Normalized defining polynomial
\( x^{20} - 6 x^{19} + 21 x^{18} - 56 x^{17} + 126 x^{16} - 252 x^{15} + 429 x^{14} - 671 x^{13} + 1012 x^{12} - 1441 x^{11} + 1837 x^{10} - 2024 x^{9} + 2266 x^{8} - 2409 x^{7} + 2453 x^{6} - 1947 x^{5} + 1452 x^{4} - 968 x^{3} + 726 x^{2} - 363 x + 121 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(58966631474860149560041954113=3^{5}\cdot 11^{17}\cdot 21911^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.45$ | 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: | $3, 11, 21911$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{1}{11} a^{10}$, $\frac{1}{11} a^{16} - \frac{4}{11} a^{14} + \frac{4}{11} a^{13} - \frac{1}{11} a^{12} - \frac{2}{11} a^{11} - \frac{5}{11} a^{10}$, $\frac{1}{11} a^{17} + \frac{2}{11} a^{14} - \frac{5}{11} a^{13} + \frac{5}{11} a^{12} + \frac{4}{11} a^{11} + \frac{4}{11} a^{10}$, $\frac{1}{253} a^{18} + \frac{10}{253} a^{17} - \frac{8}{253} a^{16} - \frac{4}{253} a^{15} - \frac{82}{253} a^{14} - \frac{49}{253} a^{13} + \frac{101}{253} a^{12} + \frac{63}{253} a^{11} - \frac{3}{253} a^{10} - \frac{3}{23} a^{9} - \frac{3}{23} a^{8} - \frac{10}{23} a^{7} - \frac{8}{23} a^{6} + \frac{2}{23} a^{5} - \frac{4}{23} a^{4} + \frac{2}{23} a^{3} - \frac{6}{23} a - \frac{7}{23}$, $\frac{1}{8257662653319407647} a^{19} - \frac{15129665941831352}{8257662653319407647} a^{18} + \frac{307937174539297563}{8257662653319407647} a^{17} + \frac{337116253073430345}{8257662653319407647} a^{16} - \frac{60849428439787419}{8257662653319407647} a^{15} + \frac{60413352567549066}{359028811013887289} a^{14} - \frac{753737172435235107}{8257662653319407647} a^{13} + \frac{1395619914913124266}{8257662653319407647} a^{12} - \frac{1815421472035088999}{8257662653319407647} a^{11} + \frac{2770039147724697864}{8257662653319407647} a^{10} + \frac{6059397349749759}{750696604847218877} a^{9} - \frac{126829525476524320}{750696604847218877} a^{8} - \frac{15788179799163409}{750696604847218877} a^{7} - \frac{12573180334477183}{750696604847218877} a^{6} - \frac{333742357503325287}{750696604847218877} a^{5} + \frac{104131541873091439}{750696604847218877} a^{4} - \frac{355044471516654510}{750696604847218877} a^{3} - \frac{25607262187719558}{750696604847218877} a^{2} - \frac{227773334723317135}{750696604847218877} a + \frac{260032406009559489}{750696604847218877}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 1531449.0983 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for t20n427 are not computed |
| Character table for t20n427 is not computed |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.4.4696817441591.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ | $20$ | R | $20$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | $20$ | $20$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.5.0.1 | $x^{5} - x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 3.5.0.1 | $x^{5} - x + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| $11$ | 11.5.4.4 | $x^{5} - 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} - 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.10.9.1 | $x^{10} - 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| 21911 | Data not computed | ||||||