Normalized defining polynomial
\( x^{25} - x - 2 \)
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: | \(12315009240091155769832362607858830802944\) \(\medspace = 2^{24}\cdot 739\cdot 12239\cdot 2507149\cdot 272308609\cdot 118872926969\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.14\) | 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: | \(2\), \(739\), \(12239\), \(2507149\), \(272308609\), \(118872926969\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{73403\!\cdots\!63009}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{11}a^{24}-\frac{3}{11}a^{23}-\frac{2}{11}a^{22}-\frac{5}{11}a^{21}+\frac{4}{11}a^{20}-\frac{1}{11}a^{19}+\frac{3}{11}a^{18}+\frac{2}{11}a^{17}+\frac{5}{11}a^{16}-\frac{4}{11}a^{15}+\frac{1}{11}a^{14}-\frac{3}{11}a^{13}-\frac{2}{11}a^{12}-\frac{5}{11}a^{11}+\frac{4}{11}a^{10}-\frac{1}{11}a^{9}+\frac{3}{11}a^{8}+\frac{2}{11}a^{7}+\frac{5}{11}a^{6}-\frac{4}{11}a^{5}+\frac{1}{11}a^{4}-\frac{3}{11}a^{3}-\frac{2}{11}a^{2}-\frac{5}{11}a+\frac{3}{11}$
Monogenic: | Not computed | |
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: | $a^{13}-a-1$, $a^{4}+a^{3}+a^{2}+a+1$, $a^{19}+a^{13}+a^{7}+a+1$, $a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}-1$, $\frac{6}{11}a^{24}+\frac{4}{11}a^{23}-\frac{1}{11}a^{22}+\frac{3}{11}a^{21}+\frac{2}{11}a^{20}-\frac{6}{11}a^{19}-\frac{4}{11}a^{18}+\frac{1}{11}a^{17}-\frac{3}{11}a^{16}-\frac{2}{11}a^{15}+\frac{6}{11}a^{14}+\frac{4}{11}a^{13}-\frac{1}{11}a^{12}+\frac{3}{11}a^{11}+\frac{13}{11}a^{10}+\frac{5}{11}a^{9}-\frac{4}{11}a^{8}+\frac{1}{11}a^{7}-\frac{3}{11}a^{6}-\frac{13}{11}a^{5}-\frac{5}{11}a^{4}+\frac{4}{11}a^{3}-\frac{1}{11}a^{2}+\frac{3}{11}a+\frac{7}{11}$, $a^{17}-a^{9}-1$, $a^{21}-a^{17}+a^{13}-a^{9}+a^{5}-a-1$, $a^{23}+a^{21}+a^{19}+a^{17}+a^{15}+a^{13}+a^{11}+a^{9}+a^{7}+a^{5}+a^{3}+a+1$, $\frac{14}{11}a^{24}-\frac{9}{11}a^{23}+\frac{5}{11}a^{22}-\frac{15}{11}a^{21}+\frac{12}{11}a^{20}-\frac{3}{11}a^{19}+\frac{9}{11}a^{18}-\frac{16}{11}a^{17}+\frac{4}{11}a^{16}-\frac{1}{11}a^{15}+\frac{14}{11}a^{14}-\frac{9}{11}a^{13}-\frac{6}{11}a^{12}-\frac{4}{11}a^{11}+\frac{12}{11}a^{10}+\frac{8}{11}a^{9}-\frac{2}{11}a^{8}-\frac{5}{11}a^{7}-\frac{7}{11}a^{6}+\frac{10}{11}a^{5}+\frac{3}{11}a^{4}+\frac{2}{11}a^{3}-\frac{17}{11}a^{2}+\frac{7}{11}a-\frac{13}{11}$, $a^{22}+a^{19}+a^{16}+a^{13}+a^{10}+a^{7}+a^{4}+a+1$, $\frac{5}{11}a^{24}-\frac{4}{11}a^{23}+\frac{1}{11}a^{22}-\frac{3}{11}a^{21}-\frac{2}{11}a^{20}+\frac{6}{11}a^{19}-\frac{7}{11}a^{18}+\frac{10}{11}a^{17}+\frac{3}{11}a^{16}+\frac{2}{11}a^{15}+\frac{5}{11}a^{14}-\frac{4}{11}a^{13}+\frac{1}{11}a^{12}-\frac{3}{11}a^{11}-\frac{2}{11}a^{10}+\frac{6}{11}a^{9}-\frac{7}{11}a^{8}+\frac{10}{11}a^{7}+\frac{3}{11}a^{6}+\frac{2}{11}a^{5}+\frac{5}{11}a^{4}-\frac{4}{11}a^{3}+\frac{12}{11}a^{2}-\frac{3}{11}a-\frac{7}{11}$, $\frac{1}{11}a^{24}-\frac{3}{11}a^{23}+\frac{9}{11}a^{22}-\frac{5}{11}a^{21}+\frac{4}{11}a^{20}+\frac{10}{11}a^{19}-\frac{8}{11}a^{18}+\frac{2}{11}a^{17}+\frac{5}{11}a^{16}-\frac{4}{11}a^{15}+\frac{1}{11}a^{14}-\frac{3}{11}a^{13}-\frac{2}{11}a^{12}-\frac{5}{11}a^{11}-\frac{7}{11}a^{10}-\frac{1}{11}a^{9}+\frac{3}{11}a^{8}-\frac{9}{11}a^{7}-\frac{6}{11}a^{6}+\frac{7}{11}a^{5}-\frac{10}{11}a^{4}-\frac{3}{11}a^{3}+\frac{9}{11}a^{2}-\frac{5}{11}a+\frac{3}{11}$ (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: | \( 29964620111.03697 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 29964620111.03697 \cdot 1}{2\cdot\sqrt{12315009240091155769832362607858830802944}}\cr\approx \mathstrut & 1.02223283180668 \end{aligned}\] (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 | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/7.10.0.1}{10} }$ | $15{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/13.9.0.1}{9} }$ | $25$ | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $25$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/41.7.0.1}{7} }$ | $15{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.8.8.13 | $x^{8} + 2 x + 2$ | $8$ | $1$ | $8$ | $C_2^3:(C_7: C_3)$ | $[8/7, 8/7, 8/7]_{7}^{3}$ | |
Deg $16$ | $8$ | $2$ | $16$ | ||||
\(739\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(12239\) | $\Q_{12239}$ | $x$ | $1$ | $1$ | $0$ | 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(2507149\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(272308609\) | $\Q_{272308609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{272308609}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{272308609}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(118872926969\) | $\Q_{118872926969}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{118872926969}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{118872926969}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ |