Normalized defining polynomial
\( x^{20} - 4 x^{19} - 2 x^{18} + 16 x^{17} - 31 x^{16} + 20 x^{15} + 24 x^{14} + 44 x^{13} + 14 x^{12} - 76 x^{11} - 44 x^{10} - 76 x^{9} + 14 x^{8} + 44 x^{7} + 24 x^{6} + 20 x^{5} - 31 x^{4} + 16 x^{3} - 2 x^{2} - 4 x + 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(49338146756019243307761664=2^{30}\cdot 11^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $19.26$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{7} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{9} + \frac{1}{16} a$, $\frac{1}{586064} a^{18} + \frac{12659}{586064} a^{17} + \frac{6205}{293032} a^{16} + \frac{16703}{293032} a^{15} - \frac{2573}{73258} a^{14} - \frac{18307}{293032} a^{13} + \frac{1713}{36629} a^{12} + \frac{3879}{293032} a^{11} - \frac{13409}{293032} a^{10} + \frac{1245}{36629} a^{9} + \frac{5805}{73258} a^{8} - \frac{69379}{293032} a^{7} + \frac{1713}{36629} a^{6} + \frac{54951}{293032} a^{5} - \frac{2573}{73258} a^{4} - \frac{56555}{293032} a^{3} + \frac{122297}{586064} a^{2} - \frac{207115}{586064} a + \frac{18315}{293032}$, $\frac{1}{586064} a^{19} + \frac{3501}{146516} a^{17} + \frac{23}{36629} a^{16} + \frac{706}{36629} a^{15} + \frac{15397}{293032} a^{14} + \frac{5167}{146516} a^{13} - \frac{113}{293032} a^{12} + \frac{1819}{293032} a^{11} + \frac{15705}{293032} a^{10} + \frac{8299}{146516} a^{9} - \frac{27005}{293032} a^{8} + \frac{7233}{73258} a^{7} + \frac{50959}{293032} a^{6} - \frac{6831}{146516} a^{5} - \frac{22851}{293032} a^{4} + \frac{154177}{586064} a^{3} - \frac{65133}{293032} a^{2} + \frac{2389}{36629} a + \frac{11985}{293032}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 156899.783975 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^4:C_5$ (as 20T40):
| A solvable group of order 160 |
| The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$ |
| Character table for $C_2\times C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.219503494144.1, 10.8.219503494144.1, 10.6.219503494144.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |