Normalized defining polynomial
\( x^{26} - 4x + 1 \)
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)
| |
Discriminant: | \(3305785123966891271739006552418919035749682642944\) \(\medspace = 2^{26}\cdot 3\cdot 149\cdot 977\cdot 3541\cdot 12747387209041\cdot 2498880811540538989\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(73.47\) | 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: | \(2\), \(3\), \(149\), \(977\), \(3541\), \(12747387209041\), \(2498880811540538989\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{49260\!\cdots\!87471}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{11}a^{25}+\frac{4}{11}a^{24}+\frac{5}{11}a^{23}-\frac{2}{11}a^{22}+\frac{3}{11}a^{21}+\frac{1}{11}a^{20}+\frac{4}{11}a^{19}+\frac{5}{11}a^{18}-\frac{2}{11}a^{17}+\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{4}{11}a^{14}+\frac{5}{11}a^{13}-\frac{2}{11}a^{12}+\frac{3}{11}a^{11}+\frac{1}{11}a^{10}+\frac{4}{11}a^{9}+\frac{5}{11}a^{8}-\frac{2}{11}a^{7}+\frac{3}{11}a^{6}+\frac{1}{11}a^{5}+\frac{4}{11}a^{4}+\frac{5}{11}a^{3}-\frac{2}{11}a^{2}+\frac{3}{11}a-\frac{3}{11}$
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)
| |
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: | $a$, $\frac{2}{11}a^{25}-\frac{3}{11}a^{24}-\frac{1}{11}a^{23}+\frac{7}{11}a^{22}+\frac{17}{11}a^{21}+\frac{2}{11}a^{20}-\frac{3}{11}a^{19}+\frac{10}{11}a^{18}+\frac{18}{11}a^{17}-\frac{5}{11}a^{16}-\frac{20}{11}a^{15}+\frac{8}{11}a^{14}+\frac{10}{11}a^{13}-\frac{15}{11}a^{12}-\frac{27}{11}a^{11}+\frac{13}{11}a^{10}+\frac{19}{11}a^{9}-\frac{12}{11}a^{8}-\frac{15}{11}a^{7}+\frac{6}{11}a^{6}+\frac{13}{11}a^{5}-\frac{25}{11}a^{4}-\frac{23}{11}a^{3}-\frac{15}{11}a^{2}+\frac{6}{11}a-\frac{6}{11}$, $\frac{1}{11}a^{25}-\frac{7}{11}a^{24}+\frac{16}{11}a^{23}-\frac{13}{11}a^{22}+\frac{3}{11}a^{21}-\frac{21}{11}a^{20}+\frac{26}{11}a^{19}-\frac{17}{11}a^{18}+\frac{9}{11}a^{17}-\frac{30}{11}a^{16}+\frac{34}{11}a^{15}-\frac{18}{11}a^{14}+\frac{16}{11}a^{13}-\frac{35}{11}a^{12}+\frac{36}{11}a^{11}-\frac{10}{11}a^{10}+\frac{15}{11}a^{9}-\frac{28}{11}a^{8}+\frac{31}{11}a^{7}+\frac{3}{11}a^{6}+\frac{1}{11}a^{5}-\frac{18}{11}a^{4}+\frac{5}{11}a^{3}+\frac{31}{11}a^{2}-\frac{30}{11}a-\frac{3}{11}$, $\frac{13}{11}a^{25}-\frac{25}{11}a^{24}+\frac{10}{11}a^{23}-\frac{15}{11}a^{22}-\frac{5}{11}a^{21}+\frac{13}{11}a^{20}+\frac{8}{11}a^{19}-\frac{1}{11}a^{18}+\frac{29}{11}a^{17}-\frac{38}{11}a^{16}+\frac{2}{11}a^{15}-\frac{3}{11}a^{14}-\frac{23}{11}a^{13}+\frac{18}{11}a^{12}+\frac{17}{11}a^{11}-\frac{9}{11}a^{10}+\frac{41}{11}a^{9}-\frac{23}{11}a^{8}-\frac{37}{11}a^{7}+\frac{28}{11}a^{6}-\frac{53}{11}a^{5}+\frac{19}{11}a^{4}+\frac{32}{11}a^{3}-\frac{4}{11}a^{2}+\frac{28}{11}a-\frac{17}{11}$, $\frac{6}{11}a^{25}-\frac{42}{11}a^{24}+\frac{30}{11}a^{23}+\frac{21}{11}a^{22}-\frac{48}{11}a^{21}+\frac{28}{11}a^{20}+\frac{35}{11}a^{19}-\frac{58}{11}a^{18}+\frac{21}{11}a^{17}+\frac{51}{11}a^{16}-\frac{71}{11}a^{15}+\frac{2}{11}a^{14}+\frac{74}{11}a^{13}-\frac{67}{11}a^{12}-\frac{15}{11}a^{11}+\frac{94}{11}a^{10}-\frac{64}{11}a^{9}-\frac{36}{11}a^{8}+\frac{120}{11}a^{7}-\frac{59}{11}a^{6}-\frac{71}{11}a^{5}+\frac{145}{11}a^{4}-\frac{47}{11}a^{3}-\frac{133}{11}a^{2}+\frac{161}{11}a-\frac{18}{11}$, $\frac{17}{11}a^{25}+\frac{13}{11}a^{24}+\frac{30}{11}a^{23}+\frac{21}{11}a^{22}+\frac{7}{11}a^{21}-\frac{16}{11}a^{20}-\frac{31}{11}a^{19}-\frac{25}{11}a^{18}-\frac{34}{11}a^{17}-\frac{15}{11}a^{16}-\frac{16}{11}a^{15}+\frac{2}{11}a^{14}+\frac{30}{11}a^{13}+\frac{54}{11}a^{12}+\frac{51}{11}a^{11}+\frac{17}{11}a^{10}+\frac{13}{11}a^{9}-\frac{3}{11}a^{8}-\frac{23}{11}a^{7}-\frac{59}{11}a^{6}-\frac{60}{11}a^{5}-\frac{64}{11}a^{4}-\frac{25}{11}a^{3}+\frac{21}{11}a^{2}+\frac{51}{11}a-\frac{7}{11}$, $a^{23}+a^{22}-a^{21}-2a^{20}+2a^{18}+a^{17}-a^{16}-a^{15}+a^{11}+a^{10}-a^{9}-2a^{8}+2a^{6}+a^{5}-a^{4}-a^{3}-1$, $\frac{4}{11}a^{25}+\frac{5}{11}a^{24}-\frac{13}{11}a^{23}-\frac{19}{11}a^{22}-\frac{10}{11}a^{21}-\frac{7}{11}a^{20}-\frac{17}{11}a^{19}-\frac{24}{11}a^{18}+\frac{3}{11}a^{17}+\frac{12}{11}a^{16}+\frac{4}{11}a^{15}-\frac{6}{11}a^{14}+\frac{9}{11}a^{13}+\frac{25}{11}a^{12}+\frac{1}{11}a^{11}-\frac{18}{11}a^{10}-\frac{6}{11}a^{9}+\frac{20}{11}a^{8}+\frac{3}{11}a^{7}-\frac{10}{11}a^{6}+\frac{4}{11}a^{5}+\frac{49}{11}a^{4}+\frac{42}{11}a^{3}-\frac{8}{11}a^{2}+\frac{12}{11}a+\frac{10}{11}$, $\frac{52}{11}a^{25}+\frac{21}{11}a^{24}+\frac{62}{11}a^{23}-\frac{49}{11}a^{22}-\frac{42}{11}a^{21}-\frac{58}{11}a^{20}+\frac{32}{11}a^{19}+\frac{73}{11}a^{18}+\frac{50}{11}a^{17}+\frac{2}{11}a^{16}-\frac{113}{11}a^{15}-\frac{34}{11}a^{14}-\frac{48}{11}a^{13}+\frac{160}{11}a^{12}+\frac{13}{11}a^{11}+\frac{96}{11}a^{10}-\frac{210}{11}a^{9}+\frac{7}{11}a^{8}-\frac{148}{11}a^{7}+\frac{255}{11}a^{6}-\frac{14}{11}a^{5}+\frac{197}{11}a^{4}-\frac{279}{11}a^{3}+\frac{6}{11}a^{2}-\frac{229}{11}a+\frac{75}{11}$, $\frac{25}{11}a^{25}+\frac{12}{11}a^{24}+\frac{26}{11}a^{23}-\frac{6}{11}a^{22}-\frac{46}{11}a^{21}+\frac{3}{11}a^{20}-\frac{21}{11}a^{19}+\frac{59}{11}a^{18}+\frac{16}{11}a^{17}-\frac{13}{11}a^{16}-\frac{19}{11}a^{15}-\frac{65}{11}a^{14}+\frac{26}{11}a^{13}+\frac{49}{11}a^{12}+\frac{20}{11}a^{11}+\frac{47}{11}a^{10}-\frac{98}{11}a^{9}-\frac{18}{11}a^{8}-\frac{17}{11}a^{7}+\frac{31}{11}a^{6}+\frac{113}{11}a^{5}-\frac{54}{11}a^{4}+\frac{4}{11}a^{3}-\frac{105}{11}a^{2}-\frac{46}{11}a+\frac{24}{11}$, $\frac{21}{11}a^{25}+\frac{7}{11}a^{24}+\frac{6}{11}a^{23}+\frac{24}{11}a^{22}-\frac{14}{11}a^{21}+\frac{10}{11}a^{20}+\frac{29}{11}a^{19}-\frac{16}{11}a^{18}+\frac{13}{11}a^{17}+\frac{8}{11}a^{16}-\frac{23}{11}a^{15}+\frac{40}{11}a^{14}-\frac{5}{11}a^{13}-\frac{31}{11}a^{12}+\frac{30}{11}a^{11}-\frac{23}{11}a^{10}+\frac{18}{11}a^{9}+\frac{50}{11}a^{8}-\frac{53}{11}a^{7}+\frac{8}{11}a^{6}+\frac{43}{11}a^{5}-\frac{15}{11}a^{4}+\frac{83}{11}a^{3}+\frac{2}{11}a^{2}-\frac{58}{11}a+\frac{25}{11}$, $\frac{19}{11}a^{25}-\frac{12}{11}a^{24}-\frac{26}{11}a^{23}+\frac{39}{11}a^{22}-\frac{9}{11}a^{21}+\frac{41}{11}a^{20}-\frac{34}{11}a^{19}-\frac{81}{11}a^{18}-\frac{5}{11}a^{17}-\frac{42}{11}a^{16}+\frac{85}{11}a^{15}+\frac{43}{11}a^{14}+\frac{18}{11}a^{13}+\frac{94}{11}a^{12}-\frac{42}{11}a^{11}+\frac{30}{11}a^{10}-\frac{111}{11}a^{9}-\frac{147}{11}a^{8}+\frac{28}{11}a^{7}-\frac{42}{11}a^{6}+\frac{173}{11}a^{5}+\frac{32}{11}a^{4}-\frac{48}{11}a^{3}+\frac{94}{11}a^{2}-\frac{119}{11}a+\frac{31}{11}$, $\frac{38}{11}a^{25}-\frac{35}{11}a^{24}-\frac{107}{11}a^{23}-\frac{87}{11}a^{22}+\frac{48}{11}a^{21}+\frac{159}{11}a^{20}+\frac{97}{11}a^{19}-\frac{85}{11}a^{18}-\frac{175}{11}a^{17}-\frac{84}{11}a^{16}+\frac{71}{11}a^{15}+\frac{152}{11}a^{14}+\frac{124}{11}a^{13}+\frac{1}{11}a^{12}-\frac{172}{11}a^{11}-\frac{226}{11}a^{10}-\frac{24}{11}a^{9}+\frac{267}{11}a^{8}+\frac{287}{11}a^{7}-\frac{40}{11}a^{6}-\frac{325}{11}a^{5}-\frac{255}{11}a^{4}+\frac{58}{11}a^{3}+\frac{254}{11}a^{2}+\frac{246}{11}a-\frac{70}{11}$ (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: | \( 149569312316316.25 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 149569312316316.25 \cdot 1}{2\cdot\sqrt{3305785123966891271739006552418919035749682642944}}\cr\approx \mathstrut & 0.622864533353178 \end{aligned}\] (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 | R | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $21{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $22{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.7.0.1}{7} }^{3}{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $26$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
Deg $24$ | $2$ | $12$ | $24$ | ||||
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
3.20.0.1 | $x^{20} + 2 x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + 2 x^{5} + 2 x^{4} + 2 x^{3} + x + 2$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(149\) | $\Q_{149}$ | $x + 147$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
149.2.1.1 | $x^{2} + 149$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
149.3.0.1 | $x^{3} + 3 x + 147$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
149.3.0.1 | $x^{3} + 3 x + 147$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
149.5.0.1 | $x^{5} + 2 x + 147$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
149.12.0.1 | $x^{12} + 121 x^{6} + 91 x^{5} + 52 x^{4} + 9 x^{3} + 104 x^{2} + 110 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
\(977\) | $\Q_{977}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(3541\) | $\Q_{3541}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{3541}$ | $x$ | $1$ | $1$ | $0$ | 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(12747387209041\) | $\Q_{12747387209041}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{12747387209041}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(2498880811540538989\) | $\Q_{2498880811540538989}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2498880811540538989}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |