Normalized defining polynomial
\( x^{25} - 4x - 1 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1501652986908320829181236929617697887525122642599\) \(\medspace = -\,7\cdot 53\cdot 3847\cdot 1012717\cdot 140141898521\cdot 847814229733\cdot 8744131694867\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(84.54\) | 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: | $7^{1/2}53^{1/2}3847^{1/2}1012717^{1/2}140141898521^{1/2}847814229733^{1/2}8744131694867^{1/2}\approx 1.2254195146594985e+24$ | ||
Ramified primes: | \(7\), \(53\), \(3847\), \(1012717\), \(140141898521\), \(847814229733\), \(8744131694867\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-15016\!\cdots\!42599}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a$, $a^{13}-2a$, $a^{24}+a^{23}-a^{21}-a^{20}+a^{18}+a^{17}-a^{15}-a^{14}+a^{12}+a^{11}-a^{9}-2a^{8}-a^{7}+2a^{6}+3a^{5}+a^{4}-2a^{3}-4a^{2}-2a-1$, $a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}-a^{16}-a^{15}-3a^{14}-3a^{13}-4a^{12}-4a^{11}-5a^{10}-5a^{9}-4a^{8}-4a^{7}-3a^{6}-3a^{5}+2a^{2}+2a+1$, $a^{24}+4a^{23}+3a^{22}-3a^{21}-5a^{20}-a^{19}+3a^{18}+4a^{17}+a^{16}-a^{15}-4a^{14}-5a^{13}+a^{12}+8a^{11}+7a^{10}-5a^{9}-11a^{8}-3a^{7}+6a^{6}+8a^{5}+3a^{4}-a^{3}-8a^{2}-12a-3$, $a^{24}+a^{23}+a^{22}+a^{21}-a^{16}-2a^{15}-2a^{14}-a^{13}+a^{11}+3a^{10}+3a^{9}+3a^{8}+a^{7}-2a^{5}-5a^{4}-6a^{3}-4a^{2}-1$, $2a^{24}+2a^{23}+3a^{22}+a^{21}-2a^{20}-a^{19}-a^{18}-4a^{17}-5a^{16}-2a^{15}+3a^{14}+4a^{13}+4a^{11}+8a^{10}+4a^{9}-2a^{8}-7a^{7}-4a^{6}-4a^{5}-10a^{4}-5a^{3}+2a^{2}+5a$, $a^{24}-12a^{23}+20a^{22}-24a^{21}+20a^{20}-13a^{19}-3a^{18}+14a^{17}-28a^{16}+30a^{15}-28a^{14}+14a^{13}+a^{12}-21a^{11}+35a^{10}-37a^{9}+35a^{8}-14a^{7}-5a^{6}+34a^{5}-44a^{4}+57a^{3}-39a^{2}+20a+10$, $a^{23}-3a^{22}+10a^{21}-9a^{20}+2a^{18}+7a^{16}-11a^{15}+4a^{14}-6a^{13}+10a^{12}+5a^{11}-17a^{10}+11a^{9}-9a^{8}+12a^{7}+a^{6}-15a^{5}+5a^{4}-3a^{3}+23a^{2}-16a-7$, $4a^{24}-2a^{23}-a^{22}+2a^{21}+a^{20}-3a^{19}+2a^{18}+2a^{17}-6a^{16}+4a^{15}+3a^{14}-10a^{13}+11a^{12}-5a^{11}-3a^{10}+6a^{9}-4a^{8}-3a^{6}+10a^{5}-14a^{4}+11a^{3}+3a^{2}-14a-3$, $4a^{24}-2a^{23}-a^{22}-2a^{21}+7a^{20}-2a^{19}-3a^{18}-4a^{17}+10a^{16}-4a^{14}-8a^{13}+11a^{12}+3a^{11}-2a^{10}-11a^{9}+9a^{8}+3a^{7}+a^{6}-10a^{5}+9a^{4}-a^{3}-7a-1$, $21a^{24}+28a^{23}+21a^{22}+17a^{21}+30a^{20}+35a^{19}+29a^{18}+32a^{17}+39a^{16}+33a^{15}+31a^{14}+52a^{13}+57a^{12}+41a^{11}+47a^{10}+62a^{9}+53a^{8}+59a^{7}+86a^{6}+82a^{5}+58a^{4}+75a^{3}+104a^{2}+91a+16$, $3a^{23}+4a^{22}+5a^{21}+6a^{20}+6a^{19}+5a^{18}+2a^{17}-2a^{16}-4a^{15}-3a^{14}+5a^{12}+8a^{11}+7a^{10}+5a^{9}+3a^{8}-a^{7}-6a^{6}-14a^{5}-25a^{4}-31a^{3}-27a^{2}-16a-1$ (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: | \( 323817690787630.7 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{11}\cdot 323817690787630.7 \cdot 1}{2\cdot\sqrt{1501652986908320829181236929617697887525122642599}}\cr\approx \mathstrut & 0.636875177723298 \end{aligned}\] (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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $20{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | R | $18{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.23.0.1 | $x^{23} + 4 x^{2} + 4 x + 4$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(53\) | $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
53.2.1.2 | $x^{2} + 106$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
53.20.0.1 | $x^{20} - x + 33$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(3847\) | $\Q_{3847}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(1012717\) | $\Q_{1012717}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1012717}$ | $x$ | $1$ | $1$ | $0$ | 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(140141898521\) | 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 $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(847814229733\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(8744131694867\) | $\Q_{8744131694867}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |