Normalized defining polynomial
\( x^{25} + 2x - 2 \)
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: | \(1534868865815055657834506241742996328415232\) \(\medspace = 2^{24}\cdot 11\cdot 83\cdot 139\cdot 177347\cdot 386150417\cdot 10526535234089689\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.69\) | 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: | $2^{24/25}11^{1/2}83^{1/2}139^{1/2}177347^{1/2}386150417^{1/2}10526535234089689^{1/2}\approx 5.883889204334063e+17$ | ||
Ramified primes: | \(2\), \(11\), \(83\), \(139\), \(177347\), \(386150417\), \(10526535234089689\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{91485\!\cdots\!53177}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a-1$, $a^{17}-a^{9}+a-1$, $a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{4}+a^{2}+3$, $a^{19}-a^{13}+a-1$, $a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{17}+a^{16}+a^{12}+a^{9}+a^{6}+a^{5}-a^{3}+a^{2}+a+1$, $a^{23}+a^{20}+a^{18}+a^{15}+a^{10}-a^{9}+a^{7}-a^{6}+2a^{5}-2a^{4}+a^{3}+a^{2}-a+1$, $a^{24}+2a^{23}+2a^{22}+a^{21}+a^{20}+a^{17}-2a^{13}-a^{12}-a^{10}-a^{8}-2a^{7}-a^{5}-a^{4}+a^{3}-a^{2}+3$, $a^{24}-a^{22}-a^{21}+a^{19}-a^{17}-a^{16}+a^{14}-a^{12}-a^{11}+a^{9}-a^{7}-a^{6}+2a^{4}+a^{3}-a^{2}-2a+1$, $a^{24}+a^{23}+a^{22}-a^{19}-2a^{18}-a^{17}-a^{16}-a^{15}-a^{14}+a^{12}+a^{10}+2a^{9}+a^{8}-a^{6}-a^{4}-2a^{3}-a+1$, $a^{24}-2a^{22}-a^{21}+a^{19}+3a^{18}+2a^{17}-2a^{14}-a^{13}+a^{12}+a^{10}+a^{9}-a^{8}+a^{7}+a^{6}-a^{5}+a^{4}-2a^{3}-2a^{2}+a+3$, $2a^{23}+a^{21}+a^{20}+a^{18}-a^{17}+a^{16}-a^{15}+2a^{10}-2a^{9}+4a^{8}-2a^{7}+2a^{6}-a^{5}+a^{4}-a^{3}-a^{2}+1$, $a^{24}+a^{22}-2a^{21}+a^{20}-2a^{19}+2a^{18}-a^{17}+a^{16}-a^{15}-a^{14}-a^{12}+2a^{11}+a^{9}-a^{8}-2a^{7}+a^{6}-a^{5}+4a^{4}-2a^{3}+a^{2}-3a+1$ (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: | \( 198284213067.61386 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 198284213067.61386 \cdot 1}{2\cdot\sqrt{1534868865815055657834506241742996328415232}}\cr\approx \mathstrut & 0.605913906192416 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $17{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $23{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $25$ | $20{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $25$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $25$ | $25$ | $1$ | $24$ | |||
\(11\) | 11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.8.0.1 | $x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
11.15.0.1 | $x^{15} + 10 x^{6} + 7 x^{5} + 5 x^{3} + 9$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | |
\(83\) | $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
83.8.0.1 | $x^{8} + x^{4} + 65 x^{3} + 23 x^{2} + 42 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
83.10.0.1 | $x^{10} + 7 x^{5} + 73 x^{3} + 53 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(139\) | 139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.7.0.1 | $x^{7} + 9 x + 137$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
139.14.0.1 | $x^{14} - 2 x + 18$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
\(177347\) | $\Q_{177347}$ | $x$ | $1$ | $1$ | $0$ | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(386150417\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(10526535234089689\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |