Normalized defining polynomial
\( x^{25} - 3x - 1 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1130059131643038744233908476350526800370111143\) \(\medspace = -\,131\cdot 13259\cdot 3933670000864773133\cdot 165394543991056112099\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(63.41\) | 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: | $131^{1/2}13259^{1/2}3933670000864773133^{1/2}165394543991056112099^{1/2}\approx 3.3616352146582453e+22$ | ||
Ramified primes: | \(131\), \(13259\), \(3933670000864773133\), \(165394543991056112099\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-11300\!\cdots\!11143}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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: | $a^{24}-3$, $a+1$, $a^{24}+a^{22}+a^{20}+a^{18}+a^{16}+a^{14}+a^{12}+a^{10}+a^{8}+a^{6}+a^{4}+a^{2}-1$, $a^{13}-2a$, $a^{20}+a^{18}+a^{13}+a^{11}+a^{10}+a^{8}-a^{7}+a^{6}+a^{4}+2a^{3}+2a$, $2a^{22}-a^{21}+2a^{20}-a^{19}-2a^{16}+a^{15}-2a^{14}+a^{13}+2a^{10}-a^{9}+2a^{8}-a^{7}-a^{6}-4a^{4}+a^{3}-3a^{2}+a+2$, $2a^{23}+a^{22}-2a^{20}-a^{19}+2a^{17}+a^{16}-2a^{14}-a^{13}+2a^{11}+a^{10}-3a^{8}-2a^{7}+4a^{5}+3a^{4}-5a^{2}-4a$, $a^{23}-a^{22}+2a^{20}-2a^{19}+a^{18}-a^{17}-a^{16}+2a^{15}-a^{14}-a^{13}+2a^{12}-2a^{11}+3a^{10}-2a^{8}+2a^{7}-a^{6}-a^{5}+2a^{4}-5a^{3}+2a^{2}+a-1$, $a^{23}-a^{20}+2a^{18}-a^{17}-a^{15}+a^{14}+a^{13}-2a^{11}+a^{9}+2a^{8}-a^{7}-3a^{6}+a^{5}+4a^{3}-2a^{2}-3a$, $2a^{23}-a^{22}+2a^{21}+2a^{20}+a^{19}+3a^{18}+6a^{16}+a^{15}+3a^{14}+3a^{13}+2a^{12}+6a^{11}-2a^{10}+5a^{9}+a^{8}+2a^{7}+a^{6}-2a^{5}+5a^{4}-a^{3}+a+1$, $a^{24}+2a^{22}-2a^{21}-2a^{19}+2a^{16}+2a^{15}-3a^{12}-a^{11}-2a^{10}+3a^{9}+5a^{7}-2a^{5}-2a^{4}-3a^{3}-a^{2}+a+1$, $a^{24}+a^{23}+2a^{22}+a^{21}+2a^{20}+a^{19}+a^{18}+2a^{17}+a^{16}+2a^{15}+a^{14}-a^{12}-2a^{11}-3a^{10}-2a^{9}-3a^{8}-2a^{7}-2a^{6}-4a^{5}-2a^{4}-4a^{3}-2a^{2}-1$, $2a^{24}+a^{23}-3a^{22}+a^{21}+3a^{20}-4a^{19}+a^{18}+4a^{17}-4a^{16}+2a^{15}+4a^{14}-5a^{13}+2a^{12}+5a^{11}-5a^{10}+6a^{8}-7a^{7}+a^{6}+6a^{5}-10a^{4}+a^{3}+6a^{2}-9a-5$ (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: | \( 6280286129297.773 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 6280286129297.773 \cdot 1}{2\cdot\sqrt{1130059131643038744233908476350526800370111143}}\cr\approx \mathstrut & 0.450264170230811 \end{aligned}\] (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 | ${\href{/padicField/2.11.0.1}{11} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(131\) | $\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{131}$ | $x + 129$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
131.2.1.2 | $x^{2} + 131$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
131.3.0.1 | $x^{3} + 3 x + 129$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
131.18.0.1 | $x^{18} - x + 126$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(13259\) | $\Q_{13259}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{13259}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(3933670000864773133\) | $\Q_{3933670000864773133}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3933670000864773133}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(165\!\cdots\!099\) | $\Q_{16\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{16\!\cdots\!99}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |