Normalized defining polynomial
\( x^{26} - 2x - 4 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1732793616956939114861181647068785799342044590637056\) \(\medspace = 2^{24}\cdot 71593\cdot 343928521\cdot 41\!\cdots\!97\) | 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: | \(93.48\) | 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: | $2^{24/25}71593^{1/2}343928521^{1/2}4194577183575531075991971526697^{1/2}\approx 1.9769799626305397e+22$ | ||
Ramified primes: | \(2\), \(71593\), \(343928521\), \(41945\!\cdots\!26697\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{10328\!\cdots\!33241}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{2}a^{25}$
Monogenic: | Not computed | |
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)
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Fundamental units: | $a+1$, $a^{25}+2a^{12}-1$, $a^{25}+a^{24}-a^{22}+a^{20}-2a^{18}-a^{17}+a^{16}-3a^{14}-2a^{13}+a^{12}+a^{11}-2a^{10}-2a^{9}+a^{8}+3a^{7}-2a^{5}+2a^{4}+6a^{3}+2a^{2}-2a-1$, $a^{25}-a^{23}+2a^{21}-a^{20}-2a^{19}+2a^{18}+a^{17}-3a^{16}-a^{15}+4a^{14}-2a^{13}-3a^{12}+3a^{11}+2a^{10}-3a^{9}-a^{8}+4a^{7}-a^{6}-3a^{5}+2a^{3}-a^{2}-4a-1$, $4a^{25}-3a^{24}+2a^{23}-a^{22}+a^{20}-2a^{19}+4a^{18}-4a^{17}+5a^{16}-6a^{15}+6a^{14}-7a^{13}+6a^{12}-4a^{11}+3a^{10}-2a^{9}+2a^{8}-2a^{6}+6a^{5}-7a^{4}+7a^{3}-7a^{2}+7a-17$, $a^{25}+2a^{24}-2a^{23}-2a^{22}+3a^{21}+2a^{20}-4a^{19}+4a^{17}-a^{16}-6a^{15}+3a^{14}+6a^{13}-4a^{12}-6a^{11}+5a^{10}+5a^{9}-7a^{8}-2a^{7}+9a^{6}+a^{5}-10a^{4}+9a^{2}-4a-9$, $a^{25}+a^{24}-a^{22}+a^{20}-a^{19}-a^{18}+2a^{17}+a^{16}-2a^{15}-2a^{14}+3a^{13}+a^{12}-4a^{11}+2a^{10}+4a^{9}-5a^{8}-3a^{7}+5a^{6}+3a^{5}-5a^{4}+5a^{2}-5a-7$, $a^{25}-5a^{24}-a^{23}-6a^{22}-11a^{21}-6a^{20}-13a^{19}-14a^{18}-8a^{17}-14a^{16}-12a^{15}-3a^{14}-9a^{13}+9a^{11}+3a^{10}+16a^{9}+22a^{8}+17a^{7}+30a^{6}+31a^{5}+19a^{4}+33a^{3}+21a^{2}+9a+13$, $2a^{25}-2a^{24}+4a^{23}-4a^{22}+2a^{21}-2a^{20}-a^{19}+4a^{18}-3a^{17}+4a^{16}-3a^{15}-a^{14}+2a^{12}-2a^{11}+7a^{10}-8a^{9}+3a^{8}-4a^{7}-a^{6}+7a^{5}-3a^{4}+5a^{3}-5a^{2}-2a-5$, $2a^{24}+2a^{23}-a^{21}+a^{20}-a^{19}+2a^{17}+3a^{16}+a^{15}+2a^{14}+a^{13}-a^{12}+a^{9}+3a^{8}+5a^{7}+2a^{6}+4a^{5}+2a^{4}-3a^{3}-3a^{2}+4a+3$, $3a^{25}+3a^{24}+5a^{23}+6a^{22}+6a^{21}+7a^{20}+7a^{19}+6a^{18}+5a^{17}+4a^{16}+a^{15}-2a^{14}-2a^{13}-6a^{12}-11a^{11}-12a^{10}-14a^{9}-17a^{8}-16a^{7}-14a^{6}-16a^{5}-14a^{4}-7a^{3}-2a^{2}+3a+5$, $5a^{25}-9a^{24}+17a^{23}-13a^{22}+14a^{21}-13a^{20}-8a^{19}+14a^{18}-12a^{17}+23a^{16}-17a^{15}+8a^{14}-11a^{13}-10a^{12}+19a^{11}-9a^{10}+28a^{9}-32a^{8}+8a^{7}-8a^{6}-15a^{5}+36a^{4}-14a^{3}+27a^{2}-40a-15$, $1156a^{25}+97a^{24}-1033a^{23}-1104a^{22}+696a^{21}+1408a^{20}+601a^{19}-1509a^{18}-1453a^{17}+344a^{16}+2119a^{15}+990a^{14}-1655a^{13}-2287a^{12}+96a^{11}+2931a^{10}+1675a^{9}-1659a^{8}-3541a^{7}-165a^{6}+3344a^{5}+3116a^{4}-2008a^{3}-4673a^{2}-1434a+2107$ (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: | \( 6568002949534731.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 6568002949534731.0 \cdot 1}{2\cdot\sqrt{1732793616956939114861181647068785799342044590637056}}\cr\approx \mathstrut & 1.19467103798012 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 403291461126605635584000000 |
The 2436 conjugacy class representatives for $S_{26}$ are not computed |
Character table for $S_{26}$ 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 | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $23{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | $25{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 $25$ | $25$ | $1$ | $24$ | ||||
\(71593\) | $\Q_{71593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{71593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{71593}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(343928521\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(419\!\cdots\!697\) | $\Q_{41\!\cdots\!97}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |