Normalized defining polynomial
\( x^{33} + 2x - 2 \)
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)
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Discriminant: | \(567077603232955945962167960613156529704687635024986127728640\) \(\medspace = 2^{32}\cdot 5\cdot 53\cdot 3229\cdot 6433737122028757573993\cdot 23983104676408544038373\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(64.67\) | 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: | $2^{32/33}5^{1/2}53^{1/2}3229^{1/2}6433737122028757573993^{1/2}23983104676408544038373^{1/2}\approx 2.2503454492475218e+25$ | ||
Ramified primes: | \(2\), \(5\), \(53\), \(3229\), \(6433737122028757573993\), \(23983104676408544038373\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{13203\!\cdots\!66465}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a-1$, $a^{4}-a^{2}+1$, $a^{22}+a^{11}-a+1$, $a^{25}-a^{17}+a-1$, $a^{31}+a^{26}+a^{23}+a^{20}+a^{18}-a^{16}+a^{15}-a^{11}+a^{9}-a^{8}-a^{5}-a^{3}+a-1$, $a^{31}+a^{30}-a^{26}-a^{25}+a^{23}+a^{22}-a^{18}-a^{17}+a^{15}+a^{14}-a^{10}-a^{9}+a^{5}+a^{4}-1$, $a^{32}-a^{30}+a^{29}+a^{28}-a^{27}+a^{25}-a^{24}-a^{23}+a^{22}-2a^{20}+a^{18}-2a^{17}-a^{16}+2a^{15}-a^{13}+2a^{12}+a^{11}-a^{10}+a^{9}+a^{8}-a^{7}+a^{5}-2a^{4}-2a^{3}+2a^{2}-1$, $2a^{32}+a^{31}+2a^{29}+3a^{28}+a^{27}-a^{26}-2a^{25}+2a^{23}-2a^{21}-a^{20}+a^{19}+3a^{18}+a^{17}-2a^{16}+2a^{14}+2a^{13}-a^{12}-4a^{11}-2a^{10}+3a^{9}+2a^{8}-a^{6}+a^{5}+5a^{4}+2a^{3}-4a^{2}-4a+3$, $a^{32}+a^{30}+a^{29}+a^{28}+a^{26}+a^{25}+a^{21}-a^{20}+a^{17}-a^{16}+a^{14}+2a^{10}-a^{9}+a^{8}+a^{6}-a^{5}+a^{3}-a^{2}+1$, $a^{32}-2a^{31}-a^{29}+a^{27}+3a^{25}-a^{24}+3a^{23}-2a^{22}+2a^{21}-3a^{20}-2a^{18}-a^{17}+a^{16}-a^{15}+4a^{14}-2a^{13}+4a^{12}-3a^{11}+4a^{10}-3a^{9}+a^{8}-2a^{7}-2a^{6}+a^{5}-3a^{4}+6a^{3}-4a^{2}+7a-3$, $a^{32}+a^{31}+2a^{30}+3a^{29}+2a^{28}+2a^{25}+2a^{24}-a^{23}-2a^{22}+2a^{20}-2a^{18}-2a^{17}-a^{16}-a^{13}-3a^{12}-2a^{11}+a^{10}+a^{9}-3a^{8}-3a^{7}+a^{6}+3a^{5}-a^{4}-4a^{3}-a^{2}+2a+5$, $a^{31}-a^{30}+a^{28}+2a^{24}+2a^{21}+a^{20}-a^{18}+2a^{17}+a^{16}-a^{15}+a^{13}+a^{12}-2a^{11}+a^{10}+a^{9}-a^{8}-a^{7}+2a^{5}-3a^{4}+a^{2}-a-1$, $4a^{32}+3a^{31}+3a^{30}+2a^{29}-a^{27}-2a^{26}-3a^{25}-3a^{24}-3a^{23}-3a^{22}-2a^{21}-a^{20}+2a^{18}+2a^{17}+2a^{16}+4a^{15}+4a^{14}+2a^{13}+2a^{12}+2a^{11}-a^{9}-2a^{8}-3a^{7}-3a^{6}-3a^{5}-4a^{4}-2a^{3}-2a^{2}-2a+9$, $5a^{32}+5a^{31}+4a^{30}+4a^{29}+3a^{28}+2a^{27}+2a^{26}+a^{25}-a^{22}-a^{21}+a^{18}+a^{17}+a^{16}+2a^{15}+2a^{14}+2a^{13}+3a^{12}+2a^{11}+2a^{10}+2a^{9}-a^{5}-a^{3}-2a^{2}-a+9$, $3a^{32}+a^{31}+2a^{30}+3a^{29}+a^{27}+3a^{26}+2a^{23}-a^{22}+3a^{20}-a^{19}-a^{18}+3a^{17}-a^{16}-a^{15}+3a^{14}-2a^{13}-a^{12}+4a^{11}-2a^{10}-3a^{9}+3a^{8}-3a^{7}-2a^{6}+3a^{5}-4a^{4}-4a^{3}+4a^{2}-3a+3$, $4a^{32}-5a^{31}+8a^{29}-5a^{28}-5a^{27}+4a^{26}-6a^{25}-5a^{24}+3a^{23}-7a^{22}-a^{21}+11a^{20}-5a^{19}-3a^{18}+13a^{17}-3a^{16}-7a^{15}+7a^{14}-7a^{13}-6a^{12}+9a^{11}-9a^{10}-6a^{9}+18a^{8}-4a^{7}-9a^{6}+16a^{5}-2a^{4}-11a^{3}+11a^{2}-10a-3$ (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: | \( 96056886306925100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{16}\cdot 96056886306925100 \cdot 1}{2\cdot\sqrt{567077603232955945962167960613156529704687635024986127728640}}\cr\approx \mathstrut & 0.752636283158186 \end{aligned}\] (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 $S_{33}$ is not computed |
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 | $26{,}\,{\href{/padicField/3.7.0.1}{7} }$ | R | $26{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | $27{,}\,{\href{/padicField/11.6.0.1}{6} }$ | $15{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | $22{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | R | $33$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $33$ | $33$ | $1$ | $32$ | |||
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.7.0.1 | $x^{7} + 3 x + 3$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
5.22.0.1 | $x^{22} + x^{12} + 3 x^{11} + 4 x^{9} + 3 x^{8} + 2 x^{6} + 2 x^{5} + 4 x^{3} + 3 x^{2} + 3 x + 2$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
53.2.1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.8.0.1 | $x^{8} + 8 x^{4} + 29 x^{3} + 18 x^{2} + x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
53.22.0.1 | $x^{22} - x + 18$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | |
\(3229\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $31$ | $1$ | $31$ | $0$ | $C_{31}$ | $[\ ]^{31}$ | ||
\(643\!\cdots\!993\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(239\!\cdots\!373\) | $\Q_{23\!\cdots\!73}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |