Normalized defining polynomial
\( x^{29} + x - 3 \)
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: | \(58740423215346262667355847452891580849174093037099352445\) \(\medspace = 5\cdot 11\cdot 16381\cdot 127865750920013\cdot 1994744457920647\cdot 255618611114956521989\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
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: | $5^{1/2}11^{1/2}16381^{1/2}127865750920013^{1/2}1994744457920647^{1/2}255618611114956521989^{1/2}\approx 7.664230112369165e+27$ | ||
Ramified primes: | \(5\), \(11\), \(16381\), \(127865750920013\), \(1994744457920647\), \(255618611114956521989\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{58740\!\cdots\!52445}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
| |
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: | $a-1$, $a^{28}+a^{27}-a^{25}-a^{24}-a^{23}-a^{22}+a^{20}+a^{19}+a^{18}+a^{17}-a^{15}-a^{14}-a^{13}-a^{12}+a^{10}+a^{9}+a^{8}+a^{7}-a^{5}-a^{4}-a^{3}-a^{2}+2$, $a^{26}-a^{24}+a^{23}-2a^{21}+a^{20}+a^{19}-2a^{18}+a^{17}+2a^{16}-2a^{15}+2a^{13}-2a^{12}-a^{11}+2a^{10}-a^{9}-a^{8}+2a^{7}-a^{5}+a^{4}-a^{2}+1$, $a^{28}+2a^{27}+2a^{26}+2a^{25}+a^{24}-a^{22}-2a^{21}-2a^{20}-2a^{19}-a^{18}-a^{17}+a^{16}+2a^{14}+2a^{12}+2a^{10}+2a^{8}-a^{7}+a^{6}-3a^{5}-a^{4}-5a^{3}-a^{2}-4a+2$, $6a^{28}+4a^{27}-a^{26}-5a^{25}-8a^{24}-9a^{23}-6a^{22}-a^{21}+4a^{20}+7a^{19}+7a^{18}+4a^{17}-a^{16}-8a^{15}-12a^{14}-10a^{13}-7a^{12}-a^{11}+7a^{10}+10a^{9}+9a^{8}+4a^{7}-4a^{6}-11a^{5}-15a^{4}-14a^{3}-7a^{2}+2a+14$, $a^{28}-4a^{27}+4a^{26}-10a^{24}+4a^{23}+9a^{22}-5a^{21}-a^{20}+6a^{19}-9a^{18}-5a^{17}+14a^{16}-9a^{14}+7a^{13}+2a^{12}-14a^{11}+10a^{10}+10a^{9}-14a^{8}-3a^{7}+16a^{6}-11a^{5}-3a^{4}+17a^{3}-8a^{2}-22a+22$, $6a^{28}-3a^{27}-7a^{26}-11a^{25}+16a^{24}+10a^{23}-10a^{22}-7a^{21}-9a^{20}+15a^{19}+17a^{18}-18a^{17}-9a^{16}-2a^{15}+11a^{14}+23a^{13}-24a^{12}-15a^{11}+9a^{10}+9a^{9}+23a^{8}-24a^{7}-27a^{6}+22a^{5}+12a^{4}+15a^{3}-17a^{2}-42a+40$, $a^{28}+3a^{27}-2a^{26}-5a^{25}+3a^{24}+3a^{23}-2a^{22}-2a^{21}+a^{20}-3a^{19}+3a^{18}+4a^{17}-5a^{16}-6a^{15}+7a^{14}+2a^{13}-4a^{12}-a^{11}-3a^{9}+6a^{8}+3a^{7}-10a^{6}-3a^{5}+10a^{4}-6a^{2}+2a-4$, $5a^{28}+2a^{27}-4a^{26}-4a^{25}+a^{24}+5a^{23}+3a^{22}-4a^{21}-6a^{20}+a^{19}+8a^{18}+3a^{17}-7a^{16}-7a^{15}+3a^{14}+10a^{13}+2a^{12}-11a^{11}-8a^{10}+8a^{9}+13a^{8}-3a^{7}-14a^{6}-4a^{5}+11a^{4}+11a^{3}-5a^{2}-14a+2$, $4a^{28}-2a^{27}+3a^{26}+4a^{25}-a^{24}-2a^{23}-7a^{22}+2a^{21}+4a^{20}-4a^{19}+8a^{18}+3a^{17}-8a^{16}-2a^{15}-4a^{14}+4a^{13}+3a^{12}-2a^{11}+12a^{10}-5a^{9}-13a^{8}+4a^{7}-3a^{6}+4a^{5}+5a^{4}+a^{3}+11a^{2}-17a-7$, $3a^{28}+15a^{27}+26a^{26}+26a^{25}+34a^{24}+18a^{23}+20a^{22}-5a^{21}-9a^{20}-28a^{19}-34a^{18}-34a^{17}-36a^{16}-16a^{15}-13a^{14}+18a^{13}+18a^{12}+47a^{11}+36a^{10}+47a^{9}+30a^{8}+11a^{7}+4a^{6}-38a^{5}-26a^{4}-66a^{3}-38a^{2}-54a-16$, $6a^{28}-4a^{27}+2a^{26}+2a^{25}-4a^{24}+2a^{23}-5a^{22}+9a^{21}-10a^{20}+9a^{19}-6a^{18}+a^{17}-a^{16}+2a^{15}+5a^{14}-11a^{13}+14a^{12}-15a^{11}+11a^{10}-9a^{9}+12a^{8}-10a^{7}+6a^{5}-13a^{4}+18a^{3}-15a^{2}+17a-17$, $3a^{28}+4a^{27}-10a^{26}+14a^{25}-15a^{24}+13a^{23}-8a^{22}-a^{21}+12a^{20}-22a^{19}+27a^{18}-24a^{17}+12a^{16}+8a^{15}-28a^{14}+41a^{13}-41a^{12}+26a^{11}-29a^{9}+50a^{8}-54a^{7}+39a^{6}-12a^{5}-20a^{4}+46a^{3}-56a^{2}+50a-28$, $3a^{28}-a^{27}-3a^{26}+2a^{25}-2a^{24}+5a^{22}-3a^{21}+2a^{19}-4a^{18}-2a^{17}+5a^{16}-3a^{15}+3a^{14}+3a^{13}-7a^{12}+a^{11}+2a^{10}-6a^{9}+4a^{8}+6a^{7}-6a^{6}+4a^{5}-a^{4}-10a^{3}+9a^{2}-2$ (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: | \( 42821205896121990 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{14}\cdot 42821205896121990 \cdot 1}{2\cdot\sqrt{58740423215346262667355847452891580849174093037099352445}}\cr\approx \mathstrut & 0.835042375123828 \end{aligned}\] (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 | $27{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | $18{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | R | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ | $20{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $27{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $24{,}\,{\href{/padicField/43.5.0.1}{5} }$ | $25{,}\,{\href{/padicField/47.4.0.1}{4} }$ | $17{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
5.23.0.1 | $x^{23} + 2 x^{2} + 3$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.5.0.1 | $x^{5} + 10 x^{2} + 9$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
11.7.0.1 | $x^{7} + 4 x + 9$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
11.14.0.1 | $x^{14} + 2 x^{7} + 9 x^{6} + 6 x^{5} + 4 x^{4} + 8 x^{3} + 6 x^{2} + 10 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(16381\) | $\Q_{16381}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(127865750920013\) | $\Q_{127865750920013}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(1994744457920647\) | $\Q_{1994744457920647}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1994744457920647}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(255\!\cdots\!989\) | $\Q_{25\!\cdots\!89}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{25\!\cdots\!89}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ |