}+2a^{21}-a^{19}+2a^{17}-3a^{15}-a^{14}+a^{13}+a^{12}-3a^{11}-a^{10}+4a^{9}+3a^{8}-3a^{6}+a^{5}+a^{4}-3a^{3}-4a^{2}-a+11$, $3a^{32}+a^{31}-2a^{29}-3a^{28}-3a^{27}-4a^{26}-5a^{25}-5a^{24}-4a^{23}-3a^{22}-a^{21}+a^{20}+a^{19}+2a^{18}+5a^{17}+4a^{16}+5a^{15}+6a^{14}+2a^{13}+2a^{12}+4a^{11}-a^{10}-a^{8}-7a^{7}-5a^{6}-3a^{5}-7a^{4}-3a^{3}-3a^{2}-6a+9$, $6a^{32}-2a^{31}-9a^{30}-12a^{29}-9a^{28}-2a^{27}+6a^{26}+12a^{25}+12a^{24}+7a^{23}-2a^{22}-11a^{21}-15a^{20}-11a^{19}-3a^{18}+9a^{17}+16a^{16}+16a^{15}+7a^{14}-5a^{13}-17a^{12}-18a^{11}-12a^{10}+a^{9}+13a^{8}+21a^{7}+16a^{6}+6a^{5}-10a^{4}-20a^{3}-21a^{2}-11a+21$, $2a^{32}+6a^{31}+9a^{30}+9a^{29}+4a^{28}-2a^{26}-5a^{25}-7a^{24}-4a^{23}+a^{22}+5a^{21}+9a^{20}+11a^{19}+12a^{18}+9a^{17}+2a^{16}-6a^{15}-12a^{14}-18a^{13}-20a^{12}-16a^{11}-11a^{10}-2a^{9}+9a^{8}+16a^{7}+17a^{6}+20a^{5}+18a^{4}+9a^{3}-a^{2}-8a-5$, $5a^{32}-5a^{31}+a^{30}+5a^{29}-6a^{28}+4a^{27}+a^{26}-7a^{25}+7a^{24}-2a^{23}-4a^{22}+9a^{21}-7a^{20}-a^{19}+8a^{18}-11a^{17}+5a^{16}+6a^{15}-12a^{14}+11a^{13}-a^{12}-12a^{11}+15a^{10}-8a^{9}-4a^{8}+17a^{7}-17a^{6}+4a^{5}+11a^{4}-22a^{3}+17a^{2}+3a-5$, $a^{32}+2a^{30}-2a^{29}-a^{28}+3a^{27}-2a^{25}-3a^{24}+3a^{23}+4a^{22}-2a^{21}-4a^{20}+4a^{18}+a^{17}-4a^{16}+2a^{14}+2a^{13}-6a^{12}+2a^{11}+4a^{10}+a^{9}-3a^{8}-10a^{7}+13a^{6}-a^{5}+a^{4}-8a^{3}-a^{2}+14a-7$, $13a^{32}+8a^{31}+2a^{30}+5a^{29}+3a^{28}-2a^{27}+3a^{26}+2a^{25}-3a^{24}+3a^{23}+2a^{22}-4a^{21}+3a^{20}+2a^{19}-5a^{18}+3a^{17}+3a^{16}-5a^{15}+2a^{14}+3a^{13}-6a^{12}+2a^{11}+4a^{10}-8a^{9}+3a^{8}+6a^{7}-9a^{6}+3a^{5}+6a^{4}-9a^{3}+3a^{2}+7a+29$, $19a^{32}+14a^{31}+10a^{30}+10a^{29}+5a^{28}+a^{27}+a^{26}-3a^{25}-7a^{24}-5a^{23}-8a^{22}-11a^{21}-5a^{20}-6a^{19}-6a^{18}+a^{16}+a^{15}+8a^{14}+9a^{13}+5a^{12}+12a^{11}+9a^{10}+3a^{9}+8a^{8}+4a^{7}-4a^{6}-a^{5}-4a^{4}-13a^{3}-5a^{2}-6a+41$, $6a^{32}+4a^{31}+a^{30}-3a^{28}-a^{27}-2a^{26}+a^{24}+a^{23}+2a^{22}+a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+3a^{16}+2a^{15}+3a^{14}+2a^{13}-a^{12}-2a^{11}-4a^{10}-2a^{9}-4a^{8}+2a^{7}-3a^{6}+3a^{5}-3a+17$, $6a^{32}+11a^{31}+9a^{30}-10a^{28}-10a^{27}-4a^{26}+5a^{25}+14a^{24}+8a^{23}-a^{22}-12a^{21}-11a^{20}-7a^{19}+10a^{18}+15a^{17}+8a^{16}-16a^{14}-14a^{13}-6a^{12}+14a^{11}+14a^{10}+14a^{9}-4a^{8}-21a^{7}-15a^{6}-5a^{5}+14a^{4}+21a^{3}+17a^{2}-12a-1$ (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: | \( 22262502678802630000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 22262502678802630000 \cdot 1}{2\cdot\sqrt{8125130643442315862419837587656427951186617662466440556546359296}}\cr\approx \mathstrut & 1.45725885207812 \end{aligned}\] (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 | R | R | $24{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $28{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $31{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
Deg $32$ | $32$ | $1$ | $32$ | ||||
\(3\) | 3.3.4.4 | $x^{3} + 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $S_3$ | $[2]^{2}$ |
3.15.15.11 | $x^{15} - 33 x^{14} + 378 x^{13} - 966 x^{12} + 7461 x^{11} + 47682 x^{10} + 1575 x^{9} + 29160 x^{8} + 30780 x^{7} + 9423 x^{6} + 24138 x^{5} + 26244 x^{4} + 6966 x^{3} - 486 x^{2} + 243$ | $3$ | $5$ | $15$ | 15T33 | $[3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
3.15.15.25 | $x^{15} - 9 x^{14} + 228 x^{13} + 2040 x^{12} + 2547 x^{11} - 22914 x^{10} - 63036 x^{9} + 193752 x^{8} - 76950 x^{7} - 3051 x^{6} + 77517 x^{5} - 19116 x^{4} - 7857 x^{3} + 7533 x^{2} - 1944 x + 243$ | $3$ | $5$ | $15$ | 15T44 | $[3/2, 3/2, 3/2, 3/2, 3/2]_{2}^{5}$ | |
\(5273\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(13873073\) | $\Q_{13873073}$ | $x$ | $1$ | $1$ | $0$ | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(79403245997\) | $\Q_{79403245997}$ | $x$ | $1$ | $1$ | $0$ | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(19528960698348533\) | $\Q_{19528960698348533}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |