Normalized defining polynomial
\( x^{25} - x - 3 \)
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: | \(25084781938832011102936096227680608444161921849\) \(\medspace = 3^{25}\cdot 2351\cdot 126764421589739\cdot 99341097532143287\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(71.78\) | 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: | not computed | ||
Ramified primes: | \(3\), \(2351\), \(126764421589739\), \(99341097532143287\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{88817\!\cdots\!51929}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{13}-a-1$, $a^{17}-a^{9}-1$, $a^{21}+a^{17}-a^{9}-a^{5}+1$, $a^{23}+a^{21}-a^{17}-a^{15}+a^{11}+a^{9}-a^{5}-a^{3}+1$, $a^{24}+a^{23}-a^{21}-a^{20}+a^{18}+a^{17}-a^{15}-a^{14}+a^{12}+a^{11}-a^{9}-a^{8}+a^{6}+a^{5}-a^{3}-a^{2}+1$, $2a^{24}-3a^{22}+3a^{21}-a^{20}-2a^{19}+2a^{18}-3a^{16}+2a^{15}+a^{14}-4a^{13}+2a^{12}+a^{11}-3a^{10}+3a^{8}-5a^{7}+a^{6}+4a^{5}-7a^{4}+2a^{3}+3a^{2}-4a-5$, $2a^{24}-a^{22}+a^{19}-2a^{18}+a^{17}-a^{16}+a^{15}-a^{13}+2a^{10}-4a^{9}+2a^{7}+2a^{6}-4a^{5}-2a^{4}+5a^{3}-a^{2}-3a-5$, $a^{24}-2a^{23}+a^{20}+a^{19}-2a^{18}-a^{17}+a^{16}+2a^{15}-a^{14}-2a^{12}+a^{11}+3a^{10}-a^{9}-a^{8}-2a^{7}+2a^{6}+2a^{5}+a^{4}-3a^{3}-2a^{2}+2a+2$, $9a^{24}-a^{23}-10a^{22}-2a^{21}+4a^{20}+11a^{19}-2a^{18}-12a^{17}-3a^{16}+6a^{15}+14a^{14}-4a^{13}-15a^{12}-3a^{11}+8a^{10}+17a^{9}-5a^{8}-20a^{7}-3a^{6}+12a^{5}+21a^{4}-9a^{3}-24a^{2}-4a+8$, $8a^{24}-a^{23}-11a^{22}+16a^{21}-6a^{20}-9a^{19}+9a^{18}+5a^{17}-17a^{16}+15a^{15}+a^{14}-15a^{13}+10a^{12}+11a^{11}-22a^{10}+8a^{9}+14a^{8}-21a^{7}+7a^{6}+16a^{5}-22a^{4}-3a^{3}+30a^{2}-26a-10$, $9a^{23}+11a^{22}-5a^{21}-5a^{20}+12a^{19}+10a^{18}+a^{17}+10a^{16}+10a^{15}-7a^{14}-2a^{13}+20a^{12}+17a^{11}+a^{10}+5a^{9}+10a^{8}-a^{7}+4a^{6}+29a^{5}+25a^{4}-6a^{3}-4a^{2}+21a+14$, $18a^{24}+10a^{23}+5a^{21}-8a^{20}-21a^{19}-9a^{18}-20a^{17}-14a^{16}+9a^{15}+a^{14}+17a^{13}+33a^{12}+17a^{11}+10a^{10}+12a^{9}-7a^{8}-40a^{7}-25a^{6}-20a^{5}-47a^{4}-4a^{3}+28a^{2}+13a+23$ (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: | \( 50040091077622.54 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 50040091077622.54 \cdot 1}{2\cdot\sqrt{25084781938832011102936096227680608444161921849}}\cr\approx \mathstrut & 1.19611062092600 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ 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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/13.9.0.1}{9} }$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $23{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
3.6.6.4 | $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(2351\) | $\Q_{2351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2351}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(126764421589739\) | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(99341097532143287\) | $\Q_{99341097532143287}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{99341097532143287}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{99341097532143287}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{99341097532143287}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |