Normalized defining polynomial
\( x^{38} - 4 x + 4 \)
Invariants
| Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(-78\!\cdots\!744\)\(\medspace = -\,2^{38}\cdot 3\cdot 1061\cdot 24469\cdot 188414113\cdot 393131311\cdot 4977192294056925650789352450614791\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $69.09$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 3, 1061, 24469, 188414113, 393131311, 4977192294056925650789352450614791$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Aut(K/\Q)|$: | $1$ | ||
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2} a^{19}$, $\frac{1}{2} a^{20}$, $\frac{1}{2} a^{21}$, $\frac{1}{2} a^{22}$, $\frac{1}{2} a^{23}$, $\frac{1}{2} a^{24}$, $\frac{1}{2} a^{25}$, $\frac{1}{2} a^{26}$, $\frac{1}{2} a^{27}$, $\frac{1}{2} a^{28}$, $\frac{1}{2} a^{29}$, $\frac{1}{2} a^{30}$, $\frac{1}{2} a^{31}$, $\frac{1}{2} a^{32}$, $\frac{1}{2} a^{33}$, $\frac{1}{2} a^{34}$, $\frac{1}{2} a^{35}$, $\frac{1}{2} a^{36}$, $\frac{1}{6} a^{37} - \frac{1}{6} a^{36} + \frac{1}{6} a^{35} - \frac{1}{6} a^{34} + \frac{1}{6} a^{33} - \frac{1}{6} a^{32} + \frac{1}{6} a^{31} - \frac{1}{6} a^{30} + \frac{1}{6} a^{29} - \frac{1}{6} a^{28} + \frac{1}{6} a^{27} - \frac{1}{6} a^{26} + \frac{1}{6} a^{25} - \frac{1}{6} a^{24} + \frac{1}{6} a^{23} - \frac{1}{6} a^{22} + \frac{1}{6} a^{21} - \frac{1}{6} a^{20} + \frac{1}{6} a^{19} + \frac{1}{3} a^{18} - \frac{1}{3} a^{17} + \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$
Class group and class number
not computed
Unit group
| Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$S_{38}$ (as 38T76):
| A non-solvable group of order 523022617466601111760007224100074291200000000 |
| The 26015 conjugacy class representatives for $S_{38}$ are not computed |
| Character table for $S_{38}$ 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 | $21{,}\,{\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | $35{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ | $30{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | $16{,}\,{\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ | $26{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | $37{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | $37{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | $33{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ | $37{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | $18{,}\,16{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | $33{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.13.0.1}{13} }{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | $32{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | $15{,}\,{\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| 1061 | Data not computed | ||||||
| 24469 | Data not computed | ||||||
| 188414113 | Data not computed | ||||||
| 393131311 | Data not computed | ||||||
| 4977192294056925650789352450614791 | Data not computed | ||||||