Normalized defining polynomial
\( x^{26} - x - 3 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5216009998678636041582835540850834092978325122393\) \(\medspace = 2083\cdot 1519447\cdot 16\!\cdots\!93\) | 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: | \(74.77\) | 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: | $2083^{1/2}1519447^{1/2}1648024217388452125665771097137180469093^{1/2}\approx 2.2838585767684118e+24$ | ||
Ramified primes: | \(2083\), \(1519447\), \(16480\!\cdots\!69093\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{52160\!\cdots\!22393}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
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)
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Fundamental units: | $a+1$, $a^{13}-2$, $a^{23}-a^{22}-a^{18}+a^{17}-a^{16}-a^{15}-a^{14}-2a^{12}-a^{10}-a^{9}-2a^{8}-2a^{6}-2a^{2}+a-1$, $a^{25}-a^{23}-a^{22}+2a^{21}-a^{19}+a^{17}-2a^{15}+a^{14}+a^{13}-a^{11}+a^{10}-3a^{8}+a^{7}+4a^{6}-4a^{4}-a^{3}+3a^{2}-a-1$, $a^{25}+a^{24}-a^{22}-2a^{20}+2a^{17}+2a^{16}-a^{15}+2a^{14}-2a^{13}-2a^{12}-2a^{11}+a^{10}+2a^{9}+4a^{7}+a^{6}-a^{5}-4a^{4}-2a^{3}+a^{2}-3a+4$, $a^{25}-a^{24}-a^{23}+a^{21}-a^{19}-2a^{18}+a^{16}+a^{15}+a^{14}-2a^{13}+a^{11}+2a^{10}-a^{9}-2a^{8}-a^{7}+3a^{6}+2a^{5}+a^{4}-2a^{3}-2a^{2}+a+1$, $a^{25}-a^{24}-3a^{23}+a^{22}+4a^{21}-a^{20}-3a^{19}+2a^{18}+3a^{17}-2a^{16}-3a^{15}+2a^{13}+3a^{12}-a^{11}-4a^{10}+3a^{9}+5a^{8}-5a^{7}-7a^{6}+4a^{5}+6a^{4}-2a^{3}-3a^{2}+a+2$, $a^{25}-6a^{24}-2a^{23}+8a^{22}+6a^{21}-6a^{20}-5a^{19}+6a^{18}+10a^{17}-5a^{16}-10a^{15}+a^{14}+12a^{13}-a^{12}-14a^{11}-5a^{10}+11a^{9}+5a^{8}-16a^{7}-11a^{6}+9a^{5}+14a^{4}-12a^{3}-15a^{2}+7a+22$, $2a^{25}+3a^{24}+a^{23}-4a^{22}-6a^{21}-4a^{20}+2a^{19}+5a^{18}+2a^{17}-4a^{16}-8a^{15}-5a^{14}+5a^{12}+3a^{11}-2a^{10}-8a^{9}-8a^{8}-4a^{7}+2a^{6}+5a^{5}+3a^{4}-5a^{3}-13a^{2}-13a-5$, $a^{24}+2a^{22}-3a^{21}-a^{20}-a^{18}+3a^{17}+a^{16}-a^{15}-a^{14}-5a^{13}+2a^{12}+a^{11}+2a^{9}-4a^{8}-3a^{7}-a^{6}-a^{5}+7a^{4}-a^{3}-2a^{2}-2a-7$, $6a^{25}+7a^{24}+a^{23}-7a^{22}-10a^{21}-3a^{20}+8a^{19}+13a^{18}+5a^{17}-8a^{16}-15a^{15}-7a^{14}+7a^{13}+16a^{12}+9a^{11}-5a^{10}-16a^{9}-12a^{8}+a^{7}+15a^{6}+16a^{5}+5a^{4}-12a^{3}-20a^{2}-13a+2$, $5a^{25}-4a^{24}-6a^{23}+4a^{22}+5a^{21}-5a^{20}-2a^{19}+8a^{18}+2a^{17}-9a^{16}-a^{15}+7a^{14}-3a^{13}-7a^{12}+7a^{11}+9a^{10}-7a^{9}-5a^{8}+8a^{7}-14a^{5}+2a^{4}+16a^{3}-2a^{2}-11a+5$, $a^{25}+2a^{24}-a^{23}-a^{21}-3a^{20}-3a^{19}-2a^{17}+3a^{15}+5a^{14}+a^{13}+6a^{12}+3a^{11}-3a^{9}-2a^{8}-7a^{7}-8a^{6}-5a^{5}-4a^{4}-4a^{3}+4a^{2}+2a+5$ (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: | \( 221401864429738.72 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 221401864429738.72 \cdot 1}{2\cdot\sqrt{5216009998678636041582835540850834092978325122393}}\cr\approx \mathstrut & 0.734007473969973 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 403291461126605635584000000 |
The 2436 conjugacy class representatives for $S_{26}$ |
Character table for $S_{26}$ |
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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/11.8.0.1}{8} }$ | $25{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | $24{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2083\) | $\Q_{2083}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2083}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{2083}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(1519447\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(164\!\cdots\!093\) | $\Q_{16\!\cdots\!93}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |