Normalized defining polynomial
\( x^{40} - 3 x^{38} + 8 x^{36} - 21 x^{34} + 55 x^{32} - 144 x^{30} + 377 x^{28} - 987 x^{26} + 2584 x^{24} + \cdots + 1 \)
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: | \(3241429490243539842962382172914969829546393600000000000000000000\) \(\medspace = 2^{40}\cdot 5^{20}\cdot 11^{36}\) | 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: | \(38.71\) | 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: | $2\cdot 5^{1/2}11^{9/10}\approx 38.70511966206675$ | ||
Ramified primes: | \(2\), \(5\), \(11\) | 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)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(220=2^{2}\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{220}(1,·)$, $\chi_{220}(131,·)$, $\chi_{220}(179,·)$, $\chi_{220}(129,·)$, $\chi_{220}(9,·)$, $\chi_{220}(139,·)$, $\chi_{220}(141,·)$, $\chi_{220}(19,·)$, $\chi_{220}(21,·)$, $\chi_{220}(151,·)$, $\chi_{220}(29,·)$, $\chi_{220}(159,·)$, $\chi_{220}(161,·)$, $\chi_{220}(91,·)$, $\chi_{220}(39,·)$, $\chi_{220}(41,·)$, $\chi_{220}(171,·)$, $\chi_{220}(49,·)$, $\chi_{220}(51,·)$, $\chi_{220}(181,·)$, $\chi_{220}(31,·)$, $\chi_{220}(61,·)$, $\chi_{220}(191,·)$, $\chi_{220}(69,·)$, $\chi_{220}(71,·)$, $\chi_{220}(201,·)$, $\chi_{220}(119,·)$, $\chi_{220}(79,·)$, $\chi_{220}(81,·)$, $\chi_{220}(211,·)$, $\chi_{220}(89,·)$, $\chi_{220}(219,·)$, $\chi_{220}(199,·)$, $\chi_{220}(59,·)$, $\chi_{220}(101,·)$, $\chi_{220}(109,·)$, $\chi_{220}(111,·)$, $\chi_{220}(189,·)$, $\chi_{220}(169,·)$, $\chi_{220}(149,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{17711}a^{22}-\frac{6765}{17711}$, $\frac{1}{17711}a^{23}-\frac{6765}{17711}a$, $\frac{1}{17711}a^{24}-\frac{6765}{17711}a^{2}$, $\frac{1}{17711}a^{25}-\frac{6765}{17711}a^{3}$, $\frac{1}{17711}a^{26}-\frac{6765}{17711}a^{4}$, $\frac{1}{17711}a^{27}-\frac{6765}{17711}a^{5}$, $\frac{1}{17711}a^{28}-\frac{6765}{17711}a^{6}$, $\frac{1}{17711}a^{29}-\frac{6765}{17711}a^{7}$, $\frac{1}{17711}a^{30}-\frac{6765}{17711}a^{8}$, $\frac{1}{17711}a^{31}-\frac{6765}{17711}a^{9}$, $\frac{1}{17711}a^{32}-\frac{6765}{17711}a^{10}$, $\frac{1}{17711}a^{33}-\frac{6765}{17711}a^{11}$, $\frac{1}{17711}a^{34}-\frac{6765}{17711}a^{12}$, $\frac{1}{17711}a^{35}-\frac{6765}{17711}a^{13}$, $\frac{1}{17711}a^{36}-\frac{6765}{17711}a^{14}$, $\frac{1}{17711}a^{37}-\frac{6765}{17711}a^{15}$, $\frac{1}{17711}a^{38}-\frac{6765}{17711}a^{16}$, $\frac{1}{17711}a^{39}-\frac{6765}{17711}a^{17}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{62}$, which has order $62$ (assuming GRH)
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( \frac{1597}{17711} a^{39} + \frac{63245986}{17711} a^{17} \) (order $44$) | 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: | $\frac{1}{17711}a^{23}+\frac{46368}{17711}a$, $\frac{6765}{17711}a^{38}-\frac{20672}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974160}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{2550408}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{372075}{17711}a^{6}-\frac{142065}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974168}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{54264}{17711}a^{6}-\frac{142065}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{55}{17711}a^{32}+\frac{2178309}{17711}a^{10}-1$, $\frac{144}{17711}a^{34}-\frac{8}{17711}a^{28}+\frac{5702887}{17711}a^{12}-\frac{317811}{17711}a^{6}+1$, $\frac{13}{17711}a^{29}+\frac{514229}{17711}a^{7}+1$, $\frac{34}{17711}a^{31}+\frac{21}{17711}a^{30}+\frac{1346269}{17711}a^{9}+\frac{832040}{17711}a^{8}$, $\frac{1597}{17711}a^{39}-\frac{6765}{17711}a^{38}+\frac{20295}{17711}a^{36}-\frac{54120}{17711}a^{34}+\frac{142065}{17711}a^{32}-\frac{372075}{17711}a^{30}+\frac{974160}{17711}a^{28}-\frac{2550405}{17711}a^{26}+\frac{6677055}{17711}a^{24}-\frac{17480760}{17711}a^{22}+2584a^{20}-6765a^{18}+\frac{63245986}{17711}a^{17}+\frac{45765225}{17711}a^{16}-\frac{17480760}{17711}a^{14}+\frac{6677055}{17711}a^{12}-\frac{2550405}{17711}a^{10}+\frac{974160}{17711}a^{8}-\frac{372075}{17711}a^{6}+\frac{142065}{17711}a^{4}-\frac{54120}{17711}a^{2}+\frac{20295}{17711}$, $\frac{34}{17711}a^{31}+\frac{1346269}{17711}a^{9}-1$, $\frac{10946}{17711}a^{39}-\frac{32838}{17711}a^{37}+\frac{87568}{17711}a^{35}-\frac{229866}{17711}a^{33}+\frac{602030}{17711}a^{31}-\frac{1576224}{17711}a^{29}+\frac{4126642}{17711}a^{27}+\frac{3}{17711}a^{26}-\frac{10803702}{17711}a^{25}+\frac{28284464}{17711}a^{23}-4181a^{21}+10946a^{19}-\frac{74049690}{17711}a^{17}+\frac{28284464}{17711}a^{15}-\frac{10803702}{17711}a^{13}+\frac{4126642}{17711}a^{11}-\frac{1576224}{17711}a^{9}+\frac{602030}{17711}a^{7}-\frac{229866}{17711}a^{5}+\frac{121393}{17711}a^{4}+\frac{87568}{17711}a^{3}-\frac{32838}{17711}a$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54120}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974168}{17711}a^{28}+\frac{2550405}{17711}a^{26}-\frac{6677057}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}-\frac{6677055}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{54264}{17711}a^{6}-\frac{142065}{17711}a^{4}-\frac{20905}{17711}a^{2}-\frac{20295}{17711}$, $\frac{4791}{17711}a^{38}-\frac{13153}{17711}a^{36}+\frac{33681}{17711}a^{34}-\frac{87890}{17711}a^{32}+\frac{229989}{17711}a^{30}-\frac{602077}{17711}a^{28}+\frac{1576242}{17711}a^{26}-\frac{4126649}{17711}a^{24}+\frac{10803705}{17711}a^{22}-1597a^{20}+4181a^{18}-\frac{4126648}{17711}a^{16}-\frac{13354113}{17711}a^{14}+\frac{5100818}{17711}a^{12}-\frac{1948341}{17711}a^{10}+\frac{744205}{17711}a^{8}-\frac{284274}{17711}a^{6}+\frac{108617}{17711}a^{4}-\frac{41577}{17711}a^{2}+\frac{16114}{17711}$, $\frac{6765}{17711}a^{38}-\frac{20295}{17711}a^{36}+\frac{54353}{17711}a^{34}-\frac{142065}{17711}a^{32}+\frac{372075}{17711}a^{30}-\frac{974160}{17711}a^{28}+\frac{2550408}{17711}a^{26}-\frac{6677055}{17711}a^{24}+\frac{17480760}{17711}a^{22}-2584a^{20}+6765a^{18}-\frac{45765225}{17711}a^{16}+\frac{17480760}{17711}a^{14}+\frac{2550410}{17711}a^{12}+\frac{2550405}{17711}a^{10}-\frac{974160}{17711}a^{8}+\frac{372075}{17711}a^{6}-\frac{20672}{17711}a^{4}+\frac{54120}{17711}a^{2}-\frac{20295}{17711}$, $\frac{7752}{17711}a^{39}-\frac{20672}{17711}a^{37}+\frac{54264}{17711}a^{35}-\frac{142120}{17711}a^{33}+\frac{1870}{89}a^{31}-\frac{974181}{17711}a^{29}+\frac{2550408}{17711}a^{27}-\frac{6677056}{17711}a^{25}+\frac{17480760}{17711}a^{23}-2584a^{21}+6765a^{19}-\frac{6677056}{17711}a^{17}+\frac{2550408}{17711}a^{15}-\frac{974168}{17711}a^{13}+\frac{372096}{17711}a^{11}+\frac{6051}{89}a^{9}-\frac{459965}{17711}a^{7}-\frac{20672}{17711}a^{5}+\frac{7752}{17711}a^{3}-\frac{2584}{17711}a$, $\frac{610}{17711}a^{37}-\frac{13}{17711}a^{29}-\frac{1}{17711}a^{22}+\frac{24157817}{17711}a^{15}-\frac{514229}{17711}a^{7}-\frac{10946}{17711}$, $\frac{8}{17711}a^{28}-\frac{3}{17711}a^{27}+\frac{3}{17711}a^{26}+\frac{317811}{17711}a^{6}-\frac{121393}{17711}a^{5}+\frac{121393}{17711}a^{4}$, $\frac{12543}{17711}a^{38}-\frac{33448}{17711}a^{36}+\frac{233}{17711}a^{35}+\frac{87801}{17711}a^{34}-\frac{229955}{17711}a^{32}+\frac{602064}{17711}a^{30}-\frac{1576237}{17711}a^{28}+\frac{4126647}{17711}a^{26}-\frac{10803704}{17711}a^{24}-\frac{1}{17711}a^{23}+\frac{28284465}{17711}a^{22}-4181a^{20}+10946a^{18}-\frac{10803704}{17711}a^{16}+\frac{4126647}{17711}a^{14}+\frac{9227465}{17711}a^{13}-\frac{1576237}{17711}a^{12}+\frac{602064}{17711}a^{10}-\frac{229955}{17711}a^{8}+\frac{87801}{17711}a^{6}-\frac{33448}{17711}a^{4}+\frac{12543}{17711}a^{2}-\frac{28657}{17711}a-\frac{4181}{17711}$, $\frac{377}{17711}a^{36}-\frac{55}{17711}a^{33}+\frac{21}{17711}a^{30}+\frac{14930352}{17711}a^{14}-\frac{2178309}{17711}a^{11}+\frac{832040}{17711}a^{8}$, $\frac{987}{17711}a^{37}+\frac{8}{17711}a^{28}-\frac{1}{17711}a^{24}+\frac{39088169}{17711}a^{15}+\frac{317811}{17711}a^{6}-\frac{46368}{17711}a^{2}$ (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: | \( 645826241875575.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 645826241875575.0 \cdot 62}{44\cdot\sqrt{3241429490243539842962382172914969829546393600000000000000000000}}\cr\approx \mathstrut & 0.146988642294531 \end{aligned}\] (assuming GRH)
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 | ${\href{/padicField/3.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | 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 $20$ | $2$ | $10$ | $20$ | |||
Deg $20$ | $2$ | $10$ | $20$ | ||||
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(11\) | 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 | $[\ ]_{10}^{2}$ |