Normalized defining polynomial
\( x^{39} - 10 x^{38} - 163 x^{37} + 1930 x^{36} + 10036 x^{35} - 159252 x^{34} - 244026 x^{33} + \cdots + 27293535527 \)
Invariants
| Degree: | $39$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[39, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(156\!\cdots\!969\)
\(\medspace = 7^{26}\cdot 131^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(329.46\) | 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: | $7^{2/3}131^{12/13}\approx 329.4602064654238$ | ||
| Ramified primes: |
\(7\), \(131\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $39$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(917=7\cdot 131\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{917}(1,·)$, $\chi_{917}(898,·)$, $\chi_{917}(375,·)$, $\chi_{917}(263,·)$, $\chi_{917}(394,·)$, $\chi_{917}(99,·)$, $\chi_{917}(39,·)$, $\chi_{917}(170,·)$, $\chi_{917}(176,·)$, $\chi_{917}(694,·)$, $\chi_{917}(183,·)$, $\chi_{917}(569,·)$, $\chi_{917}(60,·)$, $\chi_{917}(445,·)$, $\chi_{917}(191,·)$, $\chi_{917}(576,·)$, $\chi_{917}(193,·)$, $\chi_{917}(324,·)$, $\chi_{917}(438,·)$, $\chi_{917}(456,·)$, $\chi_{917}(715,·)$, $\chi_{917}(718,·)$, $\chi_{917}(848,·)$, $\chi_{917}(849,·)$, $\chi_{917}(211,·)$, $\chi_{917}(473,·)$, $\chi_{917}(604,·)$, $\chi_{917}(477,·)$, $\chi_{917}(739,·)$, $\chi_{917}(870,·)$, $\chi_{917}(361,·)$, $\chi_{917}(107,·)$, $\chi_{917}(492,·)$, $\chi_{917}(113,·)$, $\chi_{917}(631,·)$, $\chi_{917}(505,·)$, $\chi_{917}(506,·)$, $\chi_{917}(893,·)$, $\chi_{917}(767,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{5737727}a^{33}+\frac{149395}{5737727}a^{32}+\frac{356080}{5737727}a^{31}+\frac{2754262}{5737727}a^{30}+\frac{2229349}{5737727}a^{29}-\frac{2103741}{5737727}a^{28}-\frac{1169040}{5737727}a^{27}+\frac{1262892}{5737727}a^{26}-\frac{2722691}{5737727}a^{25}-\frac{1452528}{5737727}a^{24}-\frac{1647051}{5737727}a^{23}-\frac{2537819}{5737727}a^{22}+\frac{1319740}{5737727}a^{21}+\frac{2058212}{5737727}a^{20}+\frac{310529}{5737727}a^{19}-\frac{453193}{5737727}a^{18}-\frac{1996259}{5737727}a^{17}-\frac{629131}{5737727}a^{16}-\frac{2593685}{5737727}a^{15}+\frac{2687784}{5737727}a^{14}+\frac{2604605}{5737727}a^{13}+\frac{2017555}{5737727}a^{12}+\frac{11870}{78599}a^{11}-\frac{2793164}{5737727}a^{10}-\frac{1052838}{5737727}a^{9}-\frac{1807200}{5737727}a^{8}-\frac{428865}{5737727}a^{7}-\frac{697793}{5737727}a^{6}+\frac{823084}{5737727}a^{5}+\frac{1233456}{5737727}a^{4}-\frac{1698445}{5737727}a^{3}+\frac{1341394}{5737727}a^{2}-\frac{143744}{5737727}a-\frac{468226}{5737727}$, $\frac{1}{5737727}a^{34}+\frac{1248085}{5737727}a^{32}+\frac{649679}{5737727}a^{31}-\frac{1125790}{5737727}a^{30}+\frac{2141573}{5737727}a^{29}-\frac{2516497}{5737727}a^{28}-\frac{678461}{5737727}a^{27}+\frac{1203910}{5737727}a^{26}+\frac{1764660}{5737727}a^{25}-\frac{2061631}{5737727}a^{24}+\frac{1961658}{5737727}a^{23}+\frac{1264539}{5737727}a^{22}-\frac{723914}{5737727}a^{21}-\frac{1481281}{5737727}a^{20}-\frac{2410353}{5737727}a^{19}-\frac{45408}{108259}a^{18}+\frac{647895}{5737727}a^{17}+\frac{2463800}{5737727}a^{16}+\frac{340868}{5737727}a^{15}-\frac{1275161}{5737727}a^{14}+\frac{2485539}{5737727}a^{13}+\frac{2512049}{5737727}a^{12}-\frac{458040}{5737727}a^{11}+\frac{1749140}{5737727}a^{10}-\frac{1384441}{5737727}a^{9}-\frac{2528850}{5737727}a^{8}+\frac{2129200}{5737727}a^{7}-\frac{1153544}{5737727}a^{6}+\frac{1826613}{5737727}a^{5}-\frac{1017233}{5737727}a^{4}+\frac{1031048}{5737727}a^{3}-\frac{1847172}{5737727}a^{2}-\frac{2145507}{5737727}a+\frac{1993413}{5737727}$, $\frac{1}{5737727}a^{35}+\frac{1905423}{5737727}a^{32}+\frac{2149922}{5737727}a^{31}+\frac{1627181}{5737727}a^{30}-\frac{658144}{5737727}a^{29}-\frac{82673}{5737727}a^{28}-\frac{1319701}{5737727}a^{27}-\frac{1026171}{5737727}a^{26}+\frac{1870262}{5737727}a^{25}-\frac{376928}{5737727}a^{24}-\frac{278143}{5737727}a^{23}-\frac{1546290}{5737727}a^{22}-\frac{2676110}{5737727}a^{21}+\frac{345343}{5737727}a^{20}-\frac{2747920}{5737727}a^{19}-\frac{1094360}{5737727}a^{18}-\frac{1292849}{5737727}a^{17}+\frac{1365053}{5737727}a^{16}-\frac{1439431}{5737727}a^{15}-\frac{1104370}{5737727}a^{14}+\frac{689744}{5737727}a^{13}+\frac{1231913}{5737727}a^{12}-\frac{910615}{5737727}a^{11}-\frac{513253}{5737727}a^{10}-\frac{2500252}{5737727}a^{9}+\frac{693411}{5737727}a^{8}-\frac{2256395}{5737727}a^{7}+\frac{172596}{5737727}a^{6}+\frac{2830707}{5737727}a^{5}-\frac{1795704}{5737727}a^{4}-\frac{1359497}{5737727}a^{3}+\frac{1058971}{5737727}a^{2}-\frac{524183}{5737727}a-\frac{1647740}{5737727}$, $\frac{1}{29\!\cdots\!77}a^{36}-\frac{2274626859}{29\!\cdots\!77}a^{35}-\frac{596189750}{29\!\cdots\!77}a^{34}-\frac{163735443}{29\!\cdots\!77}a^{33}+\frac{14\!\cdots\!64}{29\!\cdots\!77}a^{32}+\frac{106768340929779}{559537848696109}a^{31}-\frac{75\!\cdots\!04}{29\!\cdots\!77}a^{30}-\frac{41\!\cdots\!94}{29\!\cdots\!77}a^{29}+\frac{11\!\cdots\!25}{29\!\cdots\!77}a^{28}-\frac{22\!\cdots\!61}{29\!\cdots\!77}a^{27}+\frac{12\!\cdots\!00}{29\!\cdots\!77}a^{26}-\frac{17\!\cdots\!43}{29\!\cdots\!77}a^{25}+\frac{82\!\cdots\!94}{29\!\cdots\!77}a^{24}-\frac{908448520328916}{29\!\cdots\!77}a^{23}+\frac{52\!\cdots\!98}{29\!\cdots\!77}a^{22}-\frac{37\!\cdots\!44}{29\!\cdots\!77}a^{21}-\frac{57\!\cdots\!80}{29\!\cdots\!77}a^{20}-\frac{37\!\cdots\!75}{29\!\cdots\!77}a^{19}+\frac{136813646936267}{406239807957449}a^{18}-\frac{14\!\cdots\!69}{29\!\cdots\!77}a^{17}+\frac{92\!\cdots\!85}{29\!\cdots\!77}a^{16}+\frac{909146947841258}{29\!\cdots\!77}a^{15}-\frac{72\!\cdots\!35}{29\!\cdots\!77}a^{14}+\frac{12\!\cdots\!30}{29\!\cdots\!77}a^{13}-\frac{65\!\cdots\!02}{29\!\cdots\!77}a^{12}+\frac{10\!\cdots\!07}{29\!\cdots\!77}a^{11}+\frac{109403544311628}{486155835752357}a^{10}+\frac{35\!\cdots\!92}{29\!\cdots\!77}a^{9}+\frac{75\!\cdots\!95}{29\!\cdots\!77}a^{8}+\frac{12\!\cdots\!12}{29\!\cdots\!77}a^{7}-\frac{57\!\cdots\!31}{29\!\cdots\!77}a^{6}+\frac{78\!\cdots\!29}{29\!\cdots\!77}a^{5}+\frac{11\!\cdots\!09}{29\!\cdots\!77}a^{4}+\frac{49\!\cdots\!89}{29\!\cdots\!77}a^{3}-\frac{11\!\cdots\!75}{29\!\cdots\!77}a^{2}-\frac{73\!\cdots\!35}{29\!\cdots\!77}a+\frac{14\!\cdots\!64}{29\!\cdots\!77}$, $\frac{1}{29\!\cdots\!77}a^{37}-\frac{738017409}{29\!\cdots\!77}a^{35}-\frac{2299576160}{29\!\cdots\!77}a^{34}-\frac{2042148948}{29\!\cdots\!77}a^{33}-\frac{11\!\cdots\!66}{29\!\cdots\!77}a^{32}+\frac{133984210184873}{29\!\cdots\!77}a^{31}-\frac{27\!\cdots\!03}{29\!\cdots\!77}a^{30}+\frac{10\!\cdots\!10}{29\!\cdots\!77}a^{29}+\frac{54\!\cdots\!18}{29\!\cdots\!77}a^{28}-\frac{66\!\cdots\!60}{29\!\cdots\!77}a^{27}-\frac{82\!\cdots\!48}{29\!\cdots\!77}a^{26}+\frac{30\!\cdots\!01}{29\!\cdots\!77}a^{25}+\frac{13\!\cdots\!33}{29\!\cdots\!77}a^{24}-\frac{48\!\cdots\!48}{29\!\cdots\!77}a^{23}-\frac{11\!\cdots\!11}{29\!\cdots\!77}a^{22}-\frac{13\!\cdots\!87}{29\!\cdots\!77}a^{21}-\frac{10\!\cdots\!86}{29\!\cdots\!77}a^{20}-\frac{95\!\cdots\!13}{29\!\cdots\!77}a^{19}+\frac{12\!\cdots\!32}{29\!\cdots\!77}a^{18}-\frac{14\!\cdots\!07}{29\!\cdots\!77}a^{17}+\frac{230893444525255}{559537848696109}a^{16}-\frac{13\!\cdots\!28}{29\!\cdots\!77}a^{15}+\frac{58\!\cdots\!68}{29\!\cdots\!77}a^{14}-\frac{94\!\cdots\!47}{29\!\cdots\!77}a^{13}-\frac{12\!\cdots\!91}{29\!\cdots\!77}a^{12}+\frac{19\!\cdots\!95}{29\!\cdots\!77}a^{11}-\frac{22\!\cdots\!62}{29\!\cdots\!77}a^{10}-\frac{12\!\cdots\!96}{29\!\cdots\!77}a^{9}-\frac{13\!\cdots\!73}{29\!\cdots\!77}a^{8}-\frac{80\!\cdots\!35}{29\!\cdots\!77}a^{7}-\frac{13\!\cdots\!43}{29\!\cdots\!77}a^{6}-\frac{73\!\cdots\!78}{29\!\cdots\!77}a^{5}-\frac{35\!\cdots\!73}{29\!\cdots\!77}a^{4}-\frac{11\!\cdots\!94}{29\!\cdots\!77}a^{3}-\frac{13\!\cdots\!61}{29\!\cdots\!77}a^{2}-\frac{38\!\cdots\!65}{29\!\cdots\!77}a-\frac{14\!\cdots\!74}{29\!\cdots\!77}$, $\frac{1}{53\!\cdots\!13}a^{38}-\frac{58\!\cdots\!64}{53\!\cdots\!13}a^{37}+\frac{81\!\cdots\!95}{53\!\cdots\!13}a^{36}+\frac{10\!\cdots\!21}{59\!\cdots\!17}a^{35}+\frac{21\!\cdots\!28}{53\!\cdots\!13}a^{34}+\frac{34\!\cdots\!83}{53\!\cdots\!13}a^{33}-\frac{13\!\cdots\!32}{30\!\cdots\!81}a^{32}-\frac{26\!\cdots\!98}{53\!\cdots\!13}a^{31}+\frac{11\!\cdots\!48}{53\!\cdots\!13}a^{30}-\frac{12\!\cdots\!76}{53\!\cdots\!13}a^{29}-\frac{40\!\cdots\!76}{53\!\cdots\!13}a^{28}+\frac{79\!\cdots\!58}{53\!\cdots\!13}a^{27}+\frac{47\!\cdots\!84}{53\!\cdots\!13}a^{26}+\frac{24\!\cdots\!59}{53\!\cdots\!13}a^{25}+\frac{21\!\cdots\!60}{53\!\cdots\!13}a^{24}-\frac{19\!\cdots\!62}{53\!\cdots\!13}a^{23}-\frac{13\!\cdots\!32}{53\!\cdots\!13}a^{22}-\frac{20\!\cdots\!24}{53\!\cdots\!13}a^{21}+\frac{87\!\cdots\!36}{53\!\cdots\!13}a^{20}+\frac{20\!\cdots\!09}{53\!\cdots\!13}a^{19}+\frac{47\!\cdots\!99}{53\!\cdots\!13}a^{18}+\frac{12\!\cdots\!82}{53\!\cdots\!13}a^{17}+\frac{24\!\cdots\!20}{53\!\cdots\!13}a^{16}+\frac{21\!\cdots\!26}{53\!\cdots\!13}a^{15}-\frac{26\!\cdots\!58}{53\!\cdots\!13}a^{14}+\frac{10\!\cdots\!42}{53\!\cdots\!13}a^{13}-\frac{26\!\cdots\!16}{53\!\cdots\!13}a^{12}+\frac{24\!\cdots\!84}{53\!\cdots\!13}a^{11}-\frac{60\!\cdots\!98}{53\!\cdots\!13}a^{10}-\frac{23\!\cdots\!19}{53\!\cdots\!13}a^{9}-\frac{24\!\cdots\!16}{53\!\cdots\!13}a^{8}+\frac{22\!\cdots\!82}{53\!\cdots\!13}a^{7}-\frac{22\!\cdots\!45}{53\!\cdots\!13}a^{6}+\frac{19\!\cdots\!84}{53\!\cdots\!13}a^{5}-\frac{23\!\cdots\!75}{53\!\cdots\!13}a^{4}-\frac{13\!\cdots\!61}{53\!\cdots\!13}a^{3}+\frac{95\!\cdots\!33}{30\!\cdots\!81}a^{2}+\frac{72\!\cdots\!45}{53\!\cdots\!13}a-\frac{21\!\cdots\!97}{50\!\cdots\!37}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $38$ | 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 39 |
| The 39 conjugacy class representatives for $C_{39}$ |
| Character table for $C_{39}$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 13.13.25542038069936263923006961.1 |
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 | $39$ | $39$ | $39$ | R | $39$ | ${\href{/padicField/13.13.0.1}{13} }^{3}$ | $39$ | $39$ | $39$ | ${\href{/padicField/29.13.0.1}{13} }^{3}$ | $39$ | $39$ | ${\href{/padicField/41.13.0.1}{13} }^{3}$ | ${\href{/padicField/43.13.0.1}{13} }^{3}$ | $39$ | ${\href{/padicField/53.3.0.1}{3} }^{13}$ | $39$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| Deg $39$ | $3$ | $13$ | $26$ | |||
|
\(131\)
| Deg $39$ | $13$ | $3$ | $36$ |