Normalized defining polynomial
\( x^{20} - 5 x^{19} + 8 x^{18} - 6 x^{16} - 34 x^{15} + 74 x^{14} + 102 x^{13} - 547 x^{12} + 775 x^{11} - 266 x^{10} - 574 x^{9} + 829 x^{8} - 424 x^{7} + 2 x^{6} + 142 x^{5} - 101 x^{4} + 22 x^{3} + 6 x^{2} - 4 x + 1 \)
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: | \(4902190169190997617688501=11^{16}\cdot 197^{2}\cdot 2749\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.16$ | 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: | $11, 197, 2749$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{965432269809331543} a^{19} - \frac{47114343665185254}{965432269809331543} a^{18} - \frac{74565956615909709}{965432269809331543} a^{17} + \frac{182602817423297853}{965432269809331543} a^{16} + \frac{228484106085546694}{965432269809331543} a^{15} - \frac{469569090462949094}{965432269809331543} a^{14} - \frac{64850405044747996}{965432269809331543} a^{13} - \frac{5682098637135149}{965432269809331543} a^{12} + \frac{371096561226547372}{965432269809331543} a^{11} + \frac{256645952254995998}{965432269809331543} a^{10} + \frac{68252948634865378}{965432269809331543} a^{9} - \frac{206479577326576404}{965432269809331543} a^{8} + \frac{192072710450190394}{965432269809331543} a^{7} + \frac{88111281661013211}{965432269809331543} a^{6} + \frac{441922051667435390}{965432269809331543} a^{5} - \frac{269431553718059258}{965432269809331543} a^{4} + \frac{327093927933756761}{965432269809331543} a^{3} - \frac{142926347166982275}{965432269809331543} a^{2} + \frac{320220900672140490}{965432269809331543} a + \frac{200618293737863392}{965432269809331543}$
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: | \( 44829.7189708 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 163840 |
| The 649 conjugacy class representatives for t20n846 are not computed |
| Character table for t20n846 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.42228699557.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 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 | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | R | $20$ | $20$ | ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| $197$ | $\Q_{197}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{197}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{197}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{197}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 197.4.2.1 | $x^{4} + 985 x^{2} + 349281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 2749 | Data not computed | ||||||