Normalized defining polynomial
\( x^{38} - 4x - 1 \)
Invariants
Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(2901373404155647104441523981239181159929184708380460029220825235930489\) \(\medspace = 11149\cdot 365159\cdot 14342233\cdot 49\!\cdots\!63\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(67.29\) | 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: | $11149^{1/2}365159^{1/2}14342233^{1/2}49689992079559593586453718567433043366296684203166163^{1/2}\approx 5.386439829939296e+34$ | ||
Ramified primes: | \(11149\), \(365159\), \(14342233\), \(49689\!\cdots\!66163\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{29013\!\cdots\!30489}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{26}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{27}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{28}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{29}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{12}$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{13}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{14}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{15}$, $\frac{1}{2}a^{35}-\frac{1}{2}a^{16}$, $\frac{1}{2}a^{36}-\frac{1}{2}a^{17}$, $\frac{1}{2}a^{37}-\frac{1}{2}a^{18}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $19$ | 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 523022617466601111760007224100074291200000000 |
The 26015 conjugacy class representatives for $S_{38}$ are not computed |
Character table for $S_{38}$ |
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 | $36{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.5.0.1}{5} }$ | $21{,}\,{\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $22{,}\,16$ | $25{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ | $22{,}\,{\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | $31{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | $21{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ | $32{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | $19^{2}$ | $28{,}\,{\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.12.0.1}{12} }^{2}$ | $36{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $30{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(11149\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $30$ | $1$ | $30$ | $0$ | $C_{30}$ | $[\ ]^{30}$ | ||
\(365159\) | $\Q_{365159}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $29$ | $1$ | $29$ | $0$ | $C_{29}$ | $[\ ]^{29}$ | ||
\(14342233\) | $\Q_{14342233}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $24$ | $1$ | $24$ | $0$ | $C_{24}$ | $[\ ]^{24}$ | ||
\(496\!\cdots\!163\) | $\Q_{49\!\cdots\!63}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
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 $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |