Normalized defining polynomial
\( x^{25} - x - 5 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(5293955920339377117843279852397478138372116036921849\) \(\medspace = 47\cdot 8641\cdot 6952808251\cdot 10464775307854799\cdot 179154737750394843163\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(117.21\) | 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: | $47^{1/2}8641^{1/2}6952808251^{1/2}10464775307854799^{1/2}179154737750394843163^{1/2}\approx 7.2759576141834256e+25$ | ||
Ramified primes: | \(47\), \(8641\), \(6952808251\), \(10464775307854799\), \(179154737750394843163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{52939\!\cdots\!21849}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $2a^{24}-2a^{23}-2a^{22}+a^{21}-a^{19}-a^{18}-2a^{17}+a^{16}+2a^{15}-4a^{14}-3a^{13}+4a^{12}-a^{11}-6a^{10}+2a^{8}-2a^{7}-a^{6}-3a^{5}-2a^{4}+3a^{3}-5a^{2}-11a+4$, $2a^{24}+8a^{23}+a^{22}-11a^{21}-5a^{20}+6a^{19}+3a^{18}+a^{17}+6a^{16}-2a^{15}-16a^{14}-2a^{13}+16a^{12}+8a^{11}-8a^{10}-a^{9}-6a^{8}-14a^{7}+6a^{6}+33a^{5}+4a^{4}-27a^{3}-15a^{2}+4a-8$, $5a^{24}-12a^{23}-3a^{22}+7a^{21}+14a^{20}-18a^{19}-3a^{18}+3a^{17}+22a^{16}-23a^{15}-5a^{14}-a^{13}+34a^{12}-25a^{11}-11a^{10}-3a^{9}+43a^{8}-20a^{7}-21a^{6}-8a^{5}+48a^{4}-10a^{3}-41a^{2}-6a+46$, $8a^{24}-8a^{23}-14a^{22}-2a^{21}-8a^{20}+10a^{19}+19a^{18}-4a^{17}+4a^{16}-3a^{14}+34a^{13}+23a^{12}+7a^{11}+11a^{10}-27a^{9}-9a^{8}+25a^{7}-a^{6}+12a^{5}-18a^{4}-70a^{3}-19a^{2}-12a-14$, $a^{24}-37a^{23}+10a^{22}+45a^{21}-35a^{20}-31a^{19}+46a^{18}+22a^{17}-71a^{16}+10a^{15}+69a^{14}-35a^{13}-69a^{12}+77a^{11}+43a^{10}-99a^{9}-6a^{8}+116a^{7}-43a^{6}-116a^{5}+96a^{4}+80a^{3}-136a^{2}-47a+184$, $4a^{24}-2a^{23}-17a^{22}-20a^{21}-8a^{20}-3a^{19}+28a^{18}+17a^{17}+25a^{16}-6a^{15}-22a^{14}-33a^{13}-30a^{12}+4a^{11}+20a^{10}+56a^{9}+27a^{8}+18a^{7}-38a^{6}-60a^{5}-47a^{4}-32a^{3}+53a^{2}+59a+89$, $8a^{24}+2a^{23}-12a^{22}-15a^{21}-11a^{20}-15a^{19}-31a^{18}-35a^{17}-27a^{16}-22a^{15}-34a^{14}-34a^{13}-14a^{12}+9a^{11}+6a^{10}+9a^{9}+39a^{8}+74a^{7}+67a^{6}+53a^{5}+74a^{4}+101a^{3}+72a^{2}+25a+19$, $6a^{24}-a^{23}+2a^{22}+6a^{21}-6a^{20}+2a^{19}+7a^{18}-4a^{17}+5a^{16}+3a^{15}-6a^{14}+14a^{13}+a^{12}-9a^{11}+14a^{10}-5a^{9}+8a^{8}+19a^{7}-22a^{6}+8a^{5}+12a^{4}-6a^{3}+32a^{2}-9a-22$, $2a^{24}-20a^{23}+19a^{22}+13a^{21}-5a^{20}+9a^{19}-41a^{18}+13a^{17}+12a^{16}-a^{15}+37a^{14}-50a^{13}+17a^{12}-21a^{11}-11a^{10}+59a^{9}-34a^{8}+47a^{7}-52a^{6}-35a^{5}+38a^{4}-25a^{3}+91a^{2}-21a-49$, $46a^{24}+39a^{23}-47a^{22}-22a^{21}+26a^{20}+18a^{19}+4a^{18}-47a^{17}-31a^{16}+92a^{15}+29a^{14}-129a^{13}+6a^{12}+142a^{11}-33a^{10}-109a^{9}+27a^{8}+59a^{7}+25a^{6}-52a^{5}-125a^{4}+95a^{3}+201a^{2}-181a-244$, $70a^{24}-60a^{23}+35a^{22}-9a^{21}-31a^{20}+62a^{19}-90a^{18}+110a^{17}-115a^{16}+107a^{15}-69a^{14}+38a^{13}+18a^{12}-78a^{11}+124a^{10}-172a^{9}+179a^{8}-184a^{7}+148a^{6}-84a^{5}+4a^{4}+90a^{3}-161a^{2}+262a-358$, $27a^{24}+66a^{23}+57a^{22}-43a^{21}-128a^{20}-79a^{19}+19a^{18}+48a^{17}+40a^{16}+99a^{15}+121a^{14}-16a^{13}-203a^{12}-185a^{11}-11a^{10}+87a^{9}+72a^{8}+140a^{7}+224a^{6}+62a^{5}-288a^{4}-385a^{3}-113a^{2}+133a+119$ (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: | \( 16941382677340960 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 16941382677340960 \cdot 1}{2\cdot\sqrt{5293955920339377117843279852397478138372116036921849}}\cr\approx \mathstrut & 0.881489436905420 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ |
Character table for $S_{25}$ |
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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.2.0.1}{2} }^{10}{,}\,{\href{/padicField/5.1.0.1}{1} }^{5}$ | $15{,}\,{\href{/padicField/7.10.0.1}{10} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | $16{,}\,{\href{/padicField/13.9.0.1}{9} }$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | R | $21{,}\,{\href{/padicField/53.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
47.2.0.1 | $x^{2} + 45 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.3.0.1 | $x^{3} + 3 x + 42$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
47.7.0.1 | $x^{7} + 12 x + 42$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
47.9.0.1 | $x^{9} + x^{3} + 19 x^{2} + x + 42$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(8641\) | $\Q_{8641}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{8641}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(6952808251\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(10464775307854799\) | $\Q_{10464775307854799}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
\(179\!\cdots\!163\) | $\Q_{17\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |