Normalized defining polynomial
\( x^{25} + 2x - 4 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(6250000011188186607572508208626560435749082103808\) \(\medspace = 2^{46}\cdot 88\!\cdots\!97\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(89.50\) | 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))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(88817\!\cdots\!48697\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{88817\!\cdots\!48697}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{2}a^{24}$
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)
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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)
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Fundamental units: | $a-1$, $a^{24}+a^{23}-a^{22}-a^{21}+2a^{20}+3a^{19}+a^{18}+3a^{16}+5a^{15}+a^{14}-2a^{13}+2a^{11}-a^{10}-6a^{9}-4a^{8}+a^{7}-5a^{5}-5a^{4}+3a^{3}+4a^{2}-4a-3$, $a^{24}+4a^{23}+2a^{22}+4a^{21}+a^{20}-4a^{18}-4a^{17}-2a^{16}-7a^{15}-3a^{14}-a^{13}+7a^{12}+a^{11}+4a^{10}+10a^{9}+7a^{8}+3a^{7}-6a^{6}+4a^{5}-8a^{4}-8a^{3}-10a^{2}-3a+3$, $2a^{24}+3a^{23}-3a^{22}+3a^{21}-2a^{20}+2a^{19}+3a^{18}-5a^{17}+7a^{16}-3a^{15}+a^{14}+3a^{13}-7a^{12}+12a^{11}-5a^{10}+4a^{8}-7a^{7}+14a^{6}-10a^{5}+2a^{4}+9a^{3}-10a^{2}+14a-9$, $5a^{24}-3a^{23}-7a^{22}+3a^{21}+6a^{20}-2a^{19}-a^{18}+7a^{17}-13a^{16}-3a^{15}+16a^{14}-7a^{13}+a^{12}+10a^{11}-13a^{10}-9a^{9}+13a^{8}+6a^{7}-10a^{6}+12a^{5}+3a^{4}-35a^{3}+19a^{2}+18a-17$, $a^{24}+2a^{23}-a^{22}-2a^{21}+2a^{20}+3a^{19}-a^{18}-3a^{17}+a^{16}+3a^{15}-2a^{14}-6a^{13}-a^{12}+6a^{11}+a^{10}-7a^{9}-3a^{8}+9a^{7}+6a^{6}-6a^{5}-8a^{4}+5a^{3}+8a^{2}-4a-11$, $4a^{24}+7a^{23}+8a^{22}+8a^{21}+9a^{20}+8a^{19}+3a^{18}+2a^{17}+4a^{16}+3a^{15}-2a^{14}-4a^{13}-6a^{12}-11a^{11}-13a^{10}-7a^{9}-7a^{8}-12a^{7}-13a^{6}-8a^{5}-8a^{4}-7a^{3}+4a^{2}+10a+11$, $3a^{24}+a^{23}+2a^{22}+2a^{21}-a^{20}+2a^{19}-2a^{18}-a^{17}-5a^{15}+2a^{14}-6a^{13}-3a^{11}-5a^{10}+a^{9}-5a^{8}+a^{7}-a^{5}+5a^{4}+2a^{2}+7a+3$, $26a^{24}+27a^{23}+22a^{22}+7a^{21}-4a^{20}-22a^{19}-31a^{18}-34a^{17}-33a^{16}-17a^{15}-2a^{14}+16a^{13}+37a^{12}+38a^{11}+47a^{10}+28a^{9}+13a^{8}-9a^{7}-37a^{6}-47a^{5}-55a^{4}-51a^{3}-21a^{2}-10a+89$, $6a^{24}-14a^{23}+4a^{22}-11a^{21}+14a^{20}-6a^{19}+14a^{18}-12a^{17}+5a^{16}-15a^{15}+10a^{14}+16a^{12}-3a^{11}-10a^{10}-13a^{9}-9a^{8}+20a^{7}+5a^{6}+21a^{5}-35a^{4}+10a^{3}-36a^{2}+53a-11$, $a^{24}-6a^{23}+4a^{22}-5a^{21}+4a^{20}-10a^{19}+8a^{18}-9a^{17}+11a^{16}-5a^{15}+13a^{14}-12a^{13}+13a^{12}-8a^{11}+13a^{10}-3a^{9}+13a^{8}-10a^{7}+5a^{6}-2a^{5}-3a^{4}+4a^{3}-2a^{2}+4a-21$, $20a^{24}+47a^{23}+67a^{22}+72a^{21}+57a^{20}+29a^{19}-9a^{18}-48a^{17}-74a^{16}-86a^{15}-81a^{14}-58a^{13}-14a^{12}+39a^{11}+82a^{10}+104a^{9}+107a^{8}+87a^{7}+40a^{6}-13a^{5}-73a^{4}-124a^{3}-144a^{2}-124a-33$ (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)]
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Regulator: | \( 647777435547311.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 647777435547311.2 \cdot 1}{2\cdot\sqrt{6250000011188186607572508208626560435749082103808}}\cr\approx \mathstrut & 0.980944027036443 \end{aligned}\] (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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 | $[\ ]$ |
Deg $24$ | $24$ | $1$ | $46$ | ||||
\(888\!\cdots\!697\) | $\Q_{88\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{88\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | 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 $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ |