Normalized defining polynomial
\( x^{38} - 399 x^{36} - 266 x^{35} + 70148 x^{34} + 87020 x^{33} - 7182209 x^{32} - 12457464 x^{31} + \cdots - 12\!\cdots\!51 \)
Invariants
| Degree: | $38$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[38, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(375\!\cdots\!928\)
\(\medspace = 2^{38}\cdot 3^{19}\cdot 19^{72}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(917.26\) | 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: | $2\cdot 3^{1/2}19^{36/19}\approx 917.2594246234627$ | ||
| Ramified primes: |
\(2\), \(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(4332=2^{2}\cdot 3\cdot 19^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{4332}(1,·)$, $\chi_{4332}(2053,·)$, $\chi_{4332}(647,·)$, $\chi_{4332}(4105,·)$, $\chi_{4332}(2699,·)$, $\chi_{4332}(913,·)$, $\chi_{4332}(2965,·)$, $\chi_{4332}(1559,·)$, $\chi_{4332}(3611,·)$, $\chi_{4332}(1825,·)$, $\chi_{4332}(419,·)$, $\chi_{4332}(3877,·)$, $\chi_{4332}(2471,·)$, $\chi_{4332}(685,·)$, $\chi_{4332}(2737,·)$, $\chi_{4332}(1331,·)$, $\chi_{4332}(3383,·)$, $\chi_{4332}(1597,·)$, $\chi_{4332}(191,·)$, $\chi_{4332}(3649,·)$, $\chi_{4332}(2243,·)$, $\chi_{4332}(4295,·)$, $\chi_{4332}(457,·)$, $\chi_{4332}(2509,·)$, $\chi_{4332}(1103,·)$, $\chi_{4332}(3155,·)$, $\chi_{4332}(1369,·)$, $\chi_{4332}(3421,·)$, $\chi_{4332}(2015,·)$, $\chi_{4332}(4067,·)$, $\chi_{4332}(229,·)$, $\chi_{4332}(2281,·)$, $\chi_{4332}(875,·)$, $\chi_{4332}(2927,·)$, $\chi_{4332}(1141,·)$, $\chi_{4332}(3193,·)$, $\chi_{4332}(1787,·)$, $\chi_{4332}(3839,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{1571}a^{35}-\frac{615}{1571}a^{34}+\frac{97}{1571}a^{33}-\frac{355}{1571}a^{32}+\frac{733}{1571}a^{31}-\frac{242}{1571}a^{30}+\frac{76}{1571}a^{29}+\frac{185}{1571}a^{28}+\frac{144}{1571}a^{27}+\frac{332}{1571}a^{26}+\frac{143}{1571}a^{25}+\frac{61}{1571}a^{24}+\frac{245}{1571}a^{23}-\frac{31}{1571}a^{22}-\frac{406}{1571}a^{21}+\frac{472}{1571}a^{20}-\frac{708}{1571}a^{19}-\frac{532}{1571}a^{18}-\frac{119}{1571}a^{17}+\frac{410}{1571}a^{16}+\frac{385}{1571}a^{15}-\frac{31}{1571}a^{14}+\frac{676}{1571}a^{13}-\frac{557}{1571}a^{12}-\frac{287}{1571}a^{11}+\frac{383}{1571}a^{10}+\frac{583}{1571}a^{9}-\frac{540}{1571}a^{8}+\frac{628}{1571}a^{7}+\frac{244}{1571}a^{6}+\frac{742}{1571}a^{5}+\frac{667}{1571}a^{4}-\frac{287}{1571}a^{3}+\frac{285}{1571}a^{2}-\frac{41}{1571}a-\frac{506}{1571}$, $\frac{1}{88\!\cdots\!89}a^{36}+\frac{24\!\cdots\!56}{88\!\cdots\!89}a^{35}+\frac{25\!\cdots\!91}{88\!\cdots\!89}a^{34}+\frac{47\!\cdots\!48}{88\!\cdots\!89}a^{33}-\frac{16\!\cdots\!28}{88\!\cdots\!89}a^{32}+\frac{42\!\cdots\!19}{88\!\cdots\!89}a^{31}-\frac{70\!\cdots\!96}{88\!\cdots\!89}a^{30}+\frac{43\!\cdots\!60}{88\!\cdots\!89}a^{29}+\frac{25\!\cdots\!02}{88\!\cdots\!89}a^{28}+\frac{41\!\cdots\!71}{88\!\cdots\!89}a^{27}-\frac{31\!\cdots\!36}{88\!\cdots\!89}a^{26}+\frac{12\!\cdots\!53}{88\!\cdots\!89}a^{25}+\frac{26\!\cdots\!69}{88\!\cdots\!89}a^{24}-\frac{12\!\cdots\!42}{88\!\cdots\!89}a^{23}+\frac{34\!\cdots\!80}{88\!\cdots\!89}a^{22}+\frac{68\!\cdots\!02}{88\!\cdots\!89}a^{21}-\frac{13\!\cdots\!57}{88\!\cdots\!89}a^{20}+\frac{19\!\cdots\!33}{88\!\cdots\!89}a^{19}+\frac{31\!\cdots\!94}{88\!\cdots\!89}a^{18}+\frac{11\!\cdots\!47}{88\!\cdots\!89}a^{17}+\frac{18\!\cdots\!06}{88\!\cdots\!89}a^{16}-\frac{16\!\cdots\!02}{88\!\cdots\!89}a^{15}-\frac{42\!\cdots\!96}{88\!\cdots\!89}a^{14}-\frac{65\!\cdots\!54}{88\!\cdots\!89}a^{13}-\frac{38\!\cdots\!75}{88\!\cdots\!89}a^{12}+\frac{60\!\cdots\!72}{88\!\cdots\!89}a^{11}-\frac{33\!\cdots\!63}{88\!\cdots\!89}a^{10}+\frac{36\!\cdots\!40}{88\!\cdots\!89}a^{9}+\frac{16\!\cdots\!43}{88\!\cdots\!89}a^{8}+\frac{42\!\cdots\!66}{88\!\cdots\!89}a^{7}-\frac{87\!\cdots\!80}{88\!\cdots\!89}a^{6}-\frac{31\!\cdots\!55}{88\!\cdots\!89}a^{5}+\frac{28\!\cdots\!29}{88\!\cdots\!89}a^{4}-\frac{80\!\cdots\!27}{88\!\cdots\!89}a^{3}+\frac{40\!\cdots\!26}{88\!\cdots\!89}a^{2}+\frac{35\!\cdots\!37}{88\!\cdots\!89}a-\frac{22\!\cdots\!95}{88\!\cdots\!89}$, $\frac{1}{58\!\cdots\!83}a^{37}+\frac{62\!\cdots\!82}{58\!\cdots\!83}a^{36}+\frac{14\!\cdots\!29}{58\!\cdots\!83}a^{35}+\frac{37\!\cdots\!88}{58\!\cdots\!83}a^{34}-\frac{19\!\cdots\!89}{58\!\cdots\!83}a^{33}+\frac{31\!\cdots\!16}{58\!\cdots\!83}a^{32}+\frac{22\!\cdots\!01}{58\!\cdots\!83}a^{31}+\frac{15\!\cdots\!01}{58\!\cdots\!83}a^{30}-\frac{12\!\cdots\!15}{58\!\cdots\!83}a^{29}+\frac{49\!\cdots\!29}{58\!\cdots\!83}a^{28}+\frac{25\!\cdots\!19}{58\!\cdots\!83}a^{27}+\frac{21\!\cdots\!76}{58\!\cdots\!83}a^{26}-\frac{15\!\cdots\!85}{58\!\cdots\!83}a^{25}-\frac{27\!\cdots\!79}{58\!\cdots\!83}a^{24}-\frac{28\!\cdots\!95}{58\!\cdots\!83}a^{23}+\frac{22\!\cdots\!77}{58\!\cdots\!83}a^{22}+\frac{27\!\cdots\!18}{58\!\cdots\!83}a^{21}+\frac{18\!\cdots\!12}{58\!\cdots\!83}a^{20}-\frac{50\!\cdots\!86}{58\!\cdots\!83}a^{19}-\frac{19\!\cdots\!21}{58\!\cdots\!83}a^{18}-\frac{14\!\cdots\!70}{58\!\cdots\!83}a^{17}+\frac{20\!\cdots\!89}{58\!\cdots\!83}a^{16}+\frac{26\!\cdots\!21}{58\!\cdots\!83}a^{15}-\frac{87\!\cdots\!01}{58\!\cdots\!83}a^{14}+\frac{18\!\cdots\!57}{58\!\cdots\!83}a^{13}+\frac{20\!\cdots\!29}{58\!\cdots\!83}a^{12}+\frac{16\!\cdots\!79}{58\!\cdots\!83}a^{11}+\frac{16\!\cdots\!24}{58\!\cdots\!83}a^{10}+\frac{11\!\cdots\!93}{58\!\cdots\!83}a^{9}-\frac{23\!\cdots\!15}{58\!\cdots\!83}a^{8}-\frac{47\!\cdots\!30}{58\!\cdots\!83}a^{7}+\frac{18\!\cdots\!33}{58\!\cdots\!83}a^{6}-\frac{29\!\cdots\!13}{58\!\cdots\!83}a^{5}+\frac{22\!\cdots\!49}{58\!\cdots\!83}a^{4}+\frac{14\!\cdots\!41}{58\!\cdots\!83}a^{3}+\frac{87\!\cdots\!47}{58\!\cdots\!83}a^{2}-\frac{12\!\cdots\!27}{58\!\cdots\!83}a-\frac{91\!\cdots\!21}{58\!\cdots\!83}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $37$ | 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 $C_{38}$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 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 | R | R | $38$ | $38$ | $19^{2}$ | $19^{2}$ | $38$ | R | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $38$ | $38$ | $19^{2}$ | $38$ | $19^{2}$ |
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 $38$ | $2$ | $19$ | $38$ | |||
|
\(3\)
| Deg $38$ | $2$ | $19$ | $19$ | |||
|
\(19\)
| Deg $38$ | $19$ | $2$ | $72$ |