Normalized defining polynomial
\( x^{38} - 209 x^{35} + 190 x^{34} + 1292 x^{33} + 11818 x^{32} - 17860 x^{31} - 124165 x^{30} + 92378 x^{29} - 315590 x^{28} + 461434 x^{27} + 19355661 x^{26} + \cdots + 5454582062023 \)
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)
| |
Discriminant: | \(-223\!\cdots\!259\) \(\medspace = -\,19^{73}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(286.12\) | 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: | $19^{73/38}\approx 286.1231319470719$ | ||
Ramified primes: | \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
$\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(361=19^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{361}(1,·)$, $\chi_{361}(132,·)$, $\chi_{361}(134,·)$, $\chi_{361}(265,·)$, $\chi_{361}(267,·)$, $\chi_{361}(18,·)$, $\chi_{361}(20,·)$, $\chi_{361}(151,·)$, $\chi_{361}(153,·)$, $\chi_{361}(284,·)$, $\chi_{361}(286,·)$, $\chi_{361}(37,·)$, $\chi_{361}(39,·)$, $\chi_{361}(170,·)$, $\chi_{361}(172,·)$, $\chi_{361}(303,·)$, $\chi_{361}(305,·)$, $\chi_{361}(56,·)$, $\chi_{361}(58,·)$, $\chi_{361}(189,·)$, $\chi_{361}(191,·)$, $\chi_{361}(322,·)$, $\chi_{361}(324,·)$, $\chi_{361}(75,·)$, $\chi_{361}(77,·)$, $\chi_{361}(208,·)$, $\chi_{361}(210,·)$, $\chi_{361}(341,·)$, $\chi_{361}(343,·)$, $\chi_{361}(94,·)$, $\chi_{361}(96,·)$, $\chi_{361}(227,·)$, $\chi_{361}(229,·)$, $\chi_{361}(360,·)$, $\chi_{361}(113,·)$, $\chi_{361}(115,·)$, $\chi_{361}(246,·)$, $\chi_{361}(248,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{262144}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{1123845388013}a^{36}+\frac{81759373168}{1123845388013}a^{35}+\frac{218586334288}{1123845388013}a^{34}-\frac{457169512039}{1123845388013}a^{33}-\frac{182026071013}{1123845388013}a^{32}-\frac{537349195266}{1123845388013}a^{31}-\frac{301263568141}{1123845388013}a^{30}-\frac{348191628129}{1123845388013}a^{29}-\frac{431806770831}{1123845388013}a^{28}-\frac{57842106166}{1123845388013}a^{27}+\frac{236862564189}{1123845388013}a^{26}-\frac{77373154571}{1123845388013}a^{25}-\frac{229247270794}{1123845388013}a^{24}+\frac{158637833776}{1123845388013}a^{23}-\frac{104028860513}{1123845388013}a^{22}-\frac{420125099130}{1123845388013}a^{21}-\frac{439884885972}{1123845388013}a^{20}-\frac{445027056863}{1123845388013}a^{19}-\frac{271202263346}{1123845388013}a^{18}-\frac{394809801340}{1123845388013}a^{17}-\frac{93952947123}{1123845388013}a^{16}-\frac{94967709896}{1123845388013}a^{15}+\frac{437454545342}{1123845388013}a^{14}-\frac{58950789284}{1123845388013}a^{13}+\frac{308122839381}{1123845388013}a^{12}+\frac{464732530922}{1123845388013}a^{11}-\frac{98374494772}{1123845388013}a^{10}-\frac{363358723340}{1123845388013}a^{9}+\frac{50081863872}{1123845388013}a^{8}+\frac{408500857090}{1123845388013}a^{7}-\frac{165018752863}{1123845388013}a^{6}+\frac{140800715710}{1123845388013}a^{5}-\frac{251153399305}{1123845388013}a^{4}-\frac{485051111513}{1123845388013}a^{3}-\frac{393710228217}{1123845388013}a^{2}-\frac{469715890334}{1123845388013}a+\frac{499511957736}{1123845388013}$, $\frac{1}{19\!\cdots\!29}a^{37}+\frac{69\!\cdots\!47}{19\!\cdots\!29}a^{36}-\frac{20\!\cdots\!78}{19\!\cdots\!29}a^{35}-\frac{22\!\cdots\!42}{19\!\cdots\!29}a^{34}+\frac{56\!\cdots\!95}{19\!\cdots\!29}a^{33}-\frac{33\!\cdots\!41}{19\!\cdots\!29}a^{32}+\frac{84\!\cdots\!56}{19\!\cdots\!29}a^{31}+\frac{87\!\cdots\!41}{19\!\cdots\!29}a^{30}-\frac{42\!\cdots\!05}{19\!\cdots\!29}a^{29}-\frac{42\!\cdots\!46}{19\!\cdots\!29}a^{28}-\frac{50\!\cdots\!21}{19\!\cdots\!29}a^{27}+\frac{25\!\cdots\!78}{19\!\cdots\!29}a^{26}-\frac{82\!\cdots\!80}{19\!\cdots\!29}a^{25}-\frac{95\!\cdots\!96}{19\!\cdots\!29}a^{24}+\frac{90\!\cdots\!94}{19\!\cdots\!29}a^{23}-\frac{30\!\cdots\!48}{19\!\cdots\!29}a^{22}-\frac{12\!\cdots\!97}{19\!\cdots\!29}a^{21}+\frac{25\!\cdots\!82}{19\!\cdots\!29}a^{20}+\frac{25\!\cdots\!98}{19\!\cdots\!29}a^{19}-\frac{87\!\cdots\!62}{19\!\cdots\!29}a^{18}+\frac{56\!\cdots\!21}{19\!\cdots\!29}a^{17}+\frac{36\!\cdots\!64}{19\!\cdots\!29}a^{16}-\frac{63\!\cdots\!25}{19\!\cdots\!29}a^{15}-\frac{42\!\cdots\!85}{19\!\cdots\!29}a^{14}+\frac{81\!\cdots\!67}{19\!\cdots\!29}a^{13}+\frac{30\!\cdots\!64}{19\!\cdots\!29}a^{12}-\frac{49\!\cdots\!81}{19\!\cdots\!29}a^{11}+\frac{54\!\cdots\!74}{19\!\cdots\!29}a^{10}+\frac{14\!\cdots\!66}{19\!\cdots\!29}a^{9}-\frac{30\!\cdots\!09}{19\!\cdots\!29}a^{8}-\frac{14\!\cdots\!45}{19\!\cdots\!29}a^{7}+\frac{75\!\cdots\!19}{19\!\cdots\!29}a^{6}-\frac{88\!\cdots\!77}{19\!\cdots\!29}a^{5}+\frac{35\!\cdots\!41}{19\!\cdots\!29}a^{4}+\frac{81\!\cdots\!04}{19\!\cdots\!29}a^{3}+\frac{91\!\cdots\!60}{19\!\cdots\!29}a^{2}-\frac{18\!\cdots\!81}{19\!\cdots\!29}a+\frac{45\!\cdots\!02}{19\!\cdots\!29}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Unit group
Rank: | $18$ | 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 cyclic group of order 38 |
The 38 conjugacy class representatives for $C_{38}$ |
Character table for $C_{38}$ is not computed |
Intermediate fields
\(\Q(\sqrt{-19}) \), 19.19.10842505080063916320800450434338728415281531281.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $19^{2}$ | $38$ | $19^{2}$ | R | $19^{2}$ | $38$ | $38$ | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | $38$ |
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 |
---|---|---|---|---|---|---|---|
\(19\) | Deg $38$ | $38$ | $1$ | $73$ |