Normalized defining polynomial
\( x^{29} + 2x - 2 \)
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: | \(707052870611244971067929012298698254247147755012096\) \(\medspace = 2^{28}\cdot 499\cdot 541\cdot 695389\cdot 2739817\cdot 5913513689\cdot 866002650952007\) | 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: | \(56.68\) | 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: | $2^{28/29}499^{1/2}541^{1/2}695389^{1/2}2739817^{1/2}5913513689^{1/2}866002650952007^{1/2}\approx 3.169243852658477e+21$ | ||
Ramified primes: | \(2\), \(499\), \(541\), \(695389\), \(2739817\), \(5913513689\), \(866002650952007\) | 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: | $\Q(\sqrt{26339\!\cdots\!08941}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $14$ | 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: | $a-1$, $a^{2}-a+1$, $a^{10}-a+1$, $a^{22}+a^{15}-a+1$, $a^{28}+a^{27}+a^{26}+a^{22}+a^{21}+a^{20}+a^{16}+a^{15}+a^{14}+a^{10}+a^{9}+a^{8}+a^{4}+a^{3}+a^{2}+1$, $2a^{28}+2a^{27}+2a^{26}+a^{25}+a^{24}+a^{23}-a^{19}-a^{18}-a^{17}-2a^{16}-a^{15}-a^{14}-2a^{13}-a^{12}-a^{11}-a^{10}-a^{7}-a^{4}+a^{3}-a+5$, $a^{28}+a^{25}+a^{24}+a^{22}+a^{21}-a^{20}-a^{17}-a^{16}+a^{15}-a^{13}+a^{12}-a^{10}+2a^{8}-a^{7}+a^{6}+2a^{5}-a^{4}-a^{3}+a^{2}-1$, $a^{28}-a^{26}+a^{25}-a^{21}+a^{20}+a^{17}-2a^{16}+a^{15}+2a^{14}-2a^{13}+a^{12}-a^{10}+2a^{9}-a^{8}-a^{5}+3a^{4}-2a^{3}-a^{2}+3a-1$, $5a^{28}+5a^{27}+4a^{26}+3a^{25}+3a^{24}+3a^{23}+4a^{22}+3a^{21}+2a^{20}+2a^{19}+3a^{18}+3a^{17}+2a^{16}+a^{15}+a^{14}+2a^{13}+2a^{12}+2a^{11}+a^{9}+2a^{8}+2a^{7}+a^{6}+a^{3}+2a^{2}+a+9$, $a^{28}+a^{26}-a^{25}+2a^{21}+a^{19}-a^{18}+a^{17}-a^{16}+2a^{15}-a^{13}+a^{12}-2a^{11}+2a^{10}+2a^{8}-2a^{7}+a^{6}-2a^{5}+2a^{3}-2a^{2}+2a-1$, $a^{27}-2a^{25}-a^{24}+a^{23}+a^{22}-a^{20}-a^{19}+a^{17}+a^{14}-a^{13}-2a^{12}+3a^{10}-a^{8}-2a^{7}+a^{5}+2a^{4}-a^{3}-a^{2}+a-1$, $2a^{28}+2a^{27}+2a^{26}+a^{25}-a^{24}-a^{23}-a^{22}+a^{20}+a^{19}-a^{17}-2a^{16}-2a^{15}+a^{13}+2a^{12}+3a^{11}-a^{9}-a^{8}-2a^{7}+2a^{5}+a^{4}-3a+1$, $2a^{28}+a^{27}+a^{24}+2a^{23}+a^{22}-2a^{20}+a^{18}+a^{17}-3a^{16}-2a^{15}+2a^{13}-a^{11}-a^{10}+a^{9}+2a^{8}+2a^{7}-a^{6}-2a^{5}+4a^{3}-3a+1$, $a^{26}+a^{25}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{17}+2a^{16}+a^{13}+a^{12}-a^{11}-a^{8}-a^{5}-a^{4}+a^{3}-2a^{2}-a+1$ (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: | \( 88931326459060.03 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 88931326459060.03 \cdot 1}{2\cdot\sqrt{707052870611244971067929012298698254247147755012096}}\cr\approx \mathstrut & 0.499858385590412 \end{aligned}\] (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 | $22{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $29$ | $21{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $28{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.7.0.1}{7} }$ | $16{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $26{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | $16{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ | $21{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 $29$ | $29$ | $1$ | $28$ | |||
\(499\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(541\) | $\Q_{541}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
\(695389\) | $\Q_{695389}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(2739817\) | $\Q_{2739817}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2739817}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(5913513689\) | $\Q_{5913513689}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
\(866002650952007\) | $\Q_{866002650952007}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ |