Normalized defining polynomial
\( x^{40} + 257x^{32} + 14016x^{24} + 105419x^{16} + 23219x^{8} + 1 \)
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: | \(2806502314667513790456400791754773417894971971642777758942872517738496\) \(\medspace = 2^{120}\cdot 11^{32}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(54.48\) | 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: | $2^{3}11^{4/5}\approx 54.47586502017841$ | ||
Ramified primes: | \(2\), \(11\) | 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: | \(176=2^{4}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(3,·)$, $\chi_{176}(133,·)$, $\chi_{176}(135,·)$, $\chi_{176}(9,·)$, $\chi_{176}(141,·)$, $\chi_{176}(15,·)$, $\chi_{176}(147,·)$, $\chi_{176}(23,·)$, $\chi_{176}(25,·)$, $\chi_{176}(155,·)$, $\chi_{176}(157,·)$, $\chi_{176}(5,·)$, $\chi_{176}(27,·)$, $\chi_{176}(37,·)$, $\chi_{176}(31,·)$, $\chi_{176}(169,·)$, $\chi_{176}(45,·)$, $\chi_{176}(47,·)$, $\chi_{176}(49,·)$, $\chi_{176}(53,·)$, $\chi_{176}(137,·)$, $\chi_{176}(159,·)$, $\chi_{176}(67,·)$, $\chi_{176}(69,·)$, $\chi_{176}(71,·)$, $\chi_{176}(75,·)$, $\chi_{176}(81,·)$, $\chi_{176}(163,·)$, $\chi_{176}(89,·)$, $\chi_{176}(91,·)$, $\chi_{176}(93,·)$, $\chi_{176}(97,·)$, $\chi_{176}(59,·)$, $\chi_{176}(103,·)$, $\chi_{176}(111,·)$, $\chi_{176}(113,·)$, $\chi_{176}(115,·)$, $\chi_{176}(119,·)$, $\chi_{176}(125,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{22148330043533}a^{32}+\frac{2726870929642}{22148330043533}a^{24}-\frac{7580340779727}{22148330043533}a^{16}+\frac{6188800161857}{22148330043533}a^{8}+\frac{6246971138049}{22148330043533}$, $\frac{1}{22148330043533}a^{33}+\frac{2726870929642}{22148330043533}a^{25}-\frac{7580340779727}{22148330043533}a^{17}+\frac{6188800161857}{22148330043533}a^{9}+\frac{6246971138049}{22148330043533}a$, $\frac{1}{22148330043533}a^{34}+\frac{2726870929642}{22148330043533}a^{26}-\frac{7580340779727}{22148330043533}a^{18}+\frac{6188800161857}{22148330043533}a^{10}+\frac{6246971138049}{22148330043533}a^{2}$, $\frac{1}{22148330043533}a^{35}+\frac{2726870929642}{22148330043533}a^{27}-\frac{7580340779727}{22148330043533}a^{19}+\frac{6188800161857}{22148330043533}a^{11}+\frac{6246971138049}{22148330043533}a^{3}$, $\frac{1}{22148330043533}a^{36}+\frac{2726870929642}{22148330043533}a^{28}-\frac{7580340779727}{22148330043533}a^{20}+\frac{6188800161857}{22148330043533}a^{12}+\frac{6246971138049}{22148330043533}a^{4}$, $\frac{1}{22148330043533}a^{37}+\frac{2726870929642}{22148330043533}a^{29}-\frac{7580340779727}{22148330043533}a^{21}+\frac{6188800161857}{22148330043533}a^{13}+\frac{6246971138049}{22148330043533}a^{5}$, $\frac{1}{22148330043533}a^{38}+\frac{2726870929642}{22148330043533}a^{30}-\frac{7580340779727}{22148330043533}a^{22}+\frac{6188800161857}{22148330043533}a^{14}+\frac{6246971138049}{22148330043533}a^{6}$, $\frac{1}{22148330043533}a^{39}+\frac{2726870929642}{22148330043533}a^{31}-\frac{7580340779727}{22148330043533}a^{23}+\frac{6188800161857}{22148330043533}a^{15}+\frac{6246971138049}{22148330043533}a^{7}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2605}$, which has order $2605$ (assuming GRH)
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{43114401572}{22148330043533} a^{35} + \frac{11079877784514}{22148330043533} a^{27} + \frac{604158913402484}{22148330043533} a^{19} + \frac{4538168045437461}{22148330043533} a^{11} + \frac{960702298577351}{22148330043533} a^{3} \) (order $16$) | 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: | $\frac{2378192679}{22148330043533}a^{33}+\frac{611758976686}{22148330043533}a^{25}+\frac{33479942029207}{22148330043533}a^{17}+\frac{258956167086371}{22148330043533}a^{9}+\frac{101926540832058}{22148330043533}a$, $a$, $\frac{1496826812680}{22148330043533}a^{38}+\frac{384683031892851}{22148330043533}a^{30}+\frac{20\!\cdots\!32}{22148330043533}a^{22}+\frac{15\!\cdots\!67}{22148330043533}a^{14}+\frac{34\!\cdots\!26}{22148330043533}a^{6}$, $\frac{58742050677}{22148330043533}a^{35}+\frac{15096876188376}{22148330043533}a^{27}+\frac{823368847411579}{22148330043533}a^{19}+\frac{61\!\cdots\!53}{22148330043533}a^{11}+\frac{13\!\cdots\!83}{22148330043533}a^{3}$, $\frac{1791689491944}{22148330043533}a^{38}-\frac{10370405466}{22148330043533}a^{34}+\frac{460464750524328}{22148330043533}a^{30}-\frac{2665377088768}{22148330043533}a^{26}+\frac{25\!\cdots\!77}{22148330043533}a^{22}-\frac{145398998848981}{22148330043533}a^{18}+\frac{18\!\cdots\!86}{22148330043533}a^{14}-\frac{10\!\cdots\!69}{22148330043533}a^{10}+\frac{41\!\cdots\!81}{22148330043533}a^{6}-\frac{273435950860925}{22148330043533}a^{2}-1$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{513313738084}{22148330043533}a^{37}+\frac{1594541}{246385481}a^{36}+\frac{460464750524328}{22148330043533}a^{30}+\frac{131920969892455}{22148330043533}a^{29}+\frac{409801234}{246385481}a^{28}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{22350169159}{246385481}a^{20}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{168154800730}{246385481}a^{12}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{11\!\cdots\!59}{22148330043533}a^{5}+\frac{37547254457}{246385481}a^{4}$, $\frac{1791689491944}{22148330043533}a^{38}-\frac{43114401572}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}-\frac{11079877784514}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}-\frac{604158913402484}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}-\frac{960702298577351}{22148330043533}a^{3}+1$, $\frac{6308913397526}{22148330043533}a^{39}-\frac{43114401572}{22148330043533}a^{35}+\frac{16\!\cdots\!55}{22148330043533}a^{31}-\frac{11079877784514}{22148330043533}a^{27}+\frac{88\!\cdots\!96}{22148330043533}a^{23}-\frac{604158913402484}{22148330043533}a^{19}+\frac{66\!\cdots\!34}{22148330043533}a^{15}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{14\!\cdots\!90}{22148330043533}a^{7}-\frac{960702298577351}{22148330043533}a^{3}-a$, $\frac{1594541}{246385481}a^{36}-\frac{20189716212}{22148330043533}a^{34}+\frac{409801234}{246385481}a^{28}-\frac{5188940493763}{22148330043533}a^{26}+\frac{22350169159}{246385481}a^{20}-\frac{283025395871412}{22148330043533}a^{18}+\frac{168154800730}{246385481}a^{12}-\frac{21\!\cdots\!93}{22148330043533}a^{10}+\frac{37547254457}{246385481}a^{4}-\frac{460070271039662}{22148330043533}a^{2}+1$, $\frac{5231730401719}{22148330043533}a^{39}-\frac{513313738084}{22148330043533}a^{37}+\frac{4562067107}{22148330043533}a^{33}+\frac{13\!\cdots\!22}{22148330043533}a^{31}-\frac{131920969892455}{22148330043533}a^{29}+\frac{1171942089901}{22148330043533}a^{25}+\frac{73\!\cdots\!44}{22148330043533}a^{23}-\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{63815461862317}{22148330043533}a^{17}+\frac{55\!\cdots\!68}{22148330043533}a^{15}-\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{474674153821901}{22148330043533}a^{9}+\frac{12\!\cdots\!42}{22148330043533}a^{7}-\frac{11\!\cdots\!59}{22148330043533}a^{5}+\frac{77834996691430}{22148330043533}a$, $\frac{6308913397526}{22148330043533}a^{39}-\frac{43114401572}{22148330043533}a^{35}+\frac{2378192679}{22148330043533}a^{33}+\frac{16\!\cdots\!55}{22148330043533}a^{31}-\frac{11079877784514}{22148330043533}a^{27}+\frac{611758976686}{22148330043533}a^{25}+\frac{88\!\cdots\!96}{22148330043533}a^{23}-\frac{604158913402484}{22148330043533}a^{19}+\frac{33479942029207}{22148330043533}a^{17}+\frac{66\!\cdots\!34}{22148330043533}a^{15}-\frac{45\!\cdots\!61}{22148330043533}a^{11}+\frac{258956167086371}{22148330043533}a^{9}+\frac{14\!\cdots\!90}{22148330043533}a^{7}-\frac{960702298577351}{22148330043533}a^{3}+\frac{101926540832058}{22148330043533}a$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{10370405466}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}+\frac{2665377088768}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{145398998848981}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{10\!\cdots\!69}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{273435950860925}{22148330043533}a^{3}+1$, $\frac{20189716212}{22148330043533}a^{34}-\frac{2734969148}{22148330043533}a^{33}+\frac{1827097959}{22148330043533}a^{32}+\frac{5188940493763}{22148330043533}a^{26}-\frac{701996796988}{22148330043533}a^{25}+\frac{469945292913}{22148330043533}a^{24}+\frac{283025395871412}{22148330043533}a^{18}-\frac{38108121659660}{22148330043533}a^{17}+\frac{25707340202657}{22148330043533}a^{16}+\frac{21\!\cdots\!93}{22148330043533}a^{10}-\frac{276922940567475}{22148330043533}a^{9}+\frac{197751213254426}{22148330043533}a^{8}+\frac{460070271039662}{22148330043533}a^{2}-\frac{40561756498027}{22148330043533}a+\frac{37273240193403}{22148330043533}$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{664838343235}{22148330043533}a^{37}+\frac{1594541}{246385481}a^{36}+\frac{460464750524328}{22148330043533}a^{30}+\frac{170864426195970}{22148330043533}a^{29}+\frac{409801234}{246385481}a^{28}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{93\!\cdots\!30}{22148330043533}a^{21}+\frac{22350169159}{246385481}a^{20}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{70\!\cdots\!05}{22148330043533}a^{13}+\frac{168154800730}{246385481}a^{12}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{15\!\cdots\!13}{22148330043533}a^{5}+\frac{37547254457}{246385481}a^{4}$, $\frac{16916599641814}{22148330043533}a^{39}-\frac{9819310746}{22148330043533}a^{33}+\frac{551094720}{22148330043533}a^{32}+\frac{43\!\cdots\!59}{22148330043533}a^{31}-\frac{2523563404995}{22148330043533}a^{25}+\frac{141813683773}{22148330043533}a^{24}+\frac{23\!\cdots\!84}{22148330043533}a^{23}-\frac{137626397022431}{22148330043533}a^{17}+\frac{7772601826550}{22148330043533}a^{16}+\frac{17\!\cdots\!59}{22148330043533}a^{15}-\frac{10\!\cdots\!24}{22148330043533}a^{9}+\frac{61204953831945}{22148330043533}a^{8}+\frac{39\!\cdots\!85}{22148330043533}a^{7}-\frac{208782650222270}{22148330043533}a+\frac{42504970595122}{22148330043533}$, $\frac{6308913397526}{22148330043533}a^{39}+\frac{1496826812680}{22148330043533}a^{38}+\frac{513313738084}{22148330043533}a^{37}+\frac{16\!\cdots\!55}{22148330043533}a^{31}+\frac{384683031892851}{22148330043533}a^{30}+\frac{131920969892455}{22148330043533}a^{29}+\frac{88\!\cdots\!96}{22148330043533}a^{23}+\frac{20\!\cdots\!32}{22148330043533}a^{22}+\frac{71\!\cdots\!72}{22148330043533}a^{21}+\frac{66\!\cdots\!34}{22148330043533}a^{15}+\frac{15\!\cdots\!67}{22148330043533}a^{14}+\frac{54\!\cdots\!76}{22148330043533}a^{13}+\frac{14\!\cdots\!90}{22148330043533}a^{7}+\frac{34\!\cdots\!26}{22148330043533}a^{6}+\frac{11\!\cdots\!59}{22148330043533}a^{5}$, $\frac{10607686244288}{22148330043533}a^{39}-\frac{858731936987}{22148330043533}a^{37}-\frac{1594541}{246385481}a^{36}+\frac{22924685360}{22148330043533}a^{34}+\frac{27\!\cdots\!04}{22148330043533}a^{31}-\frac{220694370736755}{22148330043533}a^{29}-\frac{409801234}{246385481}a^{28}+\frac{5890937290751}{22148330043533}a^{26}+\frac{14\!\cdots\!88}{22148330043533}a^{23}-\frac{12\!\cdots\!25}{22148330043533}a^{21}-\frac{22350169159}{246385481}a^{20}+\frac{321133517531072}{22148330043533}a^{18}+\frac{11\!\cdots\!25}{22148330043533}a^{15}-\frac{90\!\cdots\!09}{22148330043533}a^{13}-\frac{168154800730}{246385481}a^{12}+\frac{24\!\cdots\!68}{22148330043533}a^{10}+\frac{24\!\cdots\!95}{22148330043533}a^{7}-\frac{19\!\cdots\!44}{22148330043533}a^{5}-\frac{37547254457}{246385481}a^{4}+\frac{500632027537689}{22148330043533}a^{2}-1$, $\frac{1791689491944}{22148330043533}a^{38}+\frac{31111216398}{22148330043533}a^{35}+\frac{460464750524328}{22148330043533}a^{30}+\frac{7996131266304}{22148330043533}a^{27}+\frac{25\!\cdots\!77}{22148330043533}a^{22}+\frac{436196996546943}{22148330043533}a^{19}+\frac{18\!\cdots\!86}{22148330043533}a^{14}+\frac{32\!\cdots\!07}{22148330043533}a^{11}+\frac{41\!\cdots\!81}{22148330043533}a^{6}+\frac{798159522539242}{22148330043533}a^{3}+1$, $\frac{513313738084}{22148330043533}a^{37}-\frac{43114401572}{22148330043533}a^{35}-\frac{551094720}{22148330043533}a^{32}+\frac{131920969892455}{22148330043533}a^{29}-\frac{11079877784514}{22148330043533}a^{27}-\frac{141813683773}{22148330043533}a^{24}+\frac{71\!\cdots\!72}{22148330043533}a^{21}-\frac{604158913402484}{22148330043533}a^{19}-\frac{7772601826550}{22148330043533}a^{16}+\frac{54\!\cdots\!76}{22148330043533}a^{13}-\frac{45\!\cdots\!61}{22148330043533}a^{11}-\frac{61204953831945}{22148330043533}a^{8}+\frac{11\!\cdots\!59}{22148330043533}a^{5}-\frac{960702298577351}{22148330043533}a^{3}-\frac{42504970595122}{22148330043533}$ (assuming GRH) | 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: | \( 4506035972876552.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 4506035972876552.0 \cdot 2605}{16\cdot\sqrt{2806502314667513790456400791754773417894971971642777758942872517738496}}\cr\approx \mathstrut & 0.127349447243315 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{20}$ (as 40T2):
An abelian group of order 40 |
The 40 conjugacy class representatives for $C_2\times C_{20}$ |
Character table for $C_2\times C_{20}$ is not computed |
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 | $20^{2}$ | $20^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | $20^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{8}$ | $20^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | $20^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | $20^{2}$ | $20^{2}$ |
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 $40$ | $8$ | $5$ | $120$ | |||
\(11\) | 11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |
11.20.16.1 | $x^{20} + 40 x^{18} + 50 x^{17} + 650 x^{16} + 1644 x^{15} + 6440 x^{14} + 18720 x^{13} + 39010 x^{12} + 137400 x^{11} + 283734 x^{10} + 677980 x^{9} + 845540 x^{8} + 1499320 x^{7} + 2045360 x^{6} + 2492700 x^{5} + 5064740 x^{4} + 1884960 x^{3} + 9207500 x^{2} - 2109440 x + 7196441$ | $5$ | $4$ | $16$ | 20T1 | $[\ ]_{5}^{4}$ |