Normalized defining polynomial
\( x^{29} - 2x - 3 \)
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: | \(58740405420479006992414084652221434564422848468416004477\) \(\medspace = 43\cdot 907\cdot 40322333\cdot 5439157486181\cdot 68\!\cdots\!49\) | 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: | \(83.77\) | 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: | $43^{1/2}907^{1/2}40322333^{1/2}5439157486181^{1/2}6867266657725622021109001049749^{1/2}\approx 7.664228951465307e+27$ | ||
Ramified primes: | \(43\), \(907\), \(40322333\), \(5439157486181\), \(68672\!\cdots\!49749\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{58740\!\cdots\!04477}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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)
| |
Fundamental units: | $a^{27}-2a^{26}+2a^{25}-2a^{24}+a^{23}-a^{21}+2a^{20}-2a^{19}+2a^{18}-a^{17}+a^{15}-2a^{14}+2a^{13}-2a^{12}-3a^{9}+2a^{8}-4a^{7}+2a^{6}-2a^{5}-a^{4}+a^{3}-4a^{2}+3a-4$, $a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+2$, $a^{28}-2a^{26}-a^{25}+2a^{24}+a^{23}-a^{22}+a^{20}-3a^{18}-a^{17}+4a^{16}+2a^{15}-2a^{14}-2a^{13}+a^{12}+a^{11}-3a^{10}-a^{9}+5a^{8}+3a^{7}-3a^{6}-5a^{5}+4a^{3}-a^{2}-a+2$, $a^{27}-a^{26}+2a^{25}-a^{24}+2a^{23}-a^{22}+a^{21}-a^{20}+a^{19}+a^{17}+a^{16}+2a^{14}-2a^{13}+3a^{12}-3a^{11}+4a^{10}-2a^{9}+3a^{8}+3a^{5}-2a^{4}+5a^{3}-3a^{2}+4a-2$, $a^{28}-3a^{27}+3a^{26}+a^{24}-a^{23}-2a^{22}+5a^{21}-2a^{20}+4a^{19}-5a^{18}+3a^{17}+a^{16}+2a^{15}-a^{14}-3a^{13}+a^{12}+a^{11}+3a^{10}-7a^{9}+a^{8}-4a^{7}+6a^{6}-3a^{5}-5a^{4}-2a^{3}-a^{2}+7a-7$, $3a^{28}+3a^{27}+5a^{25}-4a^{23}-5a^{21}-4a^{20}+4a^{19}+4a^{17}+8a^{16}-a^{15}-a^{14}-11a^{12}-3a^{11}-2a^{9}+9a^{8}+10a^{7}+a^{6}+7a^{5}-5a^{4}-14a^{3}-4a^{2}-9a-10$, $2a^{28}+a^{27}-2a^{26}-a^{25}+a^{24}+2a^{23}-a^{22}-2a^{21}+2a^{19}+2a^{18}-2a^{17}-3a^{16}+4a^{14}+3a^{13}-4a^{12}-5a^{11}+a^{10}+7a^{9}+2a^{8}-6a^{7}-4a^{6}+2a^{5}+6a^{4}+2a^{3}-5a^{2}-5a-2$, $a^{27}+a^{26}-a^{25}+a^{24}+2a^{23}-4a^{22}+4a^{21}-a^{20}-4a^{19}+6a^{18}-6a^{17}+4a^{15}-8a^{14}+3a^{13}+a^{12}-7a^{11}+4a^{10}-6a^{8}+6a^{7}-a^{6}-4a^{5}+8a^{4}-3a^{3}+7a-2$, $a^{26}+a^{24}+2a^{23}-a^{21}-2a^{19}-a^{18}-2a^{16}-a^{15}+a^{14}+2a^{12}+2a^{11}+a^{10}+2a^{9}+a^{8}-3a^{5}-3a^{4}-3a^{3}-4a^{2}+2$, $4a^{28}-a^{27}-2a^{26}-2a^{25}+2a^{24}+4a^{23}-2a^{22}-2a^{21}-3a^{20}+6a^{19}+a^{17}-8a^{16}+4a^{15}+a^{14}+7a^{13}-9a^{12}-a^{11}-3a^{10}+12a^{9}-3a^{8}-3a^{7}-10a^{6}+10a^{5}+4a^{4}+a^{3}-13a^{2}+a-2$, $2a^{28}+4a^{27}+6a^{26}+5a^{25}+2a^{24}-2a^{23}-8a^{22}-9a^{21}-4a^{20}+2a^{18}+9a^{17}+12a^{16}+5a^{15}-3a^{14}-8a^{13}-12a^{12}-12a^{11}-6a^{10}+2a^{9}+14a^{8}+20a^{7}+12a^{6}+a^{5}-5a^{4}-17a^{3}-28a^{2}-19a-4$, $3a^{28}-a^{27}-2a^{26}+a^{25}+a^{24}-2a^{22}-2a^{21}+a^{20}+a^{19}+3a^{18}-2a^{16}-a^{15}+a^{14}+3a^{13}-5a^{12}-4a^{11}+a^{10}+9a^{9}+3a^{8}-7a^{7}-4a^{6}+5a^{5}+5a^{4}-6a^{3}-12a^{2}+4a+4$, $19a^{28}-20a^{27}+24a^{26}-23a^{25}+27a^{24}-21a^{23}+19a^{22}-7a^{21}-2a^{20}+13a^{19}-17a^{18}+21a^{17}-24a^{16}+24a^{15}-30a^{14}+24a^{13}-23a^{12}+7a^{11}-2a^{10}-17a^{9}+14a^{8}-24a^{7}+22a^{6}-30a^{5}+29a^{4}-32a^{3}+26a^{2}-10a-34$, $a^{28}+a^{27}-3a^{26}-a^{25}+2a^{24}-a^{23}+2a^{22}+2a^{21}-2a^{20}-3a^{19}+2a^{18}-2a^{16}+4a^{15}+a^{14}-2a^{13}-a^{11}-4a^{10}+3a^{9}+6a^{8}-2a^{7}-2a^{6}-5a^{4}-a^{3}+6a^{2}-2$ (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: | \( 47730899898626100 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 47730899898626100 \cdot 1}{2\cdot\sqrt{58740405420479006992414084652221434564422848468416004477}}\cr\approx \mathstrut & 0.930784858138733 \end{aligned}\] (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 | $28{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $26{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | R | $17{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(43\) | $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
43.2.1.1 | $x^{2} + 86$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
43.12.0.1 | $x^{12} + 34 x^{7} + 27 x^{6} + 16 x^{5} + 17 x^{4} + 6 x^{3} + 23 x^{2} + 38 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
43.12.0.1 | $x^{12} + 34 x^{7} + 27 x^{6} + 16 x^{5} + 17 x^{4} + 6 x^{3} + 23 x^{2} + 38 x + 3$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(907\) | $\Q_{907}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{907}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(40322333\) | $\Q_{40322333}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $26$ | $1$ | $26$ | $0$ | $C_{26}$ | $[\ ]^{26}$ | ||
\(5439157486181\) | $\Q_{5439157486181}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5439157486181}$ | $x$ | $1$ | $1$ | $0$ | 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(686\!\cdots\!749\) | 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 $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ |