Normalized defining polynomial
\( x^{29} - 3x - 1 \)
Invariants
Degree: | $29$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2274789759216463037320759498792527390442169528114066419\) \(\medspace = -\,19\cdot 157273\cdot 116400694409\cdot 65\!\cdots\!93\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(74.88\) | 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: | $19^{1/2}157273^{1/2}116400694409^{1/2}6540002603845429295048615140626894593^{1/2}\approx 1.5082406171484918e+27$ | ||
Ramified primes: | \(19\), \(157273\), \(116400694409\), \(65400\!\cdots\!94593\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-22747\!\cdots\!66419}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $15$ | 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: | $a$, $a+1$, $a^{15}-2a$, $a^{28}+a^{26}+a^{25}+a^{23}+a^{21}+a^{19}+a^{18}+a^{16}+2a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+2a^{9}+2a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+3a^{2}+a-1$, $a^{28}-2a^{26}+3a^{25}-a^{24}-a^{23}+a^{22}-a^{21}+3a^{19}-3a^{18}+2a^{16}-2a^{15}+2a^{14}-a^{13}-2a^{12}+3a^{11}+a^{10}-4a^{9}+3a^{8}-3a^{7}+a^{6}+4a^{5}-4a^{4}-a^{3}+3a^{2}-2a-2$, $2a^{28}-a^{27}-a^{26}+2a^{25}-3a^{24}+4a^{23}-3a^{22}+2a^{21}-a^{19}+3a^{18}-4a^{17}+4a^{16}-4a^{15}+3a^{14}-a^{13}-2a^{12}+4a^{11}-5a^{10}+6a^{9}-5a^{8}+2a^{7}-a^{6}-2a^{5}+4a^{4}-6a^{3}+6a^{2}-5a-2$, $a^{28}-2a^{27}-a^{26}-a^{25}-a^{23}-a^{22}-a^{21}-a^{20}+a^{19}-3a^{18}-4a^{16}+a^{15}-2a^{14}-a^{13}-3a^{12}-a^{11}-a^{10}-a^{9}-4a^{8}-4a^{7}-a^{6}-3a^{5}-7a^{3}-3a-3$, $a^{28}-a^{25}-a^{23}+2a^{22}+2a^{20}-a^{19}-a^{18}-2a^{17}+3a^{14}-a^{13}+a^{12}-3a^{11}+2a^{8}+2a^{7}+2a^{6}-2a^{5}-5a^{3}+a^{2}-a$, $a^{28}-a^{23}-a^{22}+a^{20}+a^{19}+a^{18}-a^{14}-a^{13}-a^{12}+2a^{10}+2a^{9}+a^{8}-a^{5}-2a^{4}-3a^{3}+4a$, $3a^{28}-a^{25}-2a^{24}+4a^{23}-a^{21}-a^{20}-2a^{19}+6a^{18}-a^{17}-3a^{16}+5a^{13}-a^{12}-5a^{11}+2a^{10}+a^{9}+4a^{8}-a^{7}-7a^{6}+4a^{5}+3a^{4}+3a^{3}-4a^{2}-6a-2$, $9a^{28}+9a^{27}-a^{26}-11a^{25}-9a^{24}+4a^{23}+14a^{22}+8a^{21}-8a^{20}-17a^{19}-6a^{18}+13a^{17}+19a^{16}+3a^{15}-18a^{14}-19a^{13}+2a^{12}+22a^{11}+17a^{10}-8a^{9}-25a^{8}-14a^{7}+14a^{6}+28a^{5}+12a^{4}-19a^{3}-32a^{2}-10a-2$, $a^{28}+3a^{27}+a^{26}-3a^{25}-a^{22}+2a^{21}+3a^{20}-4a^{19}-2a^{18}-a^{17}-4a^{16}+2a^{15}+6a^{14}+2a^{13}+2a^{11}-5a^{10}-3a^{9}+4a^{8}+3a^{7}-2a^{6}+3a^{5}-6a^{4}-10a^{3}+a^{2}+3a$, $3a^{28}-a^{27}-4a^{25}+a^{24}+3a^{22}-2a^{21}+a^{20}-a^{19}+3a^{18}-3a^{17}-a^{16}-3a^{15}+4a^{14}+a^{12}-5a^{11}+2a^{10}+a^{9}+5a^{8}-3a^{7}+a^{6}-a^{5}+6a^{4}-2a^{3}-4a-3$, $4a^{28}+2a^{26}+4a^{25}+2a^{23}+5a^{22}+2a^{20}+6a^{19}+3a^{17}+7a^{16}+4a^{14}+7a^{13}+a^{12}+5a^{11}+6a^{10}+3a^{9}+5a^{8}+5a^{7}+6a^{6}+5a^{5}+5a^{4}+9a^{3}+4a^{2}+6a$, $4a^{28}-5a^{27}+8a^{26}-12a^{25}+12a^{24}-17a^{23}+15a^{22}-17a^{21}+15a^{20}-13a^{19}+11a^{18}-8a^{17}+4a^{16}-a^{15}-5a^{14}+9a^{13}-13a^{12}+18a^{11}-19a^{10}+21a^{9}-24a^{8}+19a^{7}-26a^{6}+15a^{5}-18a^{4}+10a^{3}-2a^{2}+2a$ (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: | \( 6315065032809016.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{13}\cdot 6315065032809016.0 \cdot 1}{2\cdot\sqrt{2274789759216463037320759498792527390442169528114066419}}\cr\approx \mathstrut & 0.398387290598117 \end{aligned}\] (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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $28{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $29$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | R | $25{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/31.10.0.1}{10} }$ | $26{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.9.0.1}{9} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
19.20.0.1 | $x^{20} + 16 x^{11} + 13 x^{10} + 4 x^{8} + 7 x^{7} + 8 x^{6} + 6 x^{5} + 3 x^{3} + 6 x^{2} + 11 x + 2$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(157273\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(116400694409\) | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(654\!\cdots\!593\) | 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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |