Normalized defining polynomial
\( x^{40} + 20 x^{38} + 231 x^{36} + 1812 x^{34} + 10709 x^{32} + 49280 x^{30} + 181674 x^{28} + 540148 x^{26} + \cdots + 1 \)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(130305099804548492884220428175380349368393046678311823693003457545895936\)
\(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.96\) | 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: | $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(264=2^{3}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(137,·)$, $\chi_{264}(139,·)$, $\chi_{264}(19,·)$, $\chi_{264}(151,·)$, $\chi_{264}(25,·)$, $\chi_{264}(157,·)$, $\chi_{264}(133,·)$, $\chi_{264}(35,·)$, $\chi_{264}(37,·)$, $\chi_{264}(167,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(257,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(185,·)$, $\chi_{264}(181,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(215,·)$, $\chi_{264}(89,·)$, $\chi_{264}(221,·)$, $\chi_{264}(95,·)$, $\chi_{264}(263,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(229,·)$, $\chi_{264}(259,·)$, $\chi_{264}(107,·)$, $\chi_{264}(239,·)$, $\chi_{264}(113,·)$, $\chi_{264}(211,·)$, $\chi_{264}(245,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{43}a^{32}+\frac{7}{43}a^{30}+\frac{11}{43}a^{28}+\frac{14}{43}a^{26}+\frac{2}{43}a^{24}-\frac{16}{43}a^{22}+\frac{6}{43}a^{20}-\frac{4}{43}a^{18}-\frac{21}{43}a^{16}-\frac{14}{43}a^{14}-\frac{16}{43}a^{12}-\frac{10}{43}a^{10}-\frac{20}{43}a^{8}+\frac{8}{43}a^{6}-\frac{18}{43}a^{4}-\frac{11}{43}a^{2}-\frac{20}{43}$, $\frac{1}{43}a^{33}+\frac{7}{43}a^{31}+\frac{11}{43}a^{29}+\frac{14}{43}a^{27}+\frac{2}{43}a^{25}-\frac{16}{43}a^{23}+\frac{6}{43}a^{21}-\frac{4}{43}a^{19}-\frac{21}{43}a^{17}-\frac{14}{43}a^{15}-\frac{16}{43}a^{13}-\frac{10}{43}a^{11}-\frac{20}{43}a^{9}+\frac{8}{43}a^{7}-\frac{18}{43}a^{5}-\frac{11}{43}a^{3}-\frac{20}{43}a$, $\frac{1}{43}a^{34}+\frac{5}{43}a^{30}-\frac{20}{43}a^{28}-\frac{10}{43}a^{26}+\frac{13}{43}a^{24}-\frac{11}{43}a^{22}-\frac{3}{43}a^{20}+\frac{7}{43}a^{18}+\frac{4}{43}a^{16}-\frac{4}{43}a^{14}+\frac{16}{43}a^{12}+\frac{7}{43}a^{10}+\frac{19}{43}a^{8}+\frac{12}{43}a^{6}-\frac{14}{43}a^{4}+\frac{14}{43}a^{2}+\frac{11}{43}$, $\frac{1}{43}a^{35}+\frac{5}{43}a^{31}-\frac{20}{43}a^{29}-\frac{10}{43}a^{27}+\frac{13}{43}a^{25}-\frac{11}{43}a^{23}-\frac{3}{43}a^{21}+\frac{7}{43}a^{19}+\frac{4}{43}a^{17}-\frac{4}{43}a^{15}+\frac{16}{43}a^{13}+\frac{7}{43}a^{11}+\frac{19}{43}a^{9}+\frac{12}{43}a^{7}-\frac{14}{43}a^{5}+\frac{14}{43}a^{3}+\frac{11}{43}a$, $\frac{1}{43}a^{36}-\frac{12}{43}a^{30}+\frac{21}{43}a^{28}-\frac{14}{43}a^{26}-\frac{21}{43}a^{24}-\frac{9}{43}a^{22}+\frac{20}{43}a^{20}-\frac{19}{43}a^{18}+\frac{15}{43}a^{16}+\frac{1}{43}a^{12}-\frac{17}{43}a^{10}-\frac{17}{43}a^{8}-\frac{11}{43}a^{6}+\frac{18}{43}a^{4}-\frac{20}{43}a^{2}+\frac{14}{43}$, $\frac{1}{43}a^{37}-\frac{12}{43}a^{31}+\frac{21}{43}a^{29}-\frac{14}{43}a^{27}-\frac{21}{43}a^{25}-\frac{9}{43}a^{23}+\frac{20}{43}a^{21}-\frac{19}{43}a^{19}+\frac{15}{43}a^{17}+\frac{1}{43}a^{13}-\frac{17}{43}a^{11}-\frac{17}{43}a^{9}-\frac{11}{43}a^{7}+\frac{18}{43}a^{5}-\frac{20}{43}a^{3}+\frac{14}{43}a$, $\frac{1}{22\!\cdots\!29}a^{38}-\frac{23\!\cdots\!28}{22\!\cdots\!29}a^{36}-\frac{28\!\cdots\!32}{22\!\cdots\!29}a^{34}-\frac{47\!\cdots\!21}{22\!\cdots\!29}a^{32}-\frac{10\!\cdots\!91}{22\!\cdots\!29}a^{30}+\frac{24\!\cdots\!38}{97\!\cdots\!23}a^{28}-\frac{30\!\cdots\!91}{22\!\cdots\!29}a^{26}+\frac{11\!\cdots\!15}{22\!\cdots\!29}a^{24}-\frac{91\!\cdots\!99}{22\!\cdots\!29}a^{22}-\frac{77\!\cdots\!31}{22\!\cdots\!29}a^{20}+\frac{85\!\cdots\!52}{22\!\cdots\!29}a^{18}-\frac{29\!\cdots\!33}{22\!\cdots\!29}a^{16}+\frac{10\!\cdots\!59}{22\!\cdots\!29}a^{14}-\frac{23\!\cdots\!62}{22\!\cdots\!29}a^{12}-\frac{78\!\cdots\!19}{22\!\cdots\!29}a^{10}-\frac{78\!\cdots\!30}{22\!\cdots\!29}a^{8}+\frac{11\!\cdots\!43}{22\!\cdots\!29}a^{6}-\frac{31\!\cdots\!92}{22\!\cdots\!29}a^{4}+\frac{31\!\cdots\!25}{22\!\cdots\!29}a^{2}-\frac{47\!\cdots\!98}{22\!\cdots\!29}$, $\frac{1}{22\!\cdots\!29}a^{39}-\frac{23\!\cdots\!28}{22\!\cdots\!29}a^{37}-\frac{28\!\cdots\!32}{22\!\cdots\!29}a^{35}-\frac{47\!\cdots\!21}{22\!\cdots\!29}a^{33}-\frac{10\!\cdots\!91}{22\!\cdots\!29}a^{31}+\frac{24\!\cdots\!38}{97\!\cdots\!23}a^{29}-\frac{30\!\cdots\!91}{22\!\cdots\!29}a^{27}+\frac{11\!\cdots\!15}{22\!\cdots\!29}a^{25}-\frac{91\!\cdots\!99}{22\!\cdots\!29}a^{23}-\frac{77\!\cdots\!31}{22\!\cdots\!29}a^{21}+\frac{85\!\cdots\!52}{22\!\cdots\!29}a^{19}-\frac{29\!\cdots\!33}{22\!\cdots\!29}a^{17}+\frac{10\!\cdots\!59}{22\!\cdots\!29}a^{15}-\frac{23\!\cdots\!62}{22\!\cdots\!29}a^{13}-\frac{78\!\cdots\!19}{22\!\cdots\!29}a^{11}-\frac{78\!\cdots\!30}{22\!\cdots\!29}a^{9}+\frac{11\!\cdots\!43}{22\!\cdots\!29}a^{7}-\frac{31\!\cdots\!92}{22\!\cdots\!29}a^{5}+\frac{31\!\cdots\!25}{22\!\cdots\!29}a^{3}-\frac{47\!\cdots\!98}{22\!\cdots\!29}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{5080791259918300299740496912046655776}{224658185921597823206597305721673242429} a^{38} - \frac{101451924046779396192754697460220609903}{224658185921597823206597305721673242429} a^{36} - \frac{1170410703975544732317290899165648530944}{224658185921597823206597305721673242429} a^{34} - \frac{9169045072830155735773051351456043006888}{224658185921597823206597305721673242429} a^{32} - \frac{54119072247195440536245166968075232034124}{224658185921597823206597305721673242429} a^{30} - \frac{10811815539627020670491264589715384316209}{9767747213982514052460752422681445323} a^{28} - \frac{915237434801245657191272986145370561091636}{224658185921597823206597305721673242429} a^{26} - \frac{2715817121311291661864357525611831054438697}{224658185921597823206597305721673242429} a^{24} - \frac{6545746692282391260349052949038788511323256}{224658185921597823206597305721673242429} a^{22} - \frac{12728755983245911553739024764537008612303686}{224658185921597823206597305721673242429} a^{20} - \frac{19907024450807145541181119779646685184616180}{224658185921597823206597305721673242429} a^{18} - \frac{24577788814115491124269186355572593226390486}{224658185921597823206597305721673242429} a^{16} - \frac{23755874145055474657407286129028929607548560}{224658185921597823206597305721673242429} a^{14} - \frac{17348934078521056249858340015679985489178873}{224658185921597823206597305721673242429} a^{12} - \frac{9478004883568827701515967247945978499664712}{224658185921597823206597305721673242429} a^{10} - \frac{3575881750535688428642484243944196077931123}{224658185921597823206597305721673242429} a^{8} - \frac{942779388565629517685965251399746338891980}{224658185921597823206597305721673242429} a^{6} - \frac{133670352136312976626545402469902327786412}{224658185921597823206597305721673242429} a^{4} - \frac{13728457126047526199356492463843048242376}{224658185921597823206597305721673242429} a^{2} - \frac{47473013766506234347649214060219294152}{224658185921597823206597305721673242429} \)
(order $6$)
| 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
$C_2^2\times C_{10}$ (as 40T7):
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2^2\times C_{10}$ |
| Character table for $C_2^2\times C_{10}$ |
Intermediate fields
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 | ${\href{/padicField/5.10.0.1}{10} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{8}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 $40$ | $4$ | $10$ | $80$ | |||
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |