Normalized defining polynomial
\( x^{40} + 40 x^{38} + 741 x^{36} + 8436 x^{34} + 66044 x^{32} + 376960 x^{30} + 1622694 x^{28} + 5375528 x^{26} + \cdots + 1 \)
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)
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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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(5,·)$, $\chi_{264}(7,·)$, $\chi_{264}(151,·)$, $\chi_{264}(13,·)$, $\chi_{264}(17,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(47,·)$, $\chi_{264}(161,·)$, $\chi_{264}(35,·)$, $\chi_{264}(163,·)$, $\chi_{264}(41,·)$, $\chi_{264}(175,·)$, $\chi_{264}(49,·)$, $\chi_{264}(53,·)$, $\chi_{264}(61,·)$, $\chi_{264}(191,·)$, $\chi_{264}(65,·)$, $\chi_{264}(67,·)$, $\chi_{264}(71,·)$, $\chi_{264}(119,·)$, $\chi_{264}(205,·)$, $\chi_{264}(79,·)$, $\chi_{264}(83,·)$, $\chi_{264}(85,·)$, $\chi_{264}(91,·)$, $\chi_{264}(221,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(233,·)$, $\chi_{264}(107,·)$, $\chi_{264}(109,·)$, $\chi_{264}(115,·)$, $\chi_{264}(235,·)$, $\chi_{264}(245,·)$, $\chi_{264}(169,·)$, $\chi_{264}(125,·)$, $\chi_{264}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}\times C_{10}\times C_{6820}$, which has order $341000$ (assuming GRH)
Unit group
Rank: | $19$ | 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: | $a^{2}+2$, $2a^{38}+76a^{36}+1331a^{34}+14246a^{32}+104190a^{30}+551492a^{28}+2182858a^{26}+6582628a^{24}+15267031a^{22}+27296698a^{20}+37477102a^{18}+39106444a^{16}+30468217a^{14}+17242398a^{12}+6796891a^{10}+1748702a^{8}+263969a^{6}+19430a^{4}+521a^{2}+2$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24751a^{12}+27444a^{10}+19251a^{8}+7896a^{6}+1611a^{4}+108a^{2}+1$, $a^{24}+24a^{22}+252a^{20}+1520a^{18}+5814a^{16}+14688a^{14}+24752a^{12}+27456a^{10}+19305a^{8}+8008a^{6}+1716a^{4}+144a^{2}+2$, $a^{12}+12a^{10}+54a^{8}+112a^{6}+105a^{4}+36a^{2}+2$, $a^{30}+30a^{28}+405a^{26}+3250a^{24}+17250a^{22}+63756a^{20}+168245a^{18}+319770a^{16}+436049a^{14}+419886a^{12}+277057a^{10}+119130a^{8}+30646a^{6}+4004a^{4}+176a^{2}$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413705a^{22}+13123110a^{20}+17809935a^{18}+18349630a^{16}+14115100a^{14}+7904456a^{12}+3105322a^{10}+810084a^{8}+128877a^{6}+10830a^{4}+361a^{2}+2$, $a^{10}+10a^{8}+35a^{6}+50a^{4}+25a^{2}+2$, $a^{26}+26a^{24}+299a^{22}+2002a^{20}+8645a^{18}+25194a^{16}+50388a^{14}+68952a^{12}+63206a^{10}+37180a^{8}+13013a^{6}+2366a^{4}+169a^{2}+2$, $a^{39}+38a^{37}+665a^{35}+7105a^{33}+51798a^{31}+272770a^{29}+1071202a^{27}+3192670a^{25}+7277426a^{23}+12680889a^{21}+16746808a^{19}+16449914a^{17}+11607141a^{15}+5496727a^{13}+1470831a^{11}+60609a^{9}-86316a^{7}-22993a^{5}-1870a^{3}-41a$, $a$, $a^{39}+38a^{37}+665a^{35}+7105a^{33}+51798a^{31}+272770a^{29}+1071202a^{27}+3192670a^{25}+7277426a^{23}+12680889a^{21}+16746808a^{19}+16449914a^{17}+11607141a^{15}+5496727a^{13}+1470831a^{11}+60609a^{9}-86316a^{7}-22993a^{5}-1870a^{3}-40a$, $a^{37}+37a^{35}+629a^{33}+6512a^{31}+45880a^{29}+232841a^{27}+878787a^{25}+2510820a^{23}+5476185a^{21}+9126975a^{19}+11560835a^{17}+10994920a^{15}+7696444a^{13}+3848222a^{11}+1314610a^{9}+286824a^{7}+35853a^{5}+2109a^{3}+37a$, $a^{7}+7a^{5}+14a^{3}+7a$, $a^{27}+27a^{25}+324a^{23}+2277a^{21}+10395a^{19}+32320a^{17}+69785a^{15}+104771a^{13}+107848a^{11}+73865a^{9}+32010a^{7}+8085a^{5}+1023a^{3}+44a$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076102a^{26}+3223324a^{24}+7413405a^{22}+13121086a^{20}+17801081a^{18}+18323314a^{16}+14060973a^{14}+7827510a^{12}+3031183a^{10}+763687a^{8}+111481a^{6}+7514a^{4}+151a^{2}$, $a^{38}+38a^{36}+665a^{34}+7106a^{32}+51832a^{30}+273296a^{28}+1076103a^{26}+3223350a^{24}+7413705a^{22}+13123110a^{20}+17809935a^{18}+18349630a^{16}+14115100a^{14}+7904456a^{12}+3105322a^{10}+810084a^{8}+128877a^{6}+10830a^{4}+361a^{2}+1$, $a^{19}+19a^{17}+152a^{15}+665a^{13}+1729a^{11}+2717a^{9}+2508a^{7}+1254a^{5}+285a^{3}+19a$, $a^{11}+11a^{9}+44a^{7}+77a^{5}+55a^{3}+11a$ (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: | \( 45016485591237.79 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 45016485591237.79 \cdot 341000}{2\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.195529653554255 \end{aligned}\] (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 $C_2^2\times C_{10}$ 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 | R | ${\href{/padicField/5.10.0.1}{10} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | 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 $20$ | $4$ | $5$ | $40$ | |||
Deg $20$ | $4$ | $5$ | $40$ | ||||
\(3\) | 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 | $[\ ]_{2}^{10}$ | |
\(11\) | 11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
11.10.9.1 | $x^{10} + 110$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |