Normalized defining polynomial
\( x^{29} - x - 4 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(689258003388715552294905021323738295429713895620608\) \(\medspace = 2^{30}\cdot 13\cdot 3373\cdot 84163\cdot 17\!\cdots\!41\) | 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: | \(56.63\) | 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: | \(2\), \(13\), \(3373\), \(84163\), \(17394\!\cdots\!29441\) | 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: | $\Q(\sqrt{64192\!\cdots\!76667}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{14}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $14$ | 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: | $\frac{1}{2}a^{15}-\frac{1}{2}a-1$, $\frac{1}{2}a^{22}+\frac{1}{2}a^{15}+\frac{1}{2}a^{8}+\frac{1}{2}a+1$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{25}+\frac{1}{2}a^{23}-\frac{1}{2}a^{21}+\frac{1}{2}a^{19}-\frac{1}{2}a^{17}+\frac{1}{2}a^{15}-\frac{1}{2}a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}+\frac{1}{2}a^{7}-\frac{1}{2}a^{5}+\frac{1}{2}a^{3}-\frac{1}{2}a-1$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{1}{2}a-1$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{20}-\frac{1}{2}a^{18}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-a^{14}+\frac{1}{2}a^{13}-\frac{3}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-a^{7}+\frac{1}{2}a^{6}+a^{5}-\frac{1}{2}a^{4}+\frac{3}{2}a^{2}+\frac{1}{2}a-1$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+a^{24}-a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{3}{2}a^{12}-\frac{1}{2}a^{11}+a^{10}+a^{9}-a^{8}-\frac{3}{2}a^{7}+\frac{1}{2}a^{6}+\frac{5}{2}a^{5}+\frac{3}{2}a^{4}-a^{3}-\frac{3}{2}a^{2}+\frac{3}{2}a+3$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{3}{2}a^{6}+\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-1$, $\frac{1}{2}a^{26}+\frac{1}{2}a^{25}+\frac{1}{2}a^{24}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{17}-a^{16}-\frac{1}{2}a^{15}-a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}+\frac{1}{2}a^{8}+\frac{1}{2}a^{7}+a^{4}+\frac{3}{2}a^{3}+3a^{2}+\frac{3}{2}a+1$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+a^{23}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{16}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}+a^{11}-\frac{1}{2}a^{10}-a^{9}+a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+a^{5}-a^{4}+a^{3}+\frac{3}{2}a^{2}-2a+1$, $\frac{1}{2}a^{28}-a^{27}+\frac{1}{2}a^{26}-\frac{1}{2}a^{24}+a^{23}-a^{22}+\frac{1}{2}a^{20}-\frac{3}{2}a^{19}+a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+a^{15}-\frac{3}{2}a^{14}+a^{13}+\frac{1}{2}a^{12}-a^{11}+\frac{5}{2}a^{10}+2a^{7}-\frac{3}{2}a^{6}+\frac{3}{2}a^{5}-\frac{3}{2}a^{3}+\frac{5}{2}a^{2}-3a-1$, $a^{28}-\frac{1}{2}a^{27}+a^{25}-\frac{3}{2}a^{24}+a^{23}-a^{22}+\frac{3}{2}a^{21}-2a^{20}+\frac{3}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}+a^{14}-\frac{5}{2}a^{13}+3a^{12}-3a^{11}+\frac{3}{2}a^{10}+a^{8}-\frac{3}{2}a^{7}+\frac{3}{2}a^{5}-\frac{7}{2}a^{4}+\frac{9}{2}a^{3}-\frac{7}{2}a^{2}+a-3$, $2a^{28}-\frac{3}{2}a^{27}+\frac{3}{2}a^{25}-a^{24}-a^{23}+\frac{5}{2}a^{22}-\frac{3}{2}a^{21}-\frac{1}{2}a^{20}+\frac{5}{2}a^{19}-3a^{18}+2a^{17}+a^{16}-\frac{5}{2}a^{15}+a^{14}+\frac{5}{2}a^{13}-4a^{12}+\frac{3}{2}a^{11}+2a^{10}-4a^{9}+\frac{9}{2}a^{8}-\frac{3}{2}a^{7}-\frac{5}{2}a^{6}+\frac{7}{2}a^{5}-5a^{3}+5a^{2}-\frac{1}{2}a-5$, $\frac{1}{2}a^{28}+a^{27}+\frac{1}{2}a^{26}+\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{3}{2}a^{23}-\frac{5}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{7}{2}a^{19}+\frac{5}{2}a^{18}-3a^{17}+\frac{1}{2}a^{16}+2a^{15}-\frac{3}{2}a^{14}+3a^{13}+\frac{1}{2}a^{12}+\frac{3}{2}a^{11}-\frac{1}{2}a^{10}+\frac{3}{2}a^{9}-\frac{3}{2}a^{8}-\frac{7}{2}a^{7}+\frac{7}{2}a^{6}-\frac{15}{2}a^{5}+\frac{3}{2}a^{4}+a^{3}-\frac{11}{2}a^{2}+7a-1$, $\frac{3}{2}a^{28}-\frac{3}{2}a^{27}+a^{26}-\frac{3}{2}a^{25}+\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{22}+a^{20}+\frac{3}{2}a^{19}-\frac{1}{2}a^{18}-2a^{16}+a^{15}-\frac{5}{2}a^{14}+\frac{5}{2}a^{13}-2a^{12}+\frac{7}{2}a^{11}-\frac{5}{2}a^{10}+5a^{9}-\frac{7}{2}a^{8}+3a^{7}-6a^{6}+\frac{7}{2}a^{5}-\frac{11}{2}a^{4}+5a^{3}-3a^{2}+7a-5$ (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: | \( 216030725427151.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 216030725427151.9 \cdot 1}{2\cdot\sqrt{689258003388715552294905021323738295429713895620608}}\cr\approx \mathstrut & 1.22982345668480 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8841761993739701954543616000000 |
The 4565 conjugacy class representatives for $S_{29}$ |
Character table for $S_{29}$ |
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 | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | $29$ | $21{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | $15{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $20{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $29$ | $15{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/41.9.0.1}{9} }$ | $27{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $23{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | $20{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $[\ ]$ |
2.4.6.1 | $x^{4} + 2 x^{3} + 31 x^{2} + 30 x + 183$ | $2$ | $2$ | $6$ | $C_2^2$ | $[3]^{2}$ | |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
2.6.6.4 | $x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $[2, 2, 2]^{3}$ | |
2.12.12.22 | $x^{12} + 10 x^{11} - 16 x^{10} - 316 x^{9} - 364 x^{8} + 1040 x^{7} + 7648 x^{6} + 17632 x^{5} + 25008 x^{4} + 17056 x^{3} + 6016 x^{2} + 1344 x + 4544$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
13.5.0.1 | $x^{5} + 4 x + 11$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
13.6.0.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
13.12.0.1 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(3373\) | $\Q_{3373}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(84163\) | $\Q_{84163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(173\!\cdots\!441\) | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |