Normalized defining polynomial
\( x^{26} - 3x + 1 \)
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: | \(225763037443343983968209361295448843442312686849\) \(\medspace = 17\cdot 47\cdot 457\cdot 58031\cdot 959677\cdot 10678056923\cdot 1039710637886321095943\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(66.27\) | 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: | $17^{1/2}47^{1/2}457^{1/2}58031^{1/2}959677^{1/2}10678056923^{1/2}1039710637886321095943^{1/2}\approx 4.751452803546974e+23$ | ||
Ramified primes: | \(17\), \(47\), \(457\), \(58031\), \(959677\), \(10678056923\), \(1039710637886321095943\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{22576\!\cdots\!86849}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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
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: | $a^{25}-3$, $a-1$, $a^{23}-a^{21}+a^{20}+a^{19}-a^{18}+a^{16}+a^{11}-a^{9}+a^{8}+a^{7}-a^{6}+a^{4}+a-1$, $a^{23}+a^{21}-a^{17}-a^{15}+a^{11}+a^{9}-a^{5}-a^{3}+1$, $a^{25}+a^{23}+a^{21}-a^{19}+a^{18}+a^{16}+2a^{13}-a^{12}+2a^{8}-a^{7}+2a^{6}+a^{5}-a^{4}+a^{3}-2a^{2}+2a-2$, $6a^{25}-4a^{24}-8a^{23}+8a^{21}+3a^{20}-9a^{19}-11a^{18}+a^{17}+12a^{16}+7a^{15}-7a^{14}-9a^{13}+5a^{12}+15a^{11}+5a^{10}-13a^{9}-16a^{8}+a^{7}+15a^{6}+5a^{5}-15a^{4}-14a^{3}+10a^{2}+25a-8$, $a^{25}+a^{24}-a^{23}-2a^{22}-a^{20}-a^{19}-a^{17}-a^{14}-2a^{13}-a^{12}+2a^{11}+a^{10}+2a^{9}+2a^{8}-a^{7}+2a^{6}+2a^{5}-a^{4}+2a^{2}+2a-2$, $3a^{25}-4a^{23}-3a^{22}+2a^{21}+2a^{20}-3a^{19}-5a^{18}+a^{17}+7a^{16}+5a^{15}-2a^{14}-4a^{13}+4a^{11}+3a^{10}-a^{8}+a^{7}+4a^{6}+3a^{5}-4a^{4}-10a^{3}-6a^{2}+6a$, $35a^{25}+40a^{24}+41a^{23}+41a^{22}+38a^{21}+31a^{20}+23a^{19}+13a^{18}-a^{17}-14a^{16}-25a^{15}-38a^{14}-48a^{13}-54a^{12}-60a^{11}-60a^{10}-54a^{9}-48a^{8}-38a^{7}-21a^{6}-4a^{5}+13a^{4}+34a^{3}+50a^{2}+62a-27$, $4a^{25}-5a^{24}+6a^{23}-8a^{22}+8a^{21}-10a^{20}+10a^{19}-12a^{18}+12a^{17}-14a^{16}+13a^{15}-14a^{14}+14a^{13}-14a^{12}+13a^{11}-13a^{10}+12a^{9}-9a^{8}+8a^{7}-6a^{6}+4a^{5}-3a^{3}+7a^{2}-11a+2$, $6a^{25}+10a^{24}-10a^{23}-6a^{22}+17a^{21}+3a^{20}-20a^{19}+5a^{18}+14a^{17}-13a^{16}-14a^{15}+23a^{14}+9a^{13}-25a^{12}+4a^{11}+22a^{10}-15a^{9}-27a^{8}+28a^{7}+16a^{6}-31a^{5}+a^{4}+36a^{3}-14a^{2}-44a+14$, $a^{25}-a^{23}-a^{21}-a^{20}-a^{19}+3a^{16}+2a^{15}+a^{14}-2a^{11}+a^{10}+a^{9}-a^{8}-2a^{7}-a^{6}-2a^{5}+a^{4}+5a^{3}+3a^{2}+2a-2$, $a^{25}-a^{24}-2a^{23}-a^{22}+2a^{21}-2a^{19}-a^{18}+3a^{17}+2a^{16}+a^{13}-2a^{9}-2a^{8}-a^{7}+2a^{6}-a^{3}+a^{2}+a+1$ (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: | \( 32189616171752.71 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 32189616171752.71 \cdot 1}{2\cdot\sqrt{225763037443343983968209361295448843442312686849}}\cr\approx \mathstrut & 0.512953260181768 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $26$ | $24{,}\,{\href{/padicField/13.2.0.1}{2} }$ | R | ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $25{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | R | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $24{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.23.0.1 | $x^{23} + 15 x^{2} + 16 x + 14$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.2.1.2 | $x^{2} + 47$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
47.5.0.1 | $x^{5} + x + 42$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
47.18.0.1 | $x^{18} + 6 x^{11} + 41 x^{10} + 42 x^{9} + 26 x^{8} + 44 x^{7} + 24 x^{6} + 22 x^{5} + 11 x^{4} + 5 x^{3} + 45 x^{2} + 33 x + 5$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(457\) | $\Q_{457}$ | $x$ | $1$ | $1$ | $0$ | 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 $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(58031\) | $\Q_{58031}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $11$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | ||
\(959677\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(10678056923\) | $\Q_{10678056923}$ | $x$ | $1$ | $1$ | $0$ | 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 $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(103\!\cdots\!943\) | $\Q_{10\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{10\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10\!\cdots\!43}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |