Normalized defining polynomial
\( x^{26} - 2x + 4 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 13]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-1732793613976706876091650397068785799342044590637056\) \(\medspace = -\,2^{24}\cdot 3\cdot 7\cdot 331\cdot 347\cdot 1627\cdot 49019\cdot 1249111\cdot 102017147\cdot 4213315416817943\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
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: | $2^{24/25}3^{1/2}7^{1/2}331^{1/2}347^{1/2}1627^{1/2}49019^{1/2}1249111^{1/2}102017147^{1/2}4213315416817943^{1/2}\approx 1.976979960930435e+22$ | ||
Ramified primes: | \(2\), \(3\), \(7\), \(331\), \(347\), \(1627\), \(49019\), \(1249111\), \(102017147\), \(4213315416817943\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-10328\!\cdots\!01991}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{2}a^{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $a^{25}-a^{24}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{10}-a^{9}+a^{7}-a^{6}+a^{4}-a^{3}+1$, $4a^{25}-2a^{24}+4a^{23}+2a^{22}-3a^{21}+4a^{20}-2a^{19}-4a^{18}+3a^{17}-6a^{16}-3a^{15}+3a^{14}-8a^{13}+2a^{12}+3a^{11}-7a^{10}+10a^{9}-3a^{7}+15a^{6}-7a^{5}+3a^{4}+12a^{3}-16a^{2}+10a-7$, $3a^{25}-2a^{23}-3a^{22}-5a^{21}-5a^{20}+3a^{18}+2a^{17}+4a^{16}+7a^{15}+3a^{14}-2a^{13}-3a^{12}-5a^{11}-8a^{10}-5a^{9}+2a^{8}+2a^{7}+3a^{6}+11a^{5}+10a^{4}-2a^{3}-4a^{2}-2a-15$, $2a^{25}+7a^{24}-4a^{23}-4a^{22}+a^{21}-2a^{20}+7a^{19}+2a^{18}-8a^{17}+3a^{16}-2a^{15}-2a^{14}+8a^{13}-6a^{12}+a^{11}+6a^{10}-13a^{9}+3a^{8}+5a^{7}-4a^{6}+14a^{5}-12a^{4}-17a^{3}+16a^{2}-a+5$, $2a^{25}-2a^{24}-3a^{23}+3a^{21}+4a^{20}-6a^{19}-4a^{17}+12a^{16}-7a^{15}+4a^{14}-13a^{13}+11a^{12}+5a^{10}-10a^{9}-4a^{8}+9a^{7}+2a^{6}+4a^{5}-16a^{4}+3a^{3}+4a^{2}+13a-13$, $4a^{25}+5a^{24}+5a^{23}+2a^{22}+7a^{21}+6a^{20}+9a^{18}+4a^{17}+a^{16}+8a^{15}+2a^{14}+2a^{13}+3a^{12}+4a^{11}-a^{10}-a^{9}+6a^{8}-9a^{7}+a^{6}+3a^{5}-16a^{4}+3a^{3}-4a^{2}-14a-11$, $a^{25}-a^{24}-a^{23}+a^{21}-3a^{19}+2a^{18}-2a^{15}-2a^{14}+5a^{13}-5a^{12}-2a^{10}+a^{9}+2a^{8}-9a^{7}+4a^{6}-3a^{3}-3a^{2}+7a-3$, $5a^{24}-18a^{23}+20a^{22}-7a^{21}-2a^{20}+21a^{19}-16a^{18}+13a^{17}-10a^{16}-2a^{15}+a^{14}-20a^{13}+34a^{12}-28a^{11}+16a^{10}+27a^{9}-47a^{8}+45a^{7}-28a^{6}-14a^{5}+18a^{4}-37a^{3}+39a^{2}-42a+45$, $2a^{25}-10a^{24}+a^{23}+5a^{22}+9a^{21}-a^{20}+6a^{19}-10a^{18}-14a^{17}+11a^{16}-18a^{15}+27a^{14}+a^{13}+a^{12}+5a^{11}-20a^{10}-7a^{9}-2a^{8}+12a^{7}-4a^{6}+43a^{5}-36a^{4}+13a^{3}-14a^{2}-43a+45$, $3a^{25}+8a^{24}+11a^{23}+8a^{22}+5a^{21}+3a^{20}-a^{19}-7a^{18}-6a^{17}-2a^{16}-3a^{15}-2a^{14}+3a^{13}+4a^{12}-2a^{11}-6a^{10}-6a^{9}-9a^{8}-9a^{7}-a^{6}+3a^{5}+10a^{4}+19a^{3}+24a^{2}+13a-1$, $11a^{25}-7a^{24}-6a^{23}+13a^{22}-9a^{21}+3a^{20}-9a^{19}-a^{18}+11a^{17}-23a^{16}+3a^{15}-8a^{14}+a^{13}+a^{12}-36a^{11}+24a^{10}-11a^{9}-19a^{8}+3a^{7}-27a^{6}+32a^{5}-31a^{4}-17a^{3}+21a^{2}-26a+9$, $125a^{25}+83a^{24}+85a^{23}+51a^{22}-20a^{21}-24a^{20}-89a^{19}-137a^{18}-112a^{17}-164a^{16}-136a^{15}-70a^{14}-92a^{13}+16a^{12}+74a^{11}+70a^{10}+197a^{9}+179a^{8}+151a^{7}+231a^{6}+86a^{5}+62a^{4}+59a^{3}-148a^{2}-116a-417$ (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: | \( 6042375351216186.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 6042375351216186.0 \cdot 1}{2\cdot\sqrt{1732793613976706876091650397068785799342044590637056}}\cr\approx \mathstrut & 1.72640455695576 \end{aligned}\] (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 | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }$ | $25{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/29.3.0.1}{3} }$ | $22{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $26$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $25$ | $25$ | $1$ | $24$ | ||||
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
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 | $[\ ]^{20}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(331\) | 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(347\) | $\Q_{347}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(1627\) | $\Q_{1627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1627}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $15$ | $1$ | $15$ | $0$ | $C_{15}$ | $[\ ]^{15}$ | ||
\(49019\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(1249111\) | $\Q_{1249111}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(102017147\) | 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(4213315416817943\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |