Normalized defining polynomial
\( x^{26} - x + 4 \)
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: | \(-6931174461867291893087822118262619963477644915282599\) \(\medspace = -\,3\cdot 19\cdot 31\cdot 71\cdot 197\cdot 3701\cdot 5317453\cdot 31322273\cdot 135060203\cdot 3368544526633678033\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(98.60\) | 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: | $3^{1/2}19^{1/2}31^{1/2}71^{1/2}197^{1/2}3701^{1/2}5317453^{1/2}31322273^{1/2}135060203^{1/2}3368544526633678033^{1/2}\approx 8.32536753655194e+25$ | ||
Ramified primes: | \(3\), \(19\), \(31\), \(71\), \(197\), \(3701\), \(5317453\), \(31322273\), \(135060203\), \(3368544526633678033\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-69311\!\cdots\!82599}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (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: | $a^{24}+2a^{23}+2a^{22}+a^{21}+a^{18}+2a^{17}+2a^{16}+a^{15}+a^{12}+2a^{11}+2a^{10}+a^{9}+a^{6}+2a^{5}+2a^{4}+a^{3}+1$, $a^{25}-a^{24}-a^{23}+a^{20}-a^{19}-2a^{18}+2a^{17}+2a^{16}-a^{13}+a^{12}+a^{11}-3a^{10}-3a^{9}-4a^{8}-3a^{7}+a^{6}-5a^{5}-7a^{4}-3$, $9a^{25}-6a^{24}-16a^{23}-17a^{22}-7a^{21}+12a^{20}+19a^{19}+20a^{18}+a^{17}-16a^{16}-26a^{15}-18a^{14}+3a^{13}+25a^{12}+28a^{11}+19a^{10}-13a^{9}-30a^{8}-35a^{7}-14a^{6}+23a^{5}+38a^{4}+41a^{3}+4a^{2}-32a-59$, $13a^{25}+5a^{24}-22a^{23}-a^{22}+20a^{21}-10a^{20}-16a^{19}+21a^{18}+8a^{17}-30a^{16}+2a^{15}+31a^{14}-16a^{13}-22a^{12}+32a^{11}+8a^{10}-47a^{9}+3a^{8}+44a^{7}-24a^{6}-26a^{5}+57a^{4}+15a^{3}-67a^{2}+6a+43$, $31a^{25}+23a^{24}-a^{23}-20a^{22}-29a^{21}-27a^{20}-3a^{19}+27a^{18}+49a^{17}+34a^{16}-16a^{15}-53a^{14}-55a^{13}-10a^{12}+36a^{11}+52a^{10}+44a^{9}+8a^{8}-30a^{7}-65a^{6}-69a^{5}-6a^{4}+67a^{3}+109a^{2}+56a-77$, $20a^{25}-37a^{24}+25a^{23}+14a^{22}-50a^{21}+56a^{20}-19a^{19}-44a^{18}+89a^{17}-82a^{16}+16a^{15}+74a^{14}-129a^{13}+107a^{12}-14a^{11}-98a^{10}+157a^{9}-117a^{8}+3a^{7}+116a^{6}-160a^{5}+93a^{4}+34a^{3}-136a^{2}+134a-39$, $7a^{25}-10a^{24}+4a^{23}+3a^{22}-9a^{21}+12a^{20}-3a^{19}-9a^{18}+14a^{17}-13a^{16}+2a^{15}+15a^{14}-19a^{13}+10a^{12}+3a^{11}-19a^{10}+22a^{9}-6a^{8}-14a^{7}+26a^{6}-26a^{5}+3a^{4}+28a^{3}-37a^{2}+24a-3$, $15a^{25}-2a^{24}-21a^{23}-20a^{22}+3a^{21}+27a^{20}+26a^{19}-a^{18}-28a^{17}-26a^{16}+3a^{15}+29a^{14}+22a^{13}-13a^{12}-39a^{11}-24a^{10}+22a^{9}+52a^{8}+32a^{7}-22a^{6}-56a^{5}-34a^{4}+24a^{3}+55a^{2}+24a-55$, $2a^{25}+a^{24}-5a^{23}+4a^{22}+3a^{21}-7a^{20}+3a^{19}+2a^{18}-5a^{17}+3a^{16}+7a^{15}-9a^{14}-a^{13}+7a^{12}-8a^{11}+4a^{10}+7a^{9}-9a^{8}-4a^{7}+13a^{6}-8a^{5}-a^{4}+15a^{3}-17a^{2}-3a+13$, $25a^{25}+23a^{24}+13a^{23}-5a^{22}-20a^{21}-32a^{20}-32a^{19}-21a^{18}+22a^{16}+39a^{15}+41a^{14}+30a^{13}+8a^{12}-23a^{11}-40a^{10}-54a^{9}-45a^{8}-17a^{7}+14a^{6}+49a^{5}+69a^{4}+61a^{3}+31a^{2}-6a-83$, $5a^{25}-13a^{24}+16a^{22}-6a^{21}-8a^{20}+7a^{19}+10a^{18}-26a^{17}+8a^{16}+12a^{15}+3a^{14}-27a^{13}+31a^{12}-7a^{11}-15a^{10}-15a^{9}+45a^{8}-30a^{7}-5a^{6}+15a^{5}+25a^{4}-58a^{3}+14a^{2}+29a-21$, $58a^{25}-66a^{24}+74a^{23}-70a^{22}+68a^{21}-70a^{20}+76a^{19}-85a^{18}+77a^{17}-68a^{16}+51a^{15}-48a^{14}+48a^{13}-41a^{12}+23a^{11}+11a^{10}-34a^{9}+50a^{8}-56a^{7}+77a^{6}-116a^{5}+163a^{4}-197a^{3}+209a^{2}-220a+185$ (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: | \( 5328092523551477.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5328092523551477.0 \cdot 2}{2\cdot\sqrt{6931174461867291893087822118262619963477644915282599}}\cr\approx \mathstrut & 1.52232237719801 \end{aligned}\] (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 | $20{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $23{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | $22{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.13.0.1}{13} }{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | R | $21{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $22{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 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 | $[\ ]^{20}$ | |
\(19\) | 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(31\) | $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
31.2.1.1 | $x^{2} + 93$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.3.0.1 | $x^{3} + x + 28$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
31.20.0.1 | $x^{20} + x^{2} - x + 3$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(71\) | $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.1.2 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
71.8.0.1 | $x^{8} + 53 x^{3} + 22 x^{2} + 19 x + 7$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
71.10.0.1 | $x^{10} + 53 x^{5} + 17 x^{4} + 26 x^{3} + x^{2} + 40 x + 7$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(197\) | 197.2.1.2 | $x^{2} + 394$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
197.4.0.1 | $x^{4} + 16 x^{2} + 124 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
197.9.0.1 | $x^{9} + 13 x^{3} + 127 x^{2} + 8 x + 195$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
197.11.0.1 | $x^{11} + 14 x + 195$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(3701\) | 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 $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(5317453\) | $\Q_{5317453}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5317453}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
\(31322273\) | $\Q_{31322273}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{31322273}$ | $x$ | $1$ | $1$ | $0$ | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(135060203\) | 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(3368544526633678033\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |