Normalized defining polynomial
\( x^{40} - 4 x^{38} + 14 x^{36} - 48 x^{34} + 164 x^{32} - 560 x^{30} + 1912 x^{28} - 6528 x^{26} + \cdots + 1024 \)
Invariants
Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(40126953504928778716867347648517224229883090465646395672541598175985664\) \(\medspace = 2^{110}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(58.22\) | 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))
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Galois root discriminant: | $2^{11/4}11^{9/10}\approx 58.22183708777889$ | ||
Ramified primes: | \(2\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(176=2^{4}\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{176}(1,·)$, $\chi_{176}(3,·)$, $\chi_{176}(129,·)$, $\chi_{176}(9,·)$, $\chi_{176}(139,·)$, $\chi_{176}(115,·)$, $\chi_{176}(17,·)$, $\chi_{176}(19,·)$, $\chi_{176}(153,·)$, $\chi_{176}(25,·)$, $\chi_{176}(155,·)$, $\chi_{176}(161,·)$, $\chi_{176}(35,·)$, $\chi_{176}(41,·)$, $\chi_{176}(171,·)$, $\chi_{176}(27,·)$, $\chi_{176}(49,·)$, $\chi_{176}(51,·)$, $\chi_{176}(137,·)$, $\chi_{176}(57,·)$, $\chi_{176}(59,·)$, $\chi_{176}(65,·)$, $\chi_{176}(83,·)$, $\chi_{176}(67,·)$, $\chi_{176}(73,·)$, $\chi_{176}(75,·)$, $\chi_{176}(81,·)$, $\chi_{176}(163,·)$, $\chi_{176}(43,·)$, $\chi_{176}(89,·)$, $\chi_{176}(91,·)$, $\chi_{176}(97,·)$, $\chi_{176}(131,·)$, $\chi_{176}(145,·)$, $\chi_{176}(105,·)$, $\chi_{176}(107,·)$, $\chi_{176}(113,·)$, $\chi_{176}(147,·)$, $\chi_{176}(169,·)$, $\chi_{176}(123,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{32}a^{20}$, $\frac{1}{32}a^{21}$, $\frac{1}{259808}a^{22}+\frac{3363}{8119}$, $\frac{1}{259808}a^{23}+\frac{3363}{8119}a$, $\frac{1}{519616}a^{24}-\frac{2378}{8119}a^{2}$, $\frac{1}{519616}a^{25}-\frac{2378}{8119}a^{3}$, $\frac{1}{519616}a^{26}+\frac{3363}{16238}a^{4}$, $\frac{1}{519616}a^{27}+\frac{3363}{16238}a^{5}$, $\frac{1}{1039232}a^{28}-\frac{1189}{8119}a^{6}$, $\frac{1}{1039232}a^{29}-\frac{1189}{8119}a^{7}$, $\frac{1}{1039232}a^{30}+\frac{3363}{32476}a^{8}$, $\frac{1}{1039232}a^{31}+\frac{3363}{32476}a^{9}$, $\frac{1}{2078464}a^{32}-\frac{1189}{16238}a^{10}$, $\frac{1}{2078464}a^{33}-\frac{1189}{16238}a^{11}$, $\frac{1}{2078464}a^{34}+\frac{3363}{64952}a^{12}$, $\frac{1}{2078464}a^{35}+\frac{3363}{64952}a^{13}$, $\frac{1}{4156928}a^{36}-\frac{1189}{32476}a^{14}$, $\frac{1}{4156928}a^{37}-\frac{1189}{32476}a^{15}$, $\frac{1}{4156928}a^{38}+\frac{3363}{129904}a^{16}$, $\frac{1}{4156928}a^{39}+\frac{3363}{129904}a^{17}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $23$ |
Class group and class number
$C_{5731}$, which has order $5731$ (assuming GRH)
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{41}{2078464} a^{32} + \frac{117708}{8119} a^{10} \) (order $22$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{41}{2078464}a^{32}+\frac{7}{1039232}a^{28}+\frac{117708}{8119}a^{10}+\frac{40391}{8119}a^{6}$, $\frac{239}{4156928}a^{36}+\frac{1372105}{32476}a^{14}-1$, $\frac{7}{1039232}a^{28}-\frac{1}{259808}a^{26}+\frac{40391}{8119}a^{6}-\frac{47321}{16238}a^{4}$, $\frac{1189}{2078464}a^{38}-\frac{9751}{4156928}a^{36}+\frac{4179}{519616}a^{34}-\frac{57113}{2078464}a^{32}+\frac{48755}{519616}a^{30}-\frac{41615}{129904}a^{28}+\frac{284171}{259808}a^{26}-\frac{1940449}{519616}a^{24}+\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{20}+\frac{1189}{8}a^{18}-\frac{1413721}{16238}a^{16}+\frac{71043}{8119}a^{14}-\frac{332927}{64952}a^{12}+\frac{48755}{16238}a^{10}-\frac{57113}{32476}a^{8}+\frac{48749}{8119}a^{6}-\frac{28536}{8119}a^{4}+\frac{2786}{8119}a^{2}-\frac{1393}{8119}$, $\frac{1}{259808}a^{22}+\frac{19601}{8119}$, $\frac{2547}{4156928}a^{38}-\frac{9611}{4156928}a^{36}+\frac{8323}{1039232}a^{34}-\frac{57113}{2078464}a^{32}+\frac{48749}{519616}a^{30}-\frac{166465}{519616}a^{28}+\frac{568343}{519616}a^{26}-\frac{970225}{259808}a^{24}+\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{20}+\frac{1189}{8}a^{18}-\frac{7428869}{129904}a^{16}+\frac{2175865}{64952}a^{14}-\frac{242556}{8119}a^{12}+\frac{48755}{16238}a^{10}-\frac{83230}{8119}a^{8}-\frac{16745}{16238}a^{6}-\frac{37471}{16238}a^{4}-\frac{2955}{8119}a^{2}-\frac{9512}{8119}$, $\frac{51}{519616}a^{38}-\frac{7}{1039232}a^{28}+\frac{1}{259808}a^{22}+\frac{9369319}{129904}a^{16}-\frac{40391}{8119}a^{6}+\frac{11482}{8119}$, $\frac{1}{259808}a^{26}+\frac{1}{259808}a^{24}+\frac{47321}{16238}a^{4}+\frac{19601}{8119}a^{2}+1$, $\frac{3363}{4156928}a^{38}-\frac{3363}{1039232}a^{36}+\frac{23541}{2078464}a^{34}-\frac{80671}{2078464}a^{32}+\frac{137871}{1039232}a^{30}-\frac{117705}{259808}a^{28}+\frac{803757}{519616}a^{26}-\frac{171513}{32476}a^{24}+\frac{4684659}{259808}a^{22}-\frac{985}{16}a^{20}+\frac{3363}{16}a^{18}-\frac{3998607}{32476}a^{16}+\frac{4684659}{64952}a^{14}-\frac{343026}{8119}a^{12}+\frac{1274589}{32476}a^{10}-\frac{746627}{32476}a^{8}+\frac{137883}{16238}a^{6}-\frac{40356}{8119}a^{4}+\frac{23541}{8119}a^{2}-\frac{13452}{8119}$, $\frac{1189}{2078464}a^{38}-\frac{1189}{519616}a^{36}-\frac{35}{1039232}a^{35}+\frac{8323}{1039232}a^{34}-\frac{3567}{129904}a^{32}+\frac{48749}{519616}a^{30}-\frac{332913}{1039232}a^{28}+\frac{284171}{259808}a^{26}-\frac{60639}{16238}a^{24}+\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{20}+\frac{1189}{8}a^{18}-\frac{1413721}{16238}a^{16}+\frac{1656277}{32476}a^{14}-\frac{1607521}{64952}a^{13}-\frac{242556}{8119}a^{12}+\frac{284171}{16238}a^{10}-\frac{83230}{8119}a^{8}+\frac{89140}{8119}a^{6}-\frac{28536}{8119}a^{4}+\frac{16646}{8119}a^{2}-\frac{9512}{8119}$, $\frac{1393}{2078464}a^{38}-\frac{9751}{4156928}a^{36}+\frac{4179}{519616}a^{34}-\frac{57113}{2078464}a^{32}+\frac{48755}{519616}a^{30}-\frac{166467}{519616}a^{28}+\frac{71043}{64952}a^{26}-\frac{1940449}{519616}a^{24}+\frac{1}{259808}a^{23}+\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{20}+\frac{1189}{8}a^{18}-\frac{1940449}{129904}a^{16}+\frac{71043}{8119}a^{14}-\frac{332927}{64952}a^{12}+\frac{48755}{16238}a^{10}-\frac{57113}{32476}a^{8}-\frac{32033}{8119}a^{6}-\frac{9751}{16238}a^{4}+\frac{2786}{8119}a^{2}+\frac{19601}{8119}a-\frac{1393}{8119}$, $\frac{239}{4156928}a^{37}-\frac{7}{1039232}a^{28}+\frac{1}{519616}a^{24}+\frac{1372105}{32476}a^{15}-\frac{40391}{8119}a^{6}+\frac{13860}{8119}a^{2}$, $\frac{3}{259808}a^{30}+\frac{5}{519616}a^{27}+\frac{1}{519616}a^{24}+\frac{275807}{32476}a^{8}+\frac{114243}{16238}a^{5}+\frac{13860}{8119}a^{2}$, $\frac{35}{1039232}a^{34}-\frac{41}{2078464}a^{33}+\frac{41}{2078464}a^{32}+\frac{1607521}{64952}a^{12}-\frac{117708}{8119}a^{11}+\frac{117708}{8119}a^{10}$, $\frac{239}{4156928}a^{36}+\frac{169}{2078464}a^{35}+\frac{35}{1039232}a^{34}+\frac{1372105}{32476}a^{14}+\frac{3880899}{64952}a^{13}+\frac{1607521}{64952}a^{12}$, $\frac{1189}{2078464}a^{39}-\frac{1393}{2078464}a^{38}-\frac{1189}{519616}a^{37}+\frac{9751}{4156928}a^{36}+\frac{8323}{1039232}a^{35}-\frac{4179}{519616}a^{34}-\frac{3567}{129904}a^{33}+\frac{57113}{2078464}a^{32}+\frac{48749}{519616}a^{31}-\frac{48755}{519616}a^{30}-\frac{41615}{129904}a^{29}+\frac{332927}{1039232}a^{28}+\frac{284171}{259808}a^{27}-\frac{71043}{64952}a^{26}-\frac{60639}{16238}a^{25}+\frac{970225}{259808}a^{24}+\frac{1656277}{129904}a^{23}-\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{21}+\frac{1393}{32}a^{20}+\frac{1189}{8}a^{19}-\frac{1189}{8}a^{18}-\frac{1413721}{16238}a^{17}+\frac{1940449}{129904}a^{16}+\frac{1656277}{32476}a^{15}-\frac{71043}{8119}a^{14}-\frac{242556}{8119}a^{13}+\frac{332927}{64952}a^{12}+\frac{284171}{16238}a^{11}-\frac{48755}{16238}a^{10}-\frac{83230}{8119}a^{9}+\frac{57113}{32476}a^{8}+\frac{48749}{8119}a^{7}-\frac{8358}{8119}a^{6}-\frac{28536}{8119}a^{5}+\frac{9751}{16238}a^{4}+\frac{16646}{8119}a^{3}+\frac{11074}{8119}a^{2}-\frac{9512}{8119}a+\frac{1393}{8119}$, $\frac{1189}{2078464}a^{38}-\frac{1189}{519616}a^{36}+\frac{8323}{1039232}a^{34}+\frac{99}{2078464}a^{33}-\frac{3567}{129904}a^{32}+\frac{48749}{519616}a^{30}-\frac{41615}{129904}a^{28}+\frac{284171}{259808}a^{26}-\frac{1940447}{519616}a^{24}+\frac{1656277}{129904}a^{22}-\frac{1393}{32}a^{20}+\frac{1189}{8}a^{18}-\frac{1413721}{16238}a^{16}+\frac{1656277}{32476}a^{14}-\frac{242556}{8119}a^{12}+\frac{1136689}{32476}a^{11}+\frac{284171}{16238}a^{10}-\frac{83230}{8119}a^{8}+\frac{48749}{8119}a^{6}-\frac{28536}{8119}a^{4}+\frac{30506}{8119}a^{2}-\frac{9512}{8119}$, $\frac{577}{4156928}a^{37}+\frac{3}{259808}a^{30}+\frac{6625109}{64952}a^{15}+\frac{275807}{32476}a^{8}-1$, $\frac{41}{2078464}a^{32}+\frac{1}{259808}a^{27}+\frac{117708}{8119}a^{10}+\frac{47321}{16238}a^{5}+1$ (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)]
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Regulator: | \( 59973097516991576 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 59973097516991576 \cdot 5731}{22\cdot\sqrt{40126953504928778716867347648517224229883090465646395672541598175985664}}\cr\approx \mathstrut & 0.717205283868401 \end{aligned}\] (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 $C_2\times C_{20}$ |
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 | $20^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/23.1.0.1}{1} }^{40}$ | $20^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | $20^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | $20^{2}$ | $20^{2}$ |
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 $40$ | $4$ | $10$ | $110$ | |||
\(11\) | Deg $40$ | $10$ | $4$ | $36$ |