Normalized defining polynomial
\( x^{38} - 5 \)
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: | \(284791482692749835536885264940522098762203880501401727087795734405517578125\) \(\medspace = 5^{37}\cdot 19^{38}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(91.06\) | 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: | $5^{37/38}19^{359/342}\approx 105.41282038213063$ | ||
Ramified primes: | \(5\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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
$C_2\times F_{19}$ (as 38T9):
A solvable group of order 684 |
The 38 conjugacy class representatives for $C_2\times F_{19}$ |
Character table for $C_2\times F_{19}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 19.1.7547072050706152302261272430419921875.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 38 sibling: | data not computed |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $18^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | R | ${\href{/padicField/7.6.0.1}{6} }^{6}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.3.0.1}{3} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | R | $18^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{19}$ | $18^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $38$ | $38$ | $1$ | $37$ | |||
\(19\) | 19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |
19.19.19.10 | $x^{19} + 19 x + 19$ | $19$ | $1$ | $19$ | $F_{19}$ | $[19/18]_{18}$ |