Normalized defining polynomial
\( x^{42} - 4x + 4 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 21]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-6368697991994218840106904466505523750060174615816448189841639477612393330114560\) \(\medspace = -\,2^{42}\cdot 5\cdot 14815048403\cdot 223886382998191\cdot 87\!\cdots\!11\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(75.21\) | 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: | \(2\), \(5\), \(14815048403\), \(223886382998191\), \(87315\!\cdots\!46911\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-14480\!\cdots\!12015}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{2}a^{25}$, $\frac{1}{2}a^{26}$, $\frac{1}{2}a^{27}$, $\frac{1}{2}a^{28}$, $\frac{1}{2}a^{29}$, $\frac{1}{2}a^{30}$, $\frac{1}{2}a^{31}$, $\frac{1}{2}a^{32}$, $\frac{1}{2}a^{33}$, $\frac{1}{2}a^{34}$, $\frac{1}{2}a^{35}$, $\frac{1}{2}a^{36}$, $\frac{1}{2}a^{37}$, $\frac{1}{2}a^{38}$, $\frac{1}{2}a^{39}$, $\frac{1}{2}a^{40}$, $\frac{1}{10}a^{41}+\frac{1}{5}a^{40}-\frac{1}{10}a^{39}-\frac{1}{5}a^{38}+\frac{1}{10}a^{37}+\frac{1}{5}a^{36}-\frac{1}{10}a^{35}-\frac{1}{5}a^{34}+\frac{1}{10}a^{33}+\frac{1}{5}a^{32}-\frac{1}{10}a^{31}-\frac{1}{5}a^{30}+\frac{1}{10}a^{29}+\frac{1}{5}a^{28}-\frac{1}{10}a^{27}-\frac{1}{5}a^{26}+\frac{1}{10}a^{25}+\frac{1}{5}a^{24}-\frac{1}{10}a^{23}-\frac{1}{5}a^{22}+\frac{1}{10}a^{21}+\frac{1}{5}a^{20}+\frac{2}{5}a^{19}-\frac{1}{5}a^{18}-\frac{2}{5}a^{17}+\frac{1}{5}a^{16}+\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{2}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $20$ | 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: | not computed | 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: | not computed | 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 $
Galois group
A non-solvable group of order 1405006117752879898543142606244511569936384000000000 |
The 53174 conjugacy class representatives for $S_{42}$ are not computed |
Character table for $S_{42}$ |
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 | $32{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | $27{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $24{,}\,18$ | $29{,}\,{\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $23{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{3}$ | $20{,}\,{\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $27{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $41{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $41{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,16{,}\,{\href{/padicField/59.8.0.1}{8} }$ |
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 $42$ | $42$ | $1$ | $42$ | |||
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
5.9.0.1 | $x^{9} + 2 x^{3} + x + 3$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
5.18.0.1 | $x^{18} + x^{12} + x^{11} + x^{10} + x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{3} + 2 x^{2} + 2$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(14815048403\) | $\Q_{14815048403}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $34$ | $1$ | $34$ | $0$ | $C_{34}$ | $[\ ]^{34}$ | ||
\(223886382998191\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $23$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | ||
\(873\!\cdots\!911\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ |