Normalized defining polynomial
\( x^{26} - 2x - 2 \)
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: | \(212525560315468668398429873236273026387935232\) \(\medspace = 2^{26}\cdot 3\cdot 67\cdot 107\cdot 5867\cdot 10536636742939\cdot 2381953897662043\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(50.69\) | 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: | \(2\), \(3\), \(67\), \(107\), \(5867\), \(10536636742939\), \(2381953897662043\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{31668\!\cdots\!67513}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a+1$, $a^{9}-a-1$, $a^{13}-a-1$, $a^{17}-a^{16}+a^{14}-a^{11}+a^{10}+a^{9}-2a^{8}+a^{6}-a^{4}-a^{3}+a^{2}-1$, $a^{25}-a^{23}+a^{22}-a^{20}+a^{19}-a^{17}+a^{16}-a^{14}+a^{12}-a^{10}+a^{9}-a^{7}+a^{6}-a^{4}+a^{3}-2a-1$, $a^{15}-a^{13}+a^{10}-a^{9}+a^{7}-a^{4}+1$, $a^{25}-a^{23}+2a^{22}-2a^{21}+2a^{20}-a^{19}+a^{17}-2a^{16}+2a^{15}-2a^{14}+a^{13}-a^{12}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}+a-3$, $a^{25}-a^{24}+a^{22}-a^{19}+a^{18}-2a^{17}+2a^{16}+a^{14}-2a^{13}+a^{12}-2a^{11}+a^{10}+a^{9}+a^{8}-a^{7}-a^{5}-a^{4}+2a^{3}+a^{2}-3$, $a^{25}-a^{24}+a^{23}-a^{19}+a^{18}-a^{17}-a^{16}+a^{15}-a^{14}+a^{13}+a^{11}+a^{10}-a^{9}+a^{8}-2a^{6}-a^{4}-a^{2}+1$, $a^{25}-a^{24}-2a^{22}+a^{19}+a^{18}+a^{17}-a^{15}-a^{14}-a^{13}-a^{12}+a^{11}+a^{10}+2a^{9}+a^{8}-2a^{6}-3a^{5}-a^{4}-a^{3}+3a^{2}+3a+1$, $a^{22}+a^{20}+a^{17}+a^{15}+a^{14}-a^{13}+a^{12}-a^{11}+a^{9}-a^{8}-a^{6}-2a^{5}-a^{3}-1$, $a^{23}-2a^{22}+2a^{21}-a^{20}-a^{19}+2a^{18}-2a^{17}+2a^{16}-2a^{15}+2a^{14}-a^{13}-a^{12}+a^{11}-2a^{9}+3a^{8}-2a^{7}+a^{6}-a^{4}+a^{3}-2a^{2}+2a-1$, $3a^{25}-a^{24}+2a^{22}-3a^{21}+5a^{20}-5a^{19}+6a^{18}-5a^{17}+5a^{16}-3a^{15}+3a^{14}-a^{13}+a^{12}+a^{11}+a^{9}+a^{7}+a^{6}+a^{4}+a^{3}+2a-7$ (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: | \( 1482708296476.7107 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 1482708296476.7107 \cdot 1}{2\cdot\sqrt{212525560315468668398429873236273026387935232}}\cr\approx \mathstrut & 0.770084795314214 \end{aligned}\] (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 | ${\href{/padicField/5.12.0.1}{12} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/19.8.0.1}{8} }$ | $26$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.13.0.1}{13} }{,}\,{\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.11.0.1}{11} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/53.11.0.1}{11} }$ | $17{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $26$ | $26$ | $1$ | $26$ | |||
\(3\) | 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 | $[\ ]^{20}$ | |
\(67\) | $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
67.3.0.1 | $x^{3} + 6 x + 65$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
67.20.0.1 | $x^{20} - 2 x + 7$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | |
\(107\) | $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
107.2.0.1 | $x^{2} + 103 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
107.8.0.1 | $x^{8} + 2 x^{4} + 105 x^{3} + 24 x^{2} + 95 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
107.8.0.1 | $x^{8} + 2 x^{4} + 105 x^{3} + 24 x^{2} + 95 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
\(5867\) | $\Q_{5867}$ | $x$ | $1$ | $1$ | $0$ | 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 $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(10536636742939\) | $\Q_{10536636742939}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(2381953897662043\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |