Normalized defining polynomial
\( x^{29} - 4x - 2 \)
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: | \(-9553545905420272548496712387718055513666469828515038494720\) \(\medspace = -\,2^{28}\cdot 5\cdot 71\!\cdots\!99\) | 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: | $2^{28/29}5^{1/2}7117946375474536827576691201119166250278405717399^{1/2}\approx 1.1649626699833909e+25$ | ||
Ramified primes: | \(2\), \(5\), \(71179\!\cdots\!17399\) | 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: | $\Q(\sqrt{-35589\!\cdots\!86995}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a+1$, $2a^{15}-4a-1$, $a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $16a^{28}-8a^{27}+5a^{26}+a^{25}+4a^{24}+2a^{23}-a^{21}-2a^{20}-a^{19}-2a^{18}-3a^{17}-5a^{16}-4a^{15}-a^{14}+2a^{13}+4a^{12}+3a^{11}+3a^{10}+2a^{9}+4a^{8}+2a^{7}-a^{6}-7a^{5}-8a^{4}-7a^{3}-3a^{2}-a-65$, $3a^{28}-6a^{26}+2a^{25}+4a^{24}-2a^{23}+2a^{22}+2a^{21}-7a^{20}+7a^{18}-3a^{17}-4a^{16}+2a^{15}-7a^{14}-3a^{13}+12a^{12}+2a^{11}-7a^{10}+6a^{9}+a^{8}-3a^{7}+15a^{6}+5a^{5}-17a^{4}+4a^{2}-10a-5$, $4a^{28}+2a^{27}+2a^{26}-3a^{25}-5a^{24}+2a^{22}+2a^{21}-2a^{19}-2a^{18}+4a^{17}+6a^{16}-6a^{14}-8a^{13}-3a^{12}+4a^{11}+12a^{10}+3a^{9}-6a^{8}-3a^{7}-3a^{6}+6a^{5}+4a^{4}-7a^{3}-13a^{2}-5a-1$, $3a^{28}-4a^{27}-2a^{26}+5a^{25}+2a^{24}-8a^{23}-4a^{22}+8a^{21}+6a^{20}-6a^{19}-6a^{18}+6a^{17}+4a^{16}-10a^{15}-5a^{14}+12a^{13}+10a^{12}-11a^{11}-13a^{10}+8a^{9}+9a^{8}-10a^{7}-8a^{6}+16a^{5}+14a^{4}-19a^{3}-22a^{2}+13a+9$, $5a^{28}+7a^{27}+8a^{26}+7a^{25}+14a^{24}+19a^{23}+8a^{22}+18a^{21}+18a^{20}+14a^{19}+6a^{18}+14a^{17}+5a^{16}-2a^{15}-11a^{14}-3a^{13}-20a^{12}-31a^{11}-23a^{10}-32a^{9}-38a^{8}-46a^{7}-26a^{6}-35a^{5}-38a^{4}-21a^{3}+3a^{2}-14a-9$, $4a^{28}-4a^{27}+6a^{26}-6a^{25}+4a^{23}-3a^{22}+2a^{21}-9a^{20}+13a^{19}-13a^{18}+7a^{17}-6a^{16}+11a^{15}-18a^{14}+10a^{13}-a^{11}-3a^{10}+8a^{8}-15a^{7}+12a^{6}-4a^{5}+13a^{4}-28a^{3}+25a^{2}-8a-15$, $a^{28}-3a^{27}-a^{26}+6a^{25}-7a^{24}+2a^{23}-a^{22}+4a^{21}-7a^{20}+2a^{19}+2a^{18}-a^{17}-5a^{16}+2a^{15}+6a^{14}-12a^{13}+5a^{12}+a^{11}-10a^{9}+10a^{8}-4a^{7}-2a^{6}-6a^{5}+9a^{4}-2a^{3}-14a^{2}+10a-1$, $3a^{28}+2a^{27}+4a^{26}+4a^{25}+4a^{24}+2a^{23}+4a^{22}+4a^{21}+4a^{20}+3a^{19}+6a^{18}+7a^{17}+7a^{16}+5a^{15}+7a^{14}+7a^{13}+7a^{12}+4a^{11}+8a^{10}+10a^{9}+11a^{8}+9a^{7}+13a^{6}+14a^{5}+12a^{4}+8a^{3}+12a^{2}+15a+5$, $9a^{28}+9a^{27}+4a^{26}-7a^{25}-8a^{24}+13a^{22}+12a^{21}+a^{20}-13a^{19}-8a^{18}+6a^{17}+20a^{16}+13a^{15}-6a^{14}-18a^{13}-6a^{12}+18a^{11}+26a^{10}+11a^{9}-18a^{8}-21a^{7}+2a^{6}+35a^{5}+31a^{4}+a^{3}-32a^{2}-22a-15$, $5a^{28}+a^{27}-5a^{26}+8a^{25}-4a^{24}-2a^{23}+8a^{22}-5a^{21}-a^{20}+11a^{19}-15a^{18}+11a^{17}+a^{16}-8a^{15}+12a^{14}-4a^{13}-5a^{12}+12a^{11}-4a^{10}-2a^{9}+13a^{8}-12a^{7}+6a^{6}+5a^{5}-3a^{4}+13a^{2}-16a-11$, $48a^{28}-34a^{27}+6a^{26}-14a^{25}-6a^{24}-13a^{23}-11a^{22}-9a^{21}-6a^{20}-7a^{19}-6a^{18}-2a^{17}+4a^{16}+9a^{15}+9a^{14}+11a^{13}+18a^{12}+24a^{11}+25a^{10}+23a^{9}+21a^{8}+25a^{7}+26a^{6}+17a^{5}+7a^{4}+a^{3}-2a^{2}-8a-217$, $5a^{28}+7a^{27}-5a^{26}-7a^{25}+4a^{24}+7a^{23}-4a^{22}-6a^{21}+4a^{20}+6a^{19}-5a^{18}-5a^{17}+7a^{16}+7a^{15}-9a^{14}-8a^{13}+12a^{12}+13a^{11}-14a^{10}-17a^{9}+15a^{8}+23a^{7}-16a^{6}-29a^{5}+13a^{4}+33a^{3}-13a^{2}-38a-13$ (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: | \( 819283530647138000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{13}\cdot 819283530647138000 \cdot 1}{2\cdot\sqrt{9553545905420272548496712387718055513666469828515038494720}}\cr\approx \mathstrut & 0.797535837477643 \end{aligned}\] (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 | R | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | R | $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $29$ | $29$ | $1$ | $28$ | |||
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.23.0.1 | $x^{23} + 2 x^{2} + 3$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(711\!\cdots\!399\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |