Normalized defining polynomial
\( x^{38} + 342 x^{36} - 912 x^{35} + 49723 x^{34} - 231002 x^{33} + 4289459 x^{32} - 23568436 x^{31} + \cdots + 40\!\cdots\!79 \)
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: |
\(-426\!\cdots\!375\)
\(\medspace = -\,5^{19}\cdot 19^{73}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(639.79\) | 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: | $5^{1/2}19^{73/38}\approx 639.7907729687945$ | ||
| 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: | \(\Q(\sqrt{-95}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $38$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1805=5\cdot 19^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1805}(1,·)$, $\chi_{1805}(1804,·)$, $\chi_{1805}(1424,·)$, $\chi_{1805}(1426,·)$, $\chi_{1805}(1044,·)$, $\chi_{1805}(1046,·)$, $\chi_{1805}(664,·)$, $\chi_{1805}(666,·)$, $\chi_{1805}(284,·)$, $\chi_{1805}(286,·)$, $\chi_{1805}(1709,·)$, $\chi_{1805}(1711,·)$, $\chi_{1805}(1329,·)$, $\chi_{1805}(1331,·)$, $\chi_{1805}(949,·)$, $\chi_{1805}(951,·)$, $\chi_{1805}(569,·)$, $\chi_{1805}(571,·)$, $\chi_{1805}(189,·)$, $\chi_{1805}(191,·)$, $\chi_{1805}(1614,·)$, $\chi_{1805}(1616,·)$, $\chi_{1805}(1234,·)$, $\chi_{1805}(1236,·)$, $\chi_{1805}(854,·)$, $\chi_{1805}(856,·)$, $\chi_{1805}(474,·)$, $\chi_{1805}(476,·)$, $\chi_{1805}(94,·)$, $\chi_{1805}(96,·)$, $\chi_{1805}(1519,·)$, $\chi_{1805}(1521,·)$, $\chi_{1805}(1139,·)$, $\chi_{1805}(1141,·)$, $\chi_{1805}(759,·)$, $\chi_{1805}(761,·)$, $\chi_{1805}(379,·)$, $\chi_{1805}(381,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{262144}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{293}a^{34}-\frac{108}{293}a^{33}-\frac{32}{293}a^{32}+\frac{10}{293}a^{31}-\frac{73}{293}a^{30}-\frac{129}{293}a^{29}+\frac{5}{293}a^{28}+\frac{95}{293}a^{27}+\frac{41}{293}a^{26}+\frac{101}{293}a^{25}+\frac{24}{293}a^{24}-\frac{26}{293}a^{23}-\frac{65}{293}a^{22}-\frac{119}{293}a^{21}-\frac{58}{293}a^{20}+\frac{75}{293}a^{19}+\frac{15}{293}a^{18}-\frac{58}{293}a^{17}+\frac{63}{293}a^{16}-\frac{95}{293}a^{15}-\frac{90}{293}a^{14}-\frac{140}{293}a^{13}-\frac{105}{293}a^{12}-\frac{25}{293}a^{11}+\frac{87}{293}a^{10}-\frac{10}{293}a^{9}-\frac{8}{293}a^{8}-\frac{27}{293}a^{7}+\frac{20}{293}a^{6}-\frac{26}{293}a^{5}-\frac{35}{293}a^{4}+\frac{62}{293}a^{3}-\frac{54}{293}a^{2}-\frac{115}{293}a+\frac{113}{293}$, $\frac{1}{16660815929}a^{35}+\frac{22345703}{16660815929}a^{34}+\frac{2788981943}{16660815929}a^{33}-\frac{1655418384}{16660815929}a^{32}-\frac{3381698933}{16660815929}a^{31}+\frac{4673071416}{16660815929}a^{30}+\frac{1427125628}{16660815929}a^{29}+\frac{896795694}{16660815929}a^{28}+\frac{3481350688}{16660815929}a^{27}+\frac{5490107885}{16660815929}a^{26}+\frac{5586883362}{16660815929}a^{25}-\frac{2832224872}{16660815929}a^{24}-\frac{5746722963}{16660815929}a^{23}+\frac{2043898784}{16660815929}a^{22}+\frac{3053856779}{16660815929}a^{21}-\frac{54146436}{131187527}a^{20}+\frac{2909057398}{16660815929}a^{19}-\frac{242605084}{16660815929}a^{18}+\frac{4635035959}{16660815929}a^{17}+\frac{6518831411}{16660815929}a^{16}+\frac{1326809735}{16660815929}a^{15}+\frac{6321501819}{16660815929}a^{14}-\frac{2565473917}{16660815929}a^{13}+\frac{3948861917}{16660815929}a^{12}+\frac{6816132582}{16660815929}a^{11}-\frac{8025083870}{16660815929}a^{10}+\frac{5722287160}{16660815929}a^{9}-\frac{2432648505}{16660815929}a^{8}-\frac{1655995047}{16660815929}a^{7}+\frac{4681639920}{16660815929}a^{6}-\frac{5680644987}{16660815929}a^{5}+\frac{7283637888}{16660815929}a^{4}+\frac{3885686580}{16660815929}a^{3}-\frac{4844254552}{16660815929}a^{2}+\frac{4023067919}{16660815929}a+\frac{13482868}{131187527}$, $\frac{1}{16660815929}a^{36}+\frac{13541073}{16660815929}a^{34}-\frac{8134007183}{16660815929}a^{33}-\frac{5428462996}{16660815929}a^{32}-\frac{5369451344}{16660815929}a^{31}-\frac{4600825464}{16660815929}a^{30}+\frac{4450639936}{16660815929}a^{29}+\frac{1093893548}{16660815929}a^{28}-\frac{5070395189}{16660815929}a^{27}-\frac{5717970340}{16660815929}a^{26}+\frac{7937045758}{16660815929}a^{25}-\frac{2863282840}{16660815929}a^{24}-\frac{92596836}{16660815929}a^{23}-\frac{3409590323}{16660815929}a^{22}-\frac{6099986354}{16660815929}a^{21}-\frac{8172381196}{16660815929}a^{20}-\frac{1047410495}{16660815929}a^{19}+\frac{1420852694}{16660815929}a^{18}+\frac{6537005861}{16660815929}a^{17}+\frac{147916739}{16660815929}a^{16}-\frac{4611513862}{16660815929}a^{15}+\frac{1908962229}{16660815929}a^{14}-\frac{1756432330}{16660815929}a^{13}+\frac{7620770822}{16660815929}a^{12}+\frac{5539622498}{16660815929}a^{11}-\frac{1093996274}{16660815929}a^{10}+\frac{1450382557}{16660815929}a^{9}+\frac{4420362854}{16660815929}a^{8}+\frac{8172131945}{16660815929}a^{7}-\frac{1153530750}{16660815929}a^{6}-\frac{5528503164}{16660815929}a^{5}+\frac{702161965}{16660815929}a^{4}-\frac{7508751396}{16660815929}a^{3}-\frac{5350816383}{16660815929}a^{2}+\frac{4785922492}{16660815929}a-\frac{12337416}{131187527}$, $\frac{1}{50\!\cdots\!03}a^{37}-\frac{64\!\cdots\!07}{50\!\cdots\!03}a^{36}-\frac{33\!\cdots\!37}{50\!\cdots\!03}a^{35}+\frac{35\!\cdots\!01}{50\!\cdots\!03}a^{34}-\frac{34\!\cdots\!85}{50\!\cdots\!03}a^{33}-\frac{98\!\cdots\!97}{50\!\cdots\!03}a^{32}+\frac{44\!\cdots\!89}{50\!\cdots\!03}a^{31}+\frac{17\!\cdots\!10}{50\!\cdots\!03}a^{30}-\frac{20\!\cdots\!23}{50\!\cdots\!03}a^{29}+\frac{21\!\cdots\!47}{50\!\cdots\!03}a^{28}+\frac{98\!\cdots\!66}{50\!\cdots\!03}a^{27}-\frac{42\!\cdots\!94}{50\!\cdots\!03}a^{26}+\frac{16\!\cdots\!88}{50\!\cdots\!03}a^{25}-\frac{10\!\cdots\!89}{50\!\cdots\!03}a^{24}+\frac{19\!\cdots\!08}{50\!\cdots\!03}a^{23}-\frac{58\!\cdots\!40}{50\!\cdots\!03}a^{22}-\frac{17\!\cdots\!27}{50\!\cdots\!03}a^{21}-\frac{11\!\cdots\!50}{50\!\cdots\!03}a^{20}+\frac{28\!\cdots\!71}{50\!\cdots\!03}a^{19}-\frac{22\!\cdots\!28}{50\!\cdots\!03}a^{18}+\frac{17\!\cdots\!56}{50\!\cdots\!03}a^{17}-\frac{77\!\cdots\!15}{50\!\cdots\!03}a^{16}-\frac{20\!\cdots\!29}{50\!\cdots\!03}a^{15}-\frac{93\!\cdots\!66}{50\!\cdots\!03}a^{14}+\frac{24\!\cdots\!21}{50\!\cdots\!03}a^{13}+\frac{13\!\cdots\!04}{50\!\cdots\!03}a^{12}+\frac{64\!\cdots\!13}{50\!\cdots\!03}a^{11}-\frac{83\!\cdots\!82}{50\!\cdots\!03}a^{10}-\frac{18\!\cdots\!44}{50\!\cdots\!03}a^{9}-\frac{86\!\cdots\!81}{50\!\cdots\!03}a^{8}+\frac{19\!\cdots\!87}{50\!\cdots\!03}a^{7}-\frac{16\!\cdots\!06}{50\!\cdots\!03}a^{6}+\frac{24\!\cdots\!37}{50\!\cdots\!03}a^{5}-\frac{17\!\cdots\!70}{50\!\cdots\!03}a^{4}-\frac{16\!\cdots\!27}{50\!\cdots\!03}a^{3}+\frac{11\!\cdots\!88}{50\!\cdots\!03}a^{2}-\frac{10\!\cdots\!00}{50\!\cdots\!03}a+\frac{53\!\cdots\!84}{13\!\cdots\!31}$
| 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 $C_{38}$ |
Intermediate fields
| \(\Q(\sqrt{-95}) \), 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 | $19^{2}$ | $19^{2}$ | R | $38$ | $19^{2}$ | $19^{2}$ | $38$ | R | $38$ | $38$ | $38$ | $19^{2}$ | $38$ | $38$ | $38$ | $19^{2}$ | $38$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $38$ | $2$ | $19$ | $19$ | |||
|
\(19\)
| Deg $38$ | $38$ | $1$ | $73$ |