Normalized defining polynomial
\( x^{40} + 5 x^{38} - 125 x^{34} - 625 x^{32} + 15625 x^{28} + 78125 x^{26} - 1953125 x^{22} + \cdots + 95367431640625 \)
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: | \(11302165783522556415463223790320401501047994110286233600000000000000000000\) \(\medspace = 2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(67.04\) | 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: | $2\cdot 3^{1/2}5^{1/2}11^{9/10}\approx 67.03923376773275$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(11\) | 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)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(660=2^{2}\cdot 3\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(259,·)$, $\chi_{660}(641,·)$, $\chi_{660}(521,·)$, $\chi_{660}(139,·)$, $\chi_{660}(499,·)$, $\chi_{660}(401,·)$, $\chi_{660}(19,·)$, $\chi_{660}(281,·)$, $\chi_{660}(541,·)$, $\chi_{660}(161,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(41,·)$, $\chi_{660}(299,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(439,·)$, $\chi_{660}(59,·)$, $\chi_{660}(61,·)$, $\chi_{660}(581,·)$, $\chi_{660}(199,·)$, $\chi_{660}(461,·)$, $\chi_{660}(79,·)$, $\chi_{660}(599,·)$, $\chi_{660}(601,·)$, $\chi_{660}(221,·)$, $\chi_{660}(479,·)$, $\chi_{660}(481,·)$, $\chi_{660}(101,·)$, $\chi_{660}(359,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(239,·)$, $\chi_{660}(241,·)$, $\chi_{660}(659,·)$, $\chi_{660}(119,·)$, $\chi_{660}(379,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{5}a^{2}$, $\frac{1}{5}a^{3}$, $\frac{1}{25}a^{4}$, $\frac{1}{25}a^{5}$, $\frac{1}{125}a^{6}$, $\frac{1}{125}a^{7}$, $\frac{1}{625}a^{8}$, $\frac{1}{625}a^{9}$, $\frac{1}{3125}a^{10}$, $\frac{1}{3125}a^{11}$, $\frac{1}{15625}a^{12}$, $\frac{1}{15625}a^{13}$, $\frac{1}{78125}a^{14}$, $\frac{1}{78125}a^{15}$, $\frac{1}{390625}a^{16}$, $\frac{1}{390625}a^{17}$, $\frac{1}{1953125}a^{18}$, $\frac{1}{1953125}a^{19}$, $\frac{1}{9765625}a^{20}$, $\frac{1}{9765625}a^{21}$, $\frac{1}{48828125}a^{22}$, $\frac{1}{48828125}a^{23}$, $\frac{1}{244140625}a^{24}$, $\frac{1}{244140625}a^{25}$, $\frac{1}{1220703125}a^{26}$, $\frac{1}{1220703125}a^{27}$, $\frac{1}{6103515625}a^{28}$, $\frac{1}{6103515625}a^{29}$, $\frac{1}{30517578125}a^{30}$, $\frac{1}{30517578125}a^{31}$, $\frac{1}{152587890625}a^{32}$, $\frac{1}{152587890625}a^{33}$, $\frac{1}{762939453125}a^{34}$, $\frac{1}{762939453125}a^{35}$, $\frac{1}{3814697265625}a^{36}$, $\frac{1}{3814697265625}a^{37}$, $\frac{1}{19073486328125}a^{38}$, $\frac{1}{19073486328125}a^{39}$
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: | \( -\frac{1}{152587890625} a^{32} \) (order $66$) | 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 | 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 $20$ | $2$ | $10$ | $20$ | ||||
\(3\) | 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(5\) | Deg $20$ | $2$ | $10$ | $10$ | |||
Deg $20$ | $2$ | $10$ | $10$ | ||||
\(11\) | 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 | $[\ ]_{10}^{2}$ |