Normalized defining polynomial
\( x^{26} - x - 1 \)
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: | \(6244937422177169834030564821847469401\) \(\medspace = 73\cdot 181\cdot 2385857\cdot 32375941061\cdot 6118709648547401\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(26.01\) | 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: | $73^{1/2}181^{1/2}2385857^{1/2}32375941061^{1/2}6118709648547401^{1/2}\approx 2.4989872793147965e+18$ | ||
Ramified primes: | \(73\), \(181\), \(2385857\), \(32375941061\), \(6118709648547401\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{62449\!\cdots\!69401}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a$, $a^{14}-a$, $a^{25}+a^{7}-1$, $a^{25}-a^{24}+a^{15}-1$, $a^{9}-a^{4}$, $a^{17}-a^{16}$, $a^{10}-a^{7}+a^{4}$, $a^{25}-a^{24}+a^{23}-a^{22}+a^{21}-a^{20}+a^{19}+a^{17}-a^{16}-a^{14}+a^{11}-1$, $a^{20}-a^{16}-a^{14}+a^{11}+a^{9}-a^{5}$, $2a^{25}-a^{24}+2a^{23}-2a^{22}+a^{21}-a^{18}+a^{16}-a^{15}+a^{12}-a^{11}+a^{10}-a^{7}+a^{6}-a^{4}+a^{2}-3$, $a^{25}-a^{24}+a^{23}-2a^{22}+a^{21}-a^{20}+a^{19}-a^{10}-a^{8}+a^{7}-1$, $2a^{25}-2a^{24}+2a^{23}-a^{22}+a^{21}-a^{20}+a^{19}-a^{18}+a^{17}+a^{15}-a^{14}+a^{13}-a^{12}-a^{8}-a^{6}+a^{3}-2$, $a^{24}-a^{23}+a^{22}-a^{21}+a^{20}-a^{19}+a^{18}-a^{17}+a^{16}-a^{15}+a^{14}-a^{13}+a^{12}-a^{8}-a^{2}+a$ (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: | \( 117671984.44253884 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 117671984.44253884 \cdot 1}{2\cdot\sqrt{6244937422177169834030564821847469401}}\cr\approx \mathstrut & 0.356531123790108 \end{aligned}\] (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 | ${\href{/padicField/2.12.0.1}{12} }{,}\,{\href{/padicField/2.9.0.1}{9} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ | $18{,}\,{\href{/padicField/11.8.0.1}{8} }$ | $21{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $22{,}\,{\href{/padicField/17.4.0.1}{4} }$ | $25{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(73\) | 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
73.5.0.1 | $x^{5} + 9 x + 68$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
73.6.0.1 | $x^{6} + 45 x^{3} + 23 x^{2} + 48 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
73.7.0.1 | $x^{7} + 10 x + 68$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(181\) | $\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{181}$ | $x + 179$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
181.2.1.2 | $x^{2} + 362$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
181.16.0.1 | $x^{16} - 2 x + 18$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | |
\(2385857\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(32375941061\) | $\Q_{32375941061}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{32375941061}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(6118709648547401\) | 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |