Normalized defining polynomial
\( x^{42} - 3x + 1 \)
Invariants
Degree: | $42$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(145\!\cdots\!905\) \(\medspace = 3^{42}\cdot 5\cdot 277\cdot 9883\cdot 24851\cdot 5267491\cdot 28880594132963764921\cdot 25718507492700353234963282659\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(112.59\) | 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: | \(3\), \(5\), \(277\), \(9883\), \(24851\), \(5267491\), \(28880594132963764921\), \(25718507492700353234963282659\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{13308\!\cdots\!05545}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $21$ | 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 $S_{42}$ 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 | $38{,}\,{\href{/padicField/2.4.0.1}{4} }$ | R | R | $35{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $25{,}\,{\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $37{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/19.7.0.1}{7} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $24{,}\,{\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.8.0.1}{8} }$ | $31{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $40{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $15{,}\,{\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $26{,}\,{\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | $21{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.6.6.2 | $x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ |
Deg $18$ | $3$ | $6$ | $18$ | ||||
Deg $18$ | $3$ | $6$ | $18$ | ||||
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
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 | $x^{18} + x^{12} + x^{11} + x^{10} + x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{3} + 2 x^{2} + 2$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | |
\(277\) | $\Q_{277}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(9883\) | $\Q_{9883}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
Deg $25$ | $1$ | $25$ | $0$ | $C_{25}$ | $[\ ]^{25}$ | ||
\(24851\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(5267491\) | $\Q_{5267491}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{5267491}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(28880594132963764921\) | $\Q_{28880594132963764921}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $27$ | $1$ | $27$ | $0$ | $C_{27}$ | $[\ ]^{27}$ | ||
\(257\!\cdots\!659\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $37$ | $1$ | $37$ | $0$ | $C_{37}$ | $[\ ]^{37}$ |