Normalized defining polynomial
\( x^{20} - x^{19} + x^{17} - x^{16} + x^{14} - x^{13} + x^{11} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
Discriminant: |
\(328307557444402776721569\)
\(\medspace = 3^{10}\cdot 11^{18}\)
| sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | \(14.99\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
$\card{ \Gal(K/\Q) }$: | $20$ | ||
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(33=3\cdot 11\) | ||
Dirichlet character group: | $\lbrace$$\chi_{33}(1,·)$, $\chi_{33}(2,·)$, $\chi_{33}(4,·)$, $\chi_{33}(5,·)$, $\chi_{33}(7,·)$, $\chi_{33}(8,·)$, $\chi_{33}(10,·)$, $\chi_{33}(13,·)$, $\chi_{33}(14,·)$, $\chi_{33}(16,·)$, $\chi_{33}(17,·)$, $\chi_{33}(19,·)$, $\chi_{33}(20,·)$, $\chi_{33}(23,·)$, $\chi_{33}(25,·)$, $\chi_{33}(26,·)$, $\chi_{33}(28,·)$, $\chi_{33}(29,·)$, $\chi_{33}(31,·)$, $\chi_{33}(32,·)$$\rbrace$ | ||
This is a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: |
\( -a \)
(order $66$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
Fundamental units: |
$a^{17}+a^{8}$, $a^{19}-a^{18}+a^{8}$, $a^{19}-a^{9}+a^{8}-a^{6}-a^{3}-1$, $a^{19}+a^{13}+a^{8}-a^{3}+a^{2}$, $a^{14}-a^{13}$, $a^{12}+a^{6}+a$, $a^{19}-a^{17}-a^{15}+a^{13}+a^{10}-a^{9}-a^{6}+a^{2}-1$, $a^{16}-a^{9}$, $a^{18}-a^{16}+a^{12}-a^{8}+a^{7}+a^{6}-a^{5}-a^{2}+a+1$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 62791.3897584 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times C_{10}$ (as 20T3):
An abelian group of order 20 |
The 20 conjugacy class representatives for $C_2\times C_{10}$ |
Character table for $C_2\times C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\zeta_{11})^+\), 10.0.52089208083.1, \(\Q(\zeta_{11})\), \(\Q(\zeta_{33})^+\) |
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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\)
| 3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
\(11\)
| 11.20.18.1 | $x^{20} - 119977 x^{10} + 34179505129$ | $10$ | $2$ | $18$ | $C_2\times C_{10}$ | $[\ ]_{10}^{2}$ |