Normalized defining polynomial
\( x^{38} - 2x^{19} + 2 \)
Invariants
Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 19]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-282043823418243542433249166060715711952450320814680588841203531776\) \(\medspace = -\,2^{56}\cdot 19^{38}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(52.77\) | 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: | $2^{28/19}19^{359/342}\approx 61.08583685562279$ | ||
Ramified primes: | \(2\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( a^{19} - 1 \) (order $4$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a-1$, $a^{19}+a-1$, $a^{19}+a^{2}-1$, $a^{21}-a^{20}+a^{19}-a^{2}+a-1$, $a^{20}+a^{2}-a-1$, $a^{37}-a^{33}-a^{32}+a^{28}+a^{27}-a^{23}-a^{22}-a^{18}+a^{17}+2a^{14}+a^{13}-a^{12}-2a^{9}-a^{8}+a^{7}+2a^{4}+a^{3}-a^{2}-1$, $a^{33}+a^{32}+a^{31}-a^{28}-a^{23}-a^{22}-a^{21}+a^{19}+a^{18}+a^{17}-2a^{14}-a^{13}-a^{12}-a^{8}-a^{7}+2a^{4}+a^{3}+a^{2}-1$, $a^{34}-a^{30}+a^{26}-a^{22}-a^{19}+a^{18}-a^{14}+a^{10}-a^{6}+a^{2}+1$, $a^{35}-a^{34}-a^{32}+a^{31}-a^{30}-a^{28}+a^{27}-a^{24}+a^{22}-a^{19}+2a^{18}-a^{16}+a^{14}+2a^{13}-a^{12}-a^{10}+2a^{9}-a^{8}-2a^{6}+2a^{5}-2a^{2}+1$, $a^{32}-a^{27}+a^{24}-a^{19}-2a^{13}+a^{8}-2a^{5}-a^{2}+1$, $a^{36}+a^{35}-a^{31}-a^{30}+a^{28}+a^{26}+a^{25}-a^{21}-a^{20}-2a^{17}+a^{15}+2a^{12}-a^{10}-a^{9}-a^{8}-2a^{7}+a^{5}+a^{3}+a^{2}-1$, $a^{36}+a^{35}+a^{31}+a^{30}+a^{29}+a^{28}-a^{27}-a^{26}-a^{25}-a^{20}-a^{19}+a^{15}-2a^{12}-2a^{11}-a^{10}-a^{9}+a^{8}-a^{5}+a^{3}+a^{2}+2a+1$, $a^{36}+a^{34}+a^{32}+2a^{28}-a^{26}+2a^{24}-a^{22}+a^{20}+a^{19}-2a^{18}-2a^{17}-2a^{14}-2a^{13}-a^{10}-4a^{9}-4a^{5}-a^{2}-2a-1$, $a^{33}-2a^{32}-a^{26}+a^{25}-a^{21}+a^{20}-a^{18}-a^{15}+3a^{13}-a^{9}+a^{8}-a^{6}+a^{4}+a^{3}+2a^{2}-3a+1$, $a^{36}+a^{34}-a^{33}+a^{32}-a^{31}+a^{30}-a^{29}-a^{27}+a^{23}+a^{21}+a^{19}-a^{17}-a^{16}-2a^{15}-2a^{13}-2a^{11}+a^{10}-a^{9}+2a^{8}+2a^{6}+a^{5}+a^{3}+a-1$, $a^{34}+a^{33}-a^{31}-a^{30}-a^{29}-a^{28}+2a^{26}+2a^{25}+a^{24}-a^{23}-2a^{22}-2a^{21}+2a^{19}+2a^{18}+a^{17}-2a^{15}-2a^{14}-a^{13}+a^{12}+2a^{11}+2a^{10}+2a^{9}-3a^{7}-3a^{6}-a^{5}+a^{4}+a^{3}+a^{2}-1$, $a^{37}+a^{36}-a^{34}+a^{32}-a^{29}-a^{28}+a^{27}-a^{23}+a^{22}-a^{20}+a^{19}-2a^{18}-2a^{13}+a^{12}+a^{9}-a^{8}+a^{7}+a^{6}-a^{5}-a^{3}+a^{2}+3a-1$, $a^{37}-a^{36}+4a^{34}-2a^{33}+2a^{31}-3a^{30}+2a^{29}+a^{28}-3a^{27}+4a^{26}+a^{25}-3a^{24}+2a^{23}-2a^{22}+5a^{20}-2a^{19}-2a^{18}+2a^{17}-3a^{16}-3a^{15}+4a^{14}-3a^{13}+3a^{11}-5a^{10}+2a^{9}+a^{8}-7a^{7}+3a^{6}+2a^{5}-3a^{4}+3a^{3}-5a^{2}-4a+5$ (assuming GRH) | 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: | \( 1117851098845078000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{19}\cdot 1117851098845078000 \cdot 1}{4\cdot\sqrt{282043823418243542433249166060715711952450320814680588841203531776}}\cr\approx \mathstrut & 0.770163919195879 \end{aligned}\] (assuming GRH)
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 $C_2\times F_{19}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 19.1.518630842213417245507316350976.1 |
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 | R | $18^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.9.0.1}{9} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }^{6}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.6.0.1}{6} }^{6}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.9.0.1}{9} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | $18^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{18}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | $18^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $18^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $38$ | $38$ | $1$ | $56$ | |||
\(19\) | Deg $38$ | $19$ | $2$ | $38$ |