Normalized defining polynomial
\( x^{40} - 6 x^{38} + 32 x^{36} - 168 x^{34} + 880 x^{32} - 4608 x^{30} + 24128 x^{28} - 126336 x^{26} + \cdots + 1048576 \)
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: | \(3398885169161610034374122849346487403986439215513600000000000000000000\) \(\medspace = 2^{60}\cdot 5^{20}\cdot 11^{36}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(54.74\) | 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: | $2^{3/2}5^{1/2}11^{9/10}\approx 54.73730515936835$ | ||
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)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(440=2^{3}\cdot 5\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(259,·)$, $\chi_{440}(51,·)$, $\chi_{440}(129,·)$, $\chi_{440}(9,·)$, $\chi_{440}(139,·)$, $\chi_{440}(401,·)$, $\chi_{440}(19,·)$, $\chi_{440}(281,·)$, $\chi_{440}(409,·)$, $\chi_{440}(411,·)$, $\chi_{440}(289,·)$, $\chi_{440}(291,·)$, $\chi_{440}(369,·)$, $\chi_{440}(41,·)$, $\chi_{440}(171,·)$, $\chi_{440}(49,·)$, $\chi_{440}(91,·)$, $\chi_{440}(179,·)$, $\chi_{440}(329,·)$, $\chi_{440}(371,·)$, $\chi_{440}(59,·)$, $\chi_{440}(321,·)$, $\chi_{440}(161,·)$, $\chi_{440}(201,·)$, $\chi_{440}(339,·)$, $\chi_{440}(331,·)$, $\chi_{440}(81,·)$, $\chi_{440}(419,·)$, $\chi_{440}(89,·)$, $\chi_{440}(219,·)$, $\chi_{440}(379,·)$, $\chi_{440}(131,·)$, $\chi_{440}(361,·)$, $\chi_{440}(241,·)$, $\chi_{440}(211,·)$, $\chi_{440}(169,·)$, $\chi_{440}(249,·)$, $\chi_{440}(251,·)$, $\chi_{440}(299,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{524288}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{36272128}a^{22}-\frac{6765}{17711}$, $\frac{1}{36272128}a^{23}-\frac{6765}{17711}a$, $\frac{1}{72544256}a^{24}-\frac{6765}{35422}a^{2}$, $\frac{1}{72544256}a^{25}-\frac{6765}{35422}a^{3}$, $\frac{1}{145088512}a^{26}-\frac{6765}{70844}a^{4}$, $\frac{1}{145088512}a^{27}-\frac{6765}{70844}a^{5}$, $\frac{1}{290177024}a^{28}-\frac{6765}{141688}a^{6}$, $\frac{1}{290177024}a^{29}-\frac{6765}{141688}a^{7}$, $\frac{1}{580354048}a^{30}-\frac{6765}{283376}a^{8}$, $\frac{1}{580354048}a^{31}-\frac{6765}{283376}a^{9}$, $\frac{1}{1160708096}a^{32}-\frac{6765}{566752}a^{10}$, $\frac{1}{1160708096}a^{33}-\frac{6765}{566752}a^{11}$, $\frac{1}{2321416192}a^{34}-\frac{6765}{1133504}a^{12}$, $\frac{1}{2321416192}a^{35}-\frac{6765}{1133504}a^{13}$, $\frac{1}{4642832384}a^{36}-\frac{6765}{2267008}a^{14}$, $\frac{1}{4642832384}a^{37}-\frac{6765}{2267008}a^{15}$, $\frac{1}{9285664768}a^{38}-\frac{6765}{4534016}a^{16}$, $\frac{1}{9285664768}a^{39}-\frac{6765}{4534016}a^{17}$
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: | \( \frac{1}{36272128} a^{28} + \frac{317811}{141688} a^{6} \) (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)
| |
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 | ${\href{/padicField/3.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{8}$ |
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 $20$ | $2$ | $10$ | $30$ | |||
Deg $20$ | $2$ | $10$ | $30$ | ||||
\(5\) | Deg $20$ | $2$ | $10$ | $10$ | |||
Deg $20$ | $2$ | $10$ | $10$ | ||||
\(11\) | 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |