Normalized defining polynomial
\( x^{25} - 2x - 5 \)
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: | \(5293955920294624372746725596413256021078201181350393\) \(\medspace = 48407\cdot 10\!\cdots\!99\) | 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: | \(117.21\) | 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: | $48407^{1/2}109363437525453433857638886863744004401805548399^{1/2}\approx 7.275957614152672e+25$ | ||
Ramified primes: | \(48407\), \(10936\!\cdots\!48399\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{52939\!\cdots\!50393}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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)
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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: | $3a^{24}+10a^{23}-8a^{22}-7a^{21}+3a^{20}+14a^{19}-7a^{18}-14a^{17}+7a^{16}+13a^{15}-25a^{13}+12a^{12}+10a^{11}+12a^{10}-38a^{9}+11a^{8}+16a^{7}+18a^{6}-46a^{5}+3a^{4}+28a^{3}+20a^{2}-48a-16$, $3a^{23}+5a^{22}+3a^{21}-4a^{20}-7a^{19}-a^{18}+2a^{17}-2a^{16}-4a^{15}-3a^{14}+3a^{13}+13a^{12}+9a^{11}-6a^{10}-7a^{9}+2a^{8}+2a^{7}-8a^{6}-19a^{5}-18a^{4}+a^{3}+19a^{2}+13a-1$, $a^{24}+4a^{23}-5a^{22}+4a^{21}+a^{20}-8a^{19}+8a^{18}-4a^{17}-2a^{16}+8a^{15}-7a^{14}+a^{13}+8a^{12}-12a^{11}+6a^{10}+a^{9}-11a^{8}+15a^{7}-3a^{6}-8a^{5}+16a^{4}-16a^{3}+a^{2}+12a-23$, $6a^{24}+17a^{23}-19a^{22}-a^{21}+17a^{20}-19a^{19}-2a^{18}+27a^{17}-16a^{16}-14a^{15}+25a^{14}-15a^{13}-18a^{12}+41a^{11}-6a^{10}-38a^{9}+36a^{8}-3a^{7}-45a^{6}+55a^{5}+15a^{4}-67a^{3}+41a^{2}+14a-86$, $3a^{24}+3a^{23}+3a^{22}+4a^{21}+5a^{20}+4a^{19}+4a^{18}+7a^{17}+8a^{16}+6a^{15}+7a^{14}+10a^{13}+9a^{12}+7a^{11}+8a^{10}+10a^{9}+12a^{8}+11a^{7}+11a^{6}+16a^{5}+20a^{4}+18a^{3}+18a^{2}+23a+18$, $6a^{24}-10a^{23}+14a^{22}-16a^{21}+20a^{20}-23a^{19}+22a^{18}-24a^{17}+23a^{16}-24a^{15}+25a^{14}-25a^{13}+33a^{12}-30a^{11}+33a^{10}-37a^{9}+27a^{8}-27a^{7}+19a^{6}-12a^{5}+7a^{4}+a^{3}+2a^{2}+6a-23$, $2a^{24}+8a^{23}-5a^{22}+3a^{21}-a^{20}-6a^{19}+5a^{17}-5a^{16}-2a^{15}+15a^{14}-5a^{13}-a^{11}-12a^{10}-6a^{9}+16a^{8}+5a^{7}-10a^{6}+18a^{5}-8a^{4}-9a^{3}+3a^{2}-11a-23$, $8a^{24}+14a^{23}+21a^{22}+14a^{21}+3a^{20}+8a^{19}+17a^{18}+a^{17}-23a^{16}-21a^{15}-2a^{14}-16a^{13}-54a^{12}-57a^{11}-21a^{10}-11a^{9}-48a^{8}-56a^{7}-8a^{6}+30a^{5}+10a^{4}+51a^{2}+96a+67$, $14a^{24}+4a^{23}-11a^{22}-17a^{21}+4a^{20}+23a^{19}+11a^{18}-11a^{17}-25a^{16}+a^{15}+26a^{14}+15a^{13}-10a^{12}-28a^{11}+2a^{10}+24a^{9}+7a^{8}-21a^{7}-28a^{6}+18a^{5}+35a^{4}-7a^{3}-52a^{2}-40a+19$, $12a^{24}+11a^{23}+17a^{22}+14a^{21}+13a^{20}+13a^{19}+9a^{18}+7a^{17}+2a^{16}+4a^{15}-2a^{14}-6a^{13}-8a^{12}-15a^{11}-23a^{10}-35a^{9}-37a^{8}-48a^{7}-53a^{6}-45a^{5}-38a^{4}-18a^{3}-6a^{2}+39a+24$, $13a^{24}+15a^{23}+8a^{22}-9a^{21}-19a^{20}-12a^{19}+13a^{17}+5a^{16}-10a^{15}-18a^{14}-5a^{13}+21a^{12}+40a^{11}+30a^{10}-9a^{9}-38a^{8}-46a^{7}-10a^{6}+23a^{5}+23a^{4}-8a^{3}-45a^{2}-31a+4$, $21a^{24}+38a^{23}+33a^{22}+11a^{21}+6a^{20}+25a^{19}+39a^{18}+21a^{17}-17a^{16}-29a^{15}-5a^{14}+11a^{13}-18a^{12}-67a^{11}-74a^{10}-28a^{9}+9a^{8}-18a^{7}-67a^{6}-50a^{5}+38a^{4}+109a^{3}+84a^{2}+25a+18$ (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: | \( 17416422916825532 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 17416422916825532 \cdot 1}{2\cdot\sqrt{5293955920294624372746725596413256021078201181350393}}\cr\approx \mathstrut & 0.906206602041866 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ is not computed |
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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.8.0.1}{8} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $21{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(48407\) | $\Q_{48407}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(109\!\cdots\!399\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |