Normalized defining polynomial
\( x^{39} - 237 x^{37} - 158 x^{36} + 24885 x^{35} + 31758 x^{34} - 1521777 x^{33} - 2797074 x^{32} + \cdots - 9364853836691 \)
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: | \(832\!\cdots\!801\) \(\medspace = 3^{52}\cdot 79^{38}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(305.58\) | 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: | $3^{4/3}79^{38/39}\approx 305.58473738571735$ | ||
Ramified primes: | \(3\), \(79\) | 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: | \(711=3^{2}\cdot 79\) | ||
Dirichlet character group: | $\lbrace$$\chi_{711}(640,·)$, $\chi_{711}(1,·)$, $\chi_{711}(130,·)$, $\chi_{711}(10,·)$, $\chi_{711}(268,·)$, $\chi_{711}(13,·)$, $\chi_{711}(655,·)$, $\chi_{711}(277,·)$, $\chi_{711}(151,·)$, $\chi_{711}(664,·)$, $\chi_{711}(541,·)$, $\chi_{711}(289,·)$, $\chi_{711}(547,·)$, $\chi_{711}(292,·)$, $\chi_{711}(421,·)$, $\chi_{711}(49,·)$, $\chi_{711}(169,·)$, $\chi_{711}(682,·)$, $\chi_{711}(46,·)$, $\chi_{711}(433,·)$, $\chi_{711}(694,·)$, $\chi_{711}(64,·)$, $\chi_{711}(652,·)$, $\chi_{711}(202,·)$, $\chi_{711}(76,·)$, $\chi_{711}(589,·)$, $\chi_{711}(334,·)$, $\chi_{711}(598,·)$, $\chi_{711}(88,·)$, $\chi_{711}(100,·)$, $\chi_{711}(490,·)$, $\chi_{711}(493,·)$, $\chi_{711}(496,·)$, $\chi_{711}(241,·)$, $\chi_{711}(499,·)$, $\chi_{711}(121,·)$, $\chi_{711}(634,·)$, $\chi_{711}(460,·)$, $\chi_{711}(637,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{23}a^{21}-\frac{2}{23}a^{20}-\frac{4}{23}a^{19}+\frac{1}{23}a^{18}+\frac{7}{23}a^{17}+\frac{1}{23}a^{16}+\frac{11}{23}a^{15}-\frac{7}{23}a^{14}-\frac{5}{23}a^{13}-\frac{3}{23}a^{12}+\frac{1}{23}a^{10}-\frac{2}{23}a^{9}-\frac{4}{23}a^{8}+\frac{1}{23}a^{7}+\frac{7}{23}a^{6}+\frac{1}{23}a^{5}+\frac{11}{23}a^{4}-\frac{7}{23}a^{3}-\frac{5}{23}a^{2}-\frac{3}{23}a$, $\frac{1}{23}a^{22}-\frac{8}{23}a^{20}-\frac{7}{23}a^{19}+\frac{9}{23}a^{18}-\frac{8}{23}a^{17}-\frac{10}{23}a^{16}-\frac{8}{23}a^{15}+\frac{4}{23}a^{14}+\frac{10}{23}a^{13}-\frac{6}{23}a^{12}+\frac{1}{23}a^{11}-\frac{8}{23}a^{9}-\frac{7}{23}a^{8}+\frac{9}{23}a^{7}-\frac{8}{23}a^{6}-\frac{10}{23}a^{5}-\frac{8}{23}a^{4}+\frac{4}{23}a^{3}+\frac{10}{23}a^{2}-\frac{6}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{23}a^{24}-\frac{1}{23}a^{2}$, $\frac{1}{23}a^{25}-\frac{1}{23}a^{3}$, $\frac{1}{23}a^{26}-\frac{1}{23}a^{4}$, $\frac{1}{23}a^{27}-\frac{1}{23}a^{5}$, $\frac{1}{23}a^{28}-\frac{1}{23}a^{6}$, $\frac{1}{23}a^{29}-\frac{1}{23}a^{7}$, $\frac{1}{23}a^{30}-\frac{1}{23}a^{8}$, $\frac{1}{529}a^{31}+\frac{9}{529}a^{30}-\frac{5}{529}a^{29}+\frac{1}{529}a^{28}+\frac{7}{529}a^{27}+\frac{11}{529}a^{26}-\frac{4}{529}a^{25}-\frac{2}{529}a^{24}+\frac{7}{529}a^{23}-\frac{6}{529}a^{22}-\frac{4}{529}a^{21}+\frac{217}{529}a^{20}+\frac{173}{529}a^{19}-\frac{219}{529}a^{18}+\frac{158}{529}a^{17}+\frac{79}{529}a^{16}-\frac{203}{529}a^{15}-\frac{157}{529}a^{14}+\frac{52}{529}a^{13}-\frac{251}{529}a^{12}-\frac{52}{529}a^{11}-\frac{257}{529}a^{10}+\frac{170}{529}a^{9}-\frac{158}{529}a^{8}-\frac{7}{529}a^{7}+\frac{134}{529}a^{6}+\frac{141}{529}a^{5}-\frac{76}{529}a^{4}-\frac{176}{529}a^{3}+\frac{215}{529}a^{2}-\frac{5}{529}a+\frac{9}{23}$, $\frac{1}{529}a^{32}+\frac{6}{529}a^{30}-\frac{2}{529}a^{28}-\frac{6}{529}a^{27}-\frac{11}{529}a^{26}+\frac{11}{529}a^{25}+\frac{2}{529}a^{24}+\frac{4}{529}a^{22}+\frac{152}{529}a^{20}+\frac{87}{529}a^{19}-\frac{125}{529}a^{18}-\frac{101}{529}a^{17}-\frac{178}{529}a^{16}-\frac{216}{529}a^{15}-\frac{122}{529}a^{14}+\frac{86}{529}a^{13}+\frac{68}{529}a^{12}+\frac{165}{529}a^{11}+\frac{114}{529}a^{10}+\frac{244}{529}a^{9}+\frac{12}{529}a^{8}+\frac{105}{529}a^{7}+\frac{177}{529}a^{6}-\frac{126}{529}a^{5}+\frac{117}{529}a^{4}+\frac{235}{529}a^{3}-\frac{54}{529}a^{2}+\frac{160}{529}a+\frac{11}{23}$, $\frac{1}{12167}a^{33}+\frac{7}{12167}a^{32}+\frac{8}{12167}a^{31}+\frac{60}{12167}a^{30}+\frac{34}{12167}a^{29}-\frac{179}{12167}a^{28}-\frac{16}{12167}a^{27}+\frac{140}{12167}a^{26}+\frac{94}{12167}a^{25}+\frac{240}{12167}a^{24}+\frac{41}{12167}a^{23}-\frac{237}{12167}a^{22}-\frac{201}{12167}a^{21}+\frac{3770}{12167}a^{20}+\frac{1336}{12167}a^{19}-\frac{4565}{12167}a^{18}+\frac{3272}{12167}a^{17}-\frac{4938}{12167}a^{16}-\frac{3282}{12167}a^{15}+\frac{1908}{12167}a^{14}-\frac{4792}{12167}a^{13}-\frac{5772}{12167}a^{12}-\frac{5436}{12167}a^{11}+\frac{6002}{12167}a^{10}+\frac{4774}{12167}a^{9}-\frac{3853}{12167}a^{8}+\frac{4049}{12167}a^{7}+\frac{4325}{12167}a^{6}+\frac{119}{529}a^{5}-\frac{2111}{12167}a^{4}-\frac{5845}{12167}a^{3}-\frac{5584}{12167}a^{2}+\frac{5480}{12167}a-\frac{158}{529}$, $\frac{1}{12167}a^{34}+\frac{5}{12167}a^{32}+\frac{4}{12167}a^{31}-\frac{110}{12167}a^{30}+\frac{112}{12167}a^{29}+\frac{87}{12167}a^{28}-\frac{24}{12167}a^{27}+\frac{195}{12167}a^{26}+\frac{88}{12167}a^{25}+\frac{40}{12167}a^{24}+\frac{5}{12167}a^{23}+\frac{55}{12167}a^{22}-\frac{113}{12167}a^{21}+\frac{5214}{12167}a^{20}-\frac{1980}{12167}a^{19}-\frac{2263}{12167}a^{18}+\frac{4013}{12167}a^{17}-\frac{2825}{12167}a^{16}+\frac{5953}{12167}a^{15}-\frac{5245}{12167}a^{14}+\frac{5807}{12167}a^{13}+\frac{2653}{12167}a^{12}+\frac{1389}{12167}a^{11}-\frac{785}{12167}a^{10}-\frac{2771}{12167}a^{9}+\frac{3006}{12167}a^{8}-\frac{2789}{12167}a^{7}+\frac{5996}{12167}a^{6}-\frac{4319}{12167}a^{5}+\frac{3734}{12167}a^{4}+\frac{3821}{12167}a^{3}+\frac{2409}{12167}a^{2}+\frac{2396}{12167}a+\frac{25}{529}$, $\frac{1}{7677377}a^{35}+\frac{123}{7677377}a^{34}+\frac{125}{7677377}a^{33}-\frac{6683}{7677377}a^{32}+\frac{2745}{7677377}a^{31}+\frac{19979}{7677377}a^{30}-\frac{92227}{7677377}a^{29}-\frac{128011}{7677377}a^{28}-\frac{104175}{7677377}a^{27}+\frac{110954}{7677377}a^{26}-\frac{142329}{7677377}a^{25}+\frac{140008}{7677377}a^{24}+\frac{95819}{7677377}a^{23}+\frac{19221}{7677377}a^{22}-\frac{54287}{7677377}a^{21}+\frac{811924}{7677377}a^{20}+\frac{301101}{7677377}a^{19}+\frac{1328295}{7677377}a^{18}-\frac{1443358}{7677377}a^{17}-\frac{3069034}{7677377}a^{16}-\frac{69895}{7677377}a^{15}+\frac{3746169}{7677377}a^{14}-\frac{36261}{7677377}a^{13}+\frac{2032335}{7677377}a^{12}-\frac{1524641}{7677377}a^{11}-\frac{2971111}{7677377}a^{10}+\frac{3544684}{7677377}a^{9}+\frac{2857076}{7677377}a^{8}-\frac{3443426}{7677377}a^{7}-\frac{3192}{333799}a^{6}+\frac{1928713}{7677377}a^{5}-\frac{85279}{333799}a^{4}+\frac{759302}{7677377}a^{3}+\frac{669350}{7677377}a^{2}-\frac{370606}{7677377}a-\frac{138314}{333799}$, $\frac{1}{176579671}a^{36}-\frac{9}{176579671}a^{35}-\frac{6646}{176579671}a^{34}+\frac{164}{176579671}a^{33}+\frac{166823}{176579671}a^{32}+\frac{56431}{176579671}a^{31}-\frac{700790}{176579671}a^{30}+\frac{3682679}{176579671}a^{29}+\frac{448484}{176579671}a^{28}-\frac{1643509}{176579671}a^{27}+\frac{122273}{176579671}a^{26}-\frac{2978360}{176579671}a^{25}+\frac{1079220}{176579671}a^{24}+\frac{3614315}{176579671}a^{23}+\frac{319975}{176579671}a^{22}+\frac{2854088}{176579671}a^{21}+\frac{7216981}{176579671}a^{20}+\frac{74899205}{176579671}a^{19}-\frac{57576719}{176579671}a^{18}+\frac{39288481}{176579671}a^{17}+\frac{86277168}{176579671}a^{16}+\frac{78856281}{176579671}a^{15}+\frac{34638651}{176579671}a^{14}-\frac{54494221}{176579671}a^{13}+\frac{86150453}{176579671}a^{12}-\frac{88147067}{176579671}a^{11}-\frac{58490228}{176579671}a^{10}+\frac{25334759}{176579671}a^{9}+\frac{56672099}{176579671}a^{8}+\frac{57335180}{176579671}a^{7}+\frac{4915250}{176579671}a^{6}-\frac{19173119}{176579671}a^{5}+\frac{85427054}{176579671}a^{4}+\frac{21206052}{176579671}a^{3}-\frac{65632730}{176579671}a^{2}-\frac{31901994}{176579671}a+\frac{147117}{7677377}$, $\frac{1}{1189970402869}a^{37}-\frac{2651}{1189970402869}a^{36}+\frac{31001}{1189970402869}a^{35}+\frac{38886359}{1189970402869}a^{34}-\frac{11115611}{1189970402869}a^{33}-\frac{161843662}{1189970402869}a^{32}+\frac{727972067}{1189970402869}a^{31}+\frac{14373059791}{1189970402869}a^{30}+\frac{2259890754}{1189970402869}a^{29}-\frac{527065190}{51737843603}a^{28}+\frac{16257557256}{1189970402869}a^{27}-\frac{3241343891}{1189970402869}a^{26}+\frac{14185669448}{1189970402869}a^{25}-\frac{8417944364}{1189970402869}a^{24}-\frac{11901792696}{1189970402869}a^{23}-\frac{12193513235}{1189970402869}a^{22}-\frac{5556251079}{1189970402869}a^{21}+\frac{69823373792}{1189970402869}a^{20}-\frac{498490732946}{1189970402869}a^{19}-\frac{2211258452}{51737843603}a^{18}-\frac{575905482177}{1189970402869}a^{17}-\frac{362725858141}{1189970402869}a^{16}+\frac{200090386717}{1189970402869}a^{15}+\frac{211630611389}{1189970402869}a^{14}+\frac{19980835178}{51737843603}a^{13}+\frac{497931299883}{1189970402869}a^{12}+\frac{514691733375}{1189970402869}a^{11}+\frac{85854937351}{1189970402869}a^{10}+\frac{393769621385}{1189970402869}a^{9}-\frac{432592910181}{1189970402869}a^{8}+\frac{219048886588}{1189970402869}a^{7}-\frac{237279314742}{1189970402869}a^{6}+\frac{294711467268}{1189970402869}a^{5}+\frac{42709660514}{1189970402869}a^{4}-\frac{44327396442}{1189970402869}a^{3}+\frac{460520901123}{1189970402869}a^{2}+\frac{439031891768}{1189970402869}a+\frac{53628003}{176579671}$, $\frac{1}{30\!\cdots\!79}a^{38}+\frac{10\!\cdots\!90}{30\!\cdots\!79}a^{37}-\frac{30\!\cdots\!80}{30\!\cdots\!79}a^{36}-\frac{63\!\cdots\!78}{30\!\cdots\!79}a^{35}-\frac{62\!\cdots\!78}{30\!\cdots\!79}a^{34}-\frac{20\!\cdots\!74}{30\!\cdots\!79}a^{33}+\frac{14\!\cdots\!93}{30\!\cdots\!79}a^{32}-\frac{68\!\cdots\!25}{30\!\cdots\!79}a^{31}+\frac{83\!\cdots\!30}{13\!\cdots\!73}a^{30}+\frac{53\!\cdots\!60}{30\!\cdots\!79}a^{29}-\frac{17\!\cdots\!59}{30\!\cdots\!79}a^{28}+\frac{43\!\cdots\!20}{30\!\cdots\!79}a^{27}-\frac{17\!\cdots\!38}{30\!\cdots\!79}a^{26}-\frac{64\!\cdots\!79}{30\!\cdots\!79}a^{25}+\frac{21\!\cdots\!01}{30\!\cdots\!79}a^{24}-\frac{29\!\cdots\!98}{30\!\cdots\!79}a^{23}+\frac{21\!\cdots\!36}{30\!\cdots\!79}a^{22}+\frac{40\!\cdots\!42}{30\!\cdots\!79}a^{21}-\frac{15\!\cdots\!62}{30\!\cdots\!79}a^{20}+\frac{10\!\cdots\!60}{30\!\cdots\!79}a^{19}+\frac{31\!\cdots\!10}{30\!\cdots\!79}a^{18}-\frac{86\!\cdots\!85}{30\!\cdots\!79}a^{17}-\frac{31\!\cdots\!07}{13\!\cdots\!73}a^{16}-\frac{10\!\cdots\!36}{30\!\cdots\!79}a^{15}-\frac{68\!\cdots\!07}{30\!\cdots\!79}a^{14}-\frac{14\!\cdots\!39}{30\!\cdots\!79}a^{13}-\frac{73\!\cdots\!15}{30\!\cdots\!79}a^{12}-\frac{70\!\cdots\!26}{30\!\cdots\!79}a^{11}+\frac{10\!\cdots\!43}{30\!\cdots\!79}a^{10}-\frac{13\!\cdots\!29}{30\!\cdots\!79}a^{9}-\frac{71\!\cdots\!74}{30\!\cdots\!79}a^{8}-\frac{68\!\cdots\!09}{30\!\cdots\!79}a^{7}+\frac{90\!\cdots\!03}{30\!\cdots\!79}a^{6}+\frac{12\!\cdots\!47}{30\!\cdots\!79}a^{5}-\frac{20\!\cdots\!95}{30\!\cdots\!79}a^{4}-\frac{92\!\cdots\!10}{30\!\cdots\!79}a^{3}-\frac{11\!\cdots\!44}{30\!\cdots\!79}a^{2}+\frac{11\!\cdots\!10}{30\!\cdots\!79}a-\frac{78\!\cdots\!48}{26\!\cdots\!51}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $23$ |
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 $C_{39}$ is not computed |
Intermediate fields
3.3.505521.1, 13.13.59091511031674153381441.1 |
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 | ${\href{/padicField/2.13.0.1}{13} }^{3}$ | R | ${\href{/padicField/5.13.0.1}{13} }^{3}$ | $39$ | $39$ | $39$ | ${\href{/padicField/17.13.0.1}{13} }^{3}$ | $39$ | ${\href{/padicField/23.1.0.1}{1} }^{39}$ | $39$ | ${\href{/padicField/31.13.0.1}{13} }^{3}$ | $39$ | $39$ | ${\href{/padicField/43.13.0.1}{13} }^{3}$ | $39$ | $39$ | $39$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | Deg $39$ | $3$ | $13$ | $52$ | |||
\(79\) | Deg $39$ | $39$ | $1$ | $38$ |