Normalized defining polynomial
\( x^{29} - 4x - 1 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-9553546594678273369526111644888637966824486353442589488115\) \(\medspace = -\,5\cdot 71\cdot 1070754779627347\cdot 25\!\cdots\!79\) | 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: | \(99.84\) | 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))
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Galois root discriminant: | $5^{1/2}71^{1/2}1070754779627347^{1/2}25133111119629637924203464052947473855979^{1/2}\approx 9.774224570101852e+28$ | ||
Ramified primes: | \(5\), \(71\), \(1070754779627347\), \(25133\!\cdots\!55979\) | 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: | $\Q(\sqrt{-95535\!\cdots\!88115}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | 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: | $a$, $a^{14}+2$, $a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a+2$, $a^{28}+a^{25}-a^{24}+2a^{23}-2a^{22}+3a^{21}-3a^{20}+4a^{19}-3a^{18}+3a^{17}-a^{16}+a^{15}+2a^{12}-2a^{11}+5a^{10}-5a^{9}+7a^{8}-7a^{7}+8a^{6}-6a^{5}+8a^{4}-6a^{3}+7a^{2}-4a-1$, $a^{27}+a^{24}+a^{18}-a^{17}-2a^{16}+a^{14}-a^{13}-a^{12}-2a^{11}-a^{10}+a^{9}+a^{8}-2a^{7}-a^{6}+a^{4}+2a^{3}+2a^{2}-2a+1$, $a^{28}+6a^{27}-10a^{26}+4a^{25}+5a^{24}-15a^{23}+14a^{22}-5a^{21}-9a^{20}+15a^{19}-14a^{18}-a^{17}+10a^{16}-12a^{15}-a^{14}+11a^{13}-19a^{12}+6a^{11}+12a^{10}-28a^{9}+24a^{8}-8a^{7}-21a^{6}+29a^{5}-21a^{4}-5a^{3}+17a^{2}-24a-8$, $a^{28}-a^{25}+a^{24}-a^{22}+2a^{20}-2a^{19}+a^{18}+a^{17}-a^{16}-a^{15}+2a^{13}-2a^{12}-a^{11}+4a^{10}-a^{9}-4a^{8}+3a^{7}+a^{5}-5a^{4}+6a^{3}+a^{2}-6a-1$, $2a^{28}+2a^{27}-6a^{26}+9a^{25}-7a^{24}+7a^{23}-9a^{22}+15a^{21}-17a^{20}+14a^{19}-6a^{18}+3a^{17}-3a^{16}+9a^{15}-8a^{14}+5a^{13}-a^{12}+6a^{11}-16a^{10}+27a^{9}-24a^{8}+19a^{7}-11a^{6}+16a^{5}-20a^{4}+20a^{3}-6a^{2}-4a-1$, $3a^{28}+a^{27}-a^{26}-a^{25}-4a^{24}-6a^{23}-4a^{22}-4a^{21}-3a^{20}+3a^{18}+6a^{17}+6a^{16}+6a^{15}+8a^{14}+3a^{13}-2a^{12}-2a^{11}-7a^{10}-12a^{9}-9a^{8}-8a^{7}-6a^{6}+a^{5}+7a^{4}+12a^{3}+15a^{2}+15a+4$, $a^{28}-2a^{27}-2a^{26}+3a^{25}+3a^{23}-8a^{22}+4a^{21}-2a^{20}+7a^{19}-6a^{18}-3a^{16}+8a^{15}-5a^{13}-a^{12}-a^{11}+11a^{10}-7a^{9}-a^{8}-10a^{7}+15a^{6}-4a^{5}+5a^{4}-15a^{3}+6a^{2}+7a$, $4a^{27}-a^{26}-a^{25}-3a^{23}-2a^{22}+4a^{21}+2a^{20}-2a^{19}+2a^{18}-3a^{17}-4a^{16}+3a^{15}+4a^{14}-2a^{13}+3a^{12}-7a^{10}-a^{9}+3a^{8}-4a^{7}+3a^{6}+7a^{5}-5a^{4}-3a^{3}+a^{2}-5a-2$, $a^{28}+8a^{27}-2a^{26}-4a^{25}+7a^{24}+4a^{23}-5a^{22}+2a^{21}+8a^{20}-a^{19}-4a^{18}+7a^{17}+8a^{16}-6a^{15}-2a^{14}+15a^{13}+3a^{12}-12a^{11}+10a^{10}+16a^{9}-11a^{8}-2a^{7}+23a^{6}-a^{5}-15a^{4}+20a^{3}+19a^{2}-19a-4$, $6a^{28}-3a^{27}-2a^{26}+6a^{25}-6a^{24}+6a^{23}-3a^{22}+5a^{21}+2a^{19}+7a^{18}-a^{17}+12a^{16}-5a^{15}+16a^{14}+2a^{13}+2a^{12}+18a^{11}-5a^{10}+15a^{9}-a^{8}+12a^{7}+2a^{6}+2a^{5}+9a^{4}-7a^{3}+15a^{2}-17a-6$, $3a^{28}+a^{27}-a^{26}+6a^{25}-16a^{24}+15a^{23}-15a^{22}+23a^{21}-28a^{20}+20a^{19}-21a^{18}+26a^{17}-19a^{16}+7a^{15}-9a^{14}+5a^{13}+11a^{12}-18a^{11}+15a^{10}-28a^{9}+40a^{8}-36a^{7}+35a^{6}-49a^{5}+42a^{4}-29a^{3}+31a^{2}-31a-11$, $18a^{28}-a^{27}-6a^{26}-19a^{25}+5a^{24}+6a^{23}+16a^{22}-10a^{21}-6a^{20}-15a^{19}+15a^{18}+11a^{17}+19a^{16}-22a^{15}-20a^{14}-21a^{13}+31a^{12}+21a^{11}+14a^{10}-35a^{9}-11a^{8}-6a^{7}+37a^{6}+7a^{5}-53a^{3}-14a^{2}+12a+4$ (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: | \( 440767023565659140 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{13}\cdot 440767023565659140 \cdot 1}{2\cdot\sqrt{9553546594678273369526111644888637966824486353442589488115}}\cr\approx \mathstrut & 0.429066948670999 \end{aligned}\] (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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | $29$ | $20{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $23{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.10.0.1}{10} }$ | $16{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $27{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | $26{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.23.0.1 | $x^{23} + 2 x^{2} + 3$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(71\) | 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.8.0.1 | $x^{8} + 53 x^{3} + 22 x^{2} + 19 x + 7$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
71.8.0.1 | $x^{8} + 53 x^{3} + 22 x^{2} + 19 x + 7$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
71.11.0.1 | $x^{11} + 48 x + 64$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(1070754779627347\) | $\Q_{1070754779627347}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(251\!\cdots\!979\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ |