Normalized defining polynomial
\( x^{26} - 4x + 4 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1201507202980564696903888106396747076468736\) \(\medspace = -\,2^{26}\cdot 11\cdot 5351\cdot 322710411593\cdot 942553212724915163\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(41.54\) | 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: | not computed | ||
Ramified primes: | \(2\), \(11\), \(5351\), \(322710411593\), \(942553212724915163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-17903\!\cdots\!13399}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{18}a^{25}-\frac{2}{9}a^{24}-\frac{1}{9}a^{23}-\frac{1}{18}a^{22}+\frac{2}{9}a^{21}+\frac{1}{9}a^{20}+\frac{1}{18}a^{19}-\frac{2}{9}a^{18}-\frac{1}{9}a^{17}-\frac{1}{18}a^{16}+\frac{2}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{18}a^{13}-\frac{2}{9}a^{12}-\frac{1}{9}a^{11}+\frac{4}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{5}+\frac{4}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$
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)
| |
Fundamental units: | $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{13}-a$, $\frac{2}{9}a^{25}+\frac{1}{9}a^{24}+\frac{1}{18}a^{23}+\frac{5}{18}a^{22}+\frac{7}{18}a^{21}+\frac{4}{9}a^{20}+\frac{2}{9}a^{19}+\frac{1}{9}a^{18}+\frac{1}{18}a^{17}+\frac{5}{18}a^{16}+\frac{7}{18}a^{15}+\frac{4}{9}a^{14}+\frac{2}{9}a^{13}+\frac{1}{9}a^{12}-\frac{4}{9}a^{11}-\frac{2}{9}a^{10}-\frac{1}{9}a^{9}+\frac{4}{9}a^{8}+\frac{2}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a-\frac{7}{9}$, $\frac{1}{3}a^{25}+\frac{1}{6}a^{24}+\frac{1}{3}a^{23}+\frac{1}{6}a^{22}+\frac{1}{3}a^{21}+\frac{1}{6}a^{20}+\frac{1}{3}a^{19}+\frac{1}{6}a^{18}+\frac{1}{3}a^{17}+\frac{1}{6}a^{16}+\frac{1}{3}a^{15}+\frac{1}{6}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{5}{3}$, $\frac{1}{6}a^{25}+\frac{1}{3}a^{24}+\frac{1}{6}a^{23}+\frac{1}{3}a^{22}+\frac{1}{6}a^{21}-\frac{1}{6}a^{20}+\frac{1}{6}a^{19}-\frac{1}{6}a^{18}+\frac{1}{6}a^{17}-\frac{1}{6}a^{16}+\frac{1}{6}a^{15}-\frac{1}{6}a^{14}+\frac{1}{6}a^{13}-\frac{2}{3}a^{12}+\frac{2}{3}a^{11}-\frac{2}{3}a^{10}+\frac{2}{3}a^{9}-\frac{2}{3}a^{8}+\frac{2}{3}a^{7}-\frac{2}{3}a^{6}+\frac{2}{3}a^{5}-\frac{2}{3}a^{4}+\frac{2}{3}a^{3}-\frac{2}{3}a^{2}+\frac{2}{3}a-\frac{4}{3}$, $\frac{28}{9}a^{25}+\frac{55}{18}a^{24}+\frac{59}{18}a^{23}+\frac{26}{9}a^{22}+\frac{53}{18}a^{21}+\frac{49}{18}a^{20}+\frac{47}{18}a^{19}+\frac{23}{9}a^{18}+\frac{41}{18}a^{17}+\frac{43}{18}a^{16}+\frac{35}{18}a^{15}+\frac{20}{9}a^{14}+\frac{29}{18}a^{13}+\frac{14}{9}a^{12}+\frac{16}{9}a^{11}+\frac{8}{9}a^{10}+\frac{13}{9}a^{9}+\frac{11}{9}a^{8}+\frac{1}{9}a^{7}+\frac{14}{9}a^{6}-\frac{2}{9}a^{5}+\frac{8}{9}a^{4}+\frac{4}{9}a^{3}-\frac{7}{9}a^{2}+\frac{10}{9}a-\frac{116}{9}$, $\frac{1}{18}a^{25}+\frac{5}{18}a^{24}+\frac{7}{18}a^{23}-\frac{1}{18}a^{22}-\frac{5}{18}a^{21}+\frac{1}{9}a^{20}+\frac{1}{18}a^{19}-\frac{2}{9}a^{18}-\frac{1}{9}a^{17}-\frac{1}{18}a^{16}-\frac{5}{18}a^{15}+\frac{1}{9}a^{14}+\frac{1}{18}a^{13}-\frac{2}{9}a^{12}-\frac{1}{9}a^{11}+\frac{4}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{2}{9}a^{6}+\frac{8}{9}a^{5}+\frac{4}{9}a^{4}-\frac{7}{9}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{21}+\frac{1}{2}a^{18}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+a^{7}-a^{6}-a^{2}$, $\frac{4}{9}a^{25}+\frac{2}{9}a^{24}+\frac{11}{18}a^{23}+\frac{1}{18}a^{22}-\frac{2}{9}a^{21}-\frac{1}{9}a^{20}-\frac{5}{9}a^{19}+\frac{2}{9}a^{18}+\frac{1}{9}a^{17}+\frac{1}{18}a^{16}-\frac{2}{9}a^{15}-\frac{1}{9}a^{14}-\frac{5}{9}a^{13}+\frac{2}{9}a^{12}+\frac{1}{9}a^{11}-\frac{4}{9}a^{10}+\frac{7}{9}a^{9}-\frac{10}{9}a^{8}+\frac{4}{9}a^{7}+\frac{2}{9}a^{6}+\frac{1}{9}a^{5}+\frac{5}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a-\frac{5}{9}$, $\frac{1}{6}a^{25}-\frac{1}{6}a^{24}-\frac{1}{3}a^{23}-\frac{1}{6}a^{22}+\frac{1}{6}a^{21}+\frac{1}{3}a^{20}+\frac{1}{6}a^{19}+\frac{1}{3}a^{18}+\frac{1}{6}a^{17}+\frac{5}{6}a^{16}+\frac{1}{6}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{4}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{4}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{4}{3}a^{3}+\frac{4}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{10}{9}a^{25}+\frac{5}{9}a^{24}+\frac{5}{18}a^{23}+\frac{7}{18}a^{22}+\frac{4}{9}a^{21}+\frac{2}{9}a^{20}+\frac{1}{9}a^{19}+\frac{5}{9}a^{18}-\frac{2}{9}a^{17}-\frac{1}{9}a^{16}+\frac{4}{9}a^{15}+\frac{13}{18}a^{14}-\frac{8}{9}a^{13}-\frac{4}{9}a^{12}+\frac{7}{9}a^{11}-\frac{1}{9}a^{10}-\frac{5}{9}a^{9}-\frac{7}{9}a^{8}+\frac{10}{9}a^{7}-\frac{13}{9}a^{6}-\frac{2}{9}a^{5}+\frac{8}{9}a^{4}-\frac{5}{9}a^{3}-\frac{7}{9}a^{2}-\frac{8}{9}a-\frac{17}{9}$, $\frac{1}{9}a^{25}+\frac{5}{9}a^{24}+\frac{5}{18}a^{23}-\frac{1}{9}a^{22}-\frac{1}{18}a^{21}+\frac{2}{9}a^{20}+\frac{11}{18}a^{19}+\frac{1}{18}a^{18}-\frac{13}{18}a^{17}-\frac{1}{9}a^{16}-\frac{5}{9}a^{15}-\frac{7}{9}a^{14}-\frac{7}{18}a^{13}-\frac{4}{9}a^{12}-\frac{2}{9}a^{11}-\frac{10}{9}a^{10}-\frac{14}{9}a^{9}+\frac{2}{9}a^{8}+\frac{10}{9}a^{7}-\frac{4}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{9}a^{4}+\frac{4}{9}a^{3}+\frac{11}{9}a^{2}-\frac{8}{9}a+\frac{10}{9}$ (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: | \( 80495462904.86972 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 80495462904.86972 \cdot 1}{2\cdot\sqrt{1201507202980564696903888106396747076468736}}\cr\approx \mathstrut & 0.873406673313651 \end{aligned}\] (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 | $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | R | $25{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | $17{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $17{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $24{,}\,{\href{/padicField/41.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $26$ | $26$ | $1$ | $26$ | |||
\(11\) | 11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
11.10.0.1 | $x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(5351\) | $\Q_{5351}$ | $x$ | $1$ | $1$ | $0$ | 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(322710411593\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(942553212724915163\) | $\Q_{942553212724915163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{942553212724915163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |