Normalized defining polynomial
\( x^{37} - x - 4 \)
Invariants
Degree: | $37$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[1, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(725\!\cdots\!736\)\(\medspace = 2^{38}\cdot 40158589\cdot 859997730921167\cdot 76406102828442561567149509932891263\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $72.63$ | 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: | $2, 40158589, 859997730921167, 76406102828442561567149509932891263$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2} a^{19} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{30} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{31} - \frac{1}{2} a^{13}$, $\frac{1}{2} a^{32} - \frac{1}{2} a^{14}$, $\frac{1}{2} a^{33} - \frac{1}{2} a^{15}$, $\frac{1}{2} a^{34} - \frac{1}{2} a^{16}$, $\frac{1}{2} a^{35} - \frac{1}{2} a^{17}$, $\frac{1}{2} a^{36} - \frac{1}{2} a^{18}$
Class group and class number
not computed
Unit group
Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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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
$S_{37}$ (as 37T11):
A non-solvable group of order 13763753091226345046315979581580902400000000 |
The 21637 conjugacy class representatives for $S_{37}$ are not computed |
Character table for $S_{37}$ 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 | $29{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ | $28{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | $37$ | $33{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }$ | $37$ | $28{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | $20{,}\,{\href{/LocalNumberField/19.11.0.1}{11} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ | $25{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ | $21{,}\,{\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $29{,}\,{\href{/LocalNumberField/31.8.0.1}{8} }$ | $37$ | $32{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | $21{,}\,{\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | $17{,}\,15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ | $31{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ | $21{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.4.6.3 | $x^{4} + 2 x^{2} + 20$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
2.8.8.7 | $x^{8} + 2 x^{6} + 4 x^{5} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
2.12.12.19 | $x^{12} - 6 x^{10} + 27 x^{8} - 4 x^{6} + 7 x^{4} + 10 x^{2} + 29$ | $2$ | $6$ | $12$ | 12T105 | $[2, 2, 2, 2]^{12}$ | |
2.12.12.16 | $x^{12} - 16 x^{10} - 23 x^{8} + 24 x^{6} - 29 x^{4} - 8 x^{2} - 13$ | $2$ | $6$ | $12$ | 12T134 | $[2, 2, 2, 2, 2, 2]^{6}$ | |
40158589 | Data not computed | ||||||
859997730921167 | Data not computed | ||||||
76406102828442561567149509932891263 | Data not computed |