Normalized defining polynomial
\( x^{29} - 3x - 3 \)
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)
| |
Discriminant: | \(56465633456127198798358813489589629057788939128252989821\) \(\medspace = 3^{28}\cdot 13\cdot 41\cdot 389\cdot 1619\cdot 4483\cdot 202067\cdot 1052207903\cdot 7714356793195489\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(83.65\) | 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: | $3^{28/29}13^{1/2}41^{1/2}389^{1/2}1619^{1/2}4483^{1/2}202067^{1/2}1052207903^{1/2}7714356793195489^{1/2}\approx 4.537987903482011e+21$ | ||
Ramified primes: | \(3\), \(13\), \(41\), \(389\), \(1619\), \(4483\), \(202067\), \(1052207903\), \(7714356793195489\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{24682\!\cdots\!45261}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
Monogenic: | Yes | |
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)
| |
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: | $a+1$, $a^{28}+a^{26}-a^{25}-a^{23}+a^{20}-a^{17}+a^{14}+a^{13}+a^{12}-a^{11}-a^{10}-a^{9}-a^{8}+a^{6}+a^{5}-a^{3}-a^{2}-2$, $a^{24}-a^{22}+a^{21}+a^{19}-a^{17}+a^{16}+a^{15}+2a^{10}+a^{9}-2a^{8}+a^{7}+a^{6}+a^{5}+a^{4}-2a^{3}+a^{2}+3a+1$, $4a^{28}-a^{27}+3a^{26}-a^{25}-2a^{23}-2a^{22}-a^{21}-a^{20}+2a^{19}+a^{18}+4a^{17}+2a^{15}-4a^{14}-a^{13}-5a^{12}+a^{11}-a^{10}+6a^{9}+2a^{8}+6a^{7}-a^{6}-7a^{4}-4a^{3}-7a^{2}+a-10$, $4a^{28}-a^{27}-2a^{25}-4a^{24}-a^{23}-2a^{22}+2a^{21}+2a^{20}+a^{19}+3a^{18}-2a^{17}+a^{16}-4a^{15}-a^{14}+a^{13}+2a^{12}+7a^{11}+a^{10}+3a^{9}-4a^{8}-6a^{7}-8a^{6}-11a^{5}-2a^{4}+a^{3}+12a^{2}+13a-1$, $42a^{28}-42a^{27}+41a^{26}-39a^{25}+40a^{24}-41a^{23}+39a^{22}-35a^{21}+33a^{20}-35a^{19}+34a^{18}-31a^{17}+25a^{16}-25a^{15}+26a^{14}-26a^{13}+19a^{12}-16a^{11}+16a^{10}-18a^{9}+15a^{8}-9a^{7}+8a^{6}-9a^{5}+11a^{4}-3a^{3}+2a^{2}-2a-119$, $a^{28}-3a^{27}+a^{26}-3a^{25}+a^{24}-2a^{23}+a^{22}-a^{21}+a^{19}+2a^{17}+a^{15}+a^{14}+a^{12}-a^{11}-a^{10}-3a^{8}-a^{7}-6a^{6}-3a^{5}-6a^{4}-5a^{3}-7a^{2}-7a-10$, $2a^{28}-3a^{27}-5a^{26}-2a^{25}+5a^{24}+5a^{23}+a^{22}-6a^{21}-5a^{20}-a^{19}+8a^{18}+6a^{17}-a^{16}-10a^{15}-6a^{14}+2a^{13}+13a^{12}+6a^{11}-6a^{10}-14a^{9}-4a^{8}+8a^{7}+15a^{6}+3a^{5}-12a^{4}-17a^{3}-a^{2}+17a+14$, $3a^{28}-a^{27}+4a^{25}-6a^{24}+3a^{23}-a^{22}-a^{21}+5a^{20}-5a^{19}+a^{18}+a^{17}-4a^{16}+7a^{15}-3a^{14}+2a^{12}-8a^{11}+6a^{10}-a^{8}+6a^{7}-9a^{6}+2a^{5}+3a^{4}-2a^{3}+8a^{2}-11a-11$, $2a^{28}+3a^{27}+2a^{26}+a^{25}-a^{24}-a^{23}-4a^{22}-2a^{21}-a^{20}+2a^{18}+4a^{17}+5a^{16}+2a^{15}-a^{14}-4a^{13}-7a^{12}-8a^{11}-a^{10}+4a^{9}+9a^{8}+10a^{7}+7a^{6}-11a^{4}-13a^{3}-10a^{2}-5a-1$, $3a^{28}+5a^{27}+4a^{26}-a^{25}-5a^{24}-6a^{23}-4a^{22}+3a^{21}+7a^{20}+5a^{19}-7a^{17}-10a^{16}-5a^{15}+4a^{14}+12a^{13}+10a^{12}+4a^{11}-7a^{10}-14a^{9}-5a^{8}+7a^{7}+17a^{6}+21a^{5}+8a^{4}-11a^{3}-18a^{2}-13a-5$, $2a^{27}+2a^{26}-a^{24}-3a^{23}-a^{22}+a^{21}+2a^{20}+3a^{19}+a^{18}-2a^{17}-4a^{16}-4a^{15}+a^{14}+4a^{13}+4a^{12}+3a^{11}-2a^{10}-5a^{9}-5a^{8}-a^{7}+6a^{6}+6a^{5}+3a^{4}-2a^{3}-8a^{2}-7a-2$, $a^{28}-5a^{27}-2a^{26}+3a^{25}-5a^{24}+2a^{23}-6a^{21}-a^{20}-3a^{19}-6a^{18}+2a^{17}+3a^{16}+11a^{14}+5a^{13}+10a^{11}-4a^{10}+9a^{8}-a^{7}+10a^{6}+14a^{5}+a^{4}+6a^{3}+a^{2}-18a-11$, $a^{28}-a^{27}+2a^{26}-a^{25}+4a^{24}+a^{23}+4a^{21}-2a^{20}+3a^{19}+a^{17}-a^{16}+5a^{15}+6a^{13}+4a^{12}-a^{11}+3a^{10}-a^{9}+a^{8}+3a^{7}+5a^{6}+10a^{4}+3a^{3}+6a^{2}+7a-5$ (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: | \( 29272458686891230 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 29272458686891230 \cdot 1}{2\cdot\sqrt{56465633456127198798358813489589629057788939128252989821}}\cr\approx \mathstrut & 0.582217501804476 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 8841761993739701954543616000000 |
The 4565 conjugacy class representatives for $S_{29}$ are not computed |
Character table for $S_{29}$ 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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | $16{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.9.0.1}{9} }^{2}{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | $24{,}\,{\href{/padicField/43.5.0.1}{5} }$ | $19{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $29$ | $29$ | $1$ | $28$ | |||
\(13\) | $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | 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 | $x^{12} + x^{8} + 5 x^{7} + 8 x^{6} + 11 x^{5} + 3 x^{4} + x^{3} + x^{2} + 4 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(41\) | 41.2.1.2 | $x^{2} + 123$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
41.13.0.1 | $x^{13} + 13 x + 35$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | |
41.14.0.1 | $x^{14} + 12 x^{7} + 15 x^{6} + 4 x^{5} + 27 x^{4} + 11 x^{3} + 39 x^{2} + 10 x + 6$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(389\) | $\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(1619\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(4483\) | $\Q_{4483}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(202067\) | $\Q_{202067}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{202067}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
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 $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(1052207903\) | $\Q_{1052207903}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1052207903}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(7714356793195489\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |