Normalized defining polynomial
\( x^{20} - 4 x^{19} + 2 x^{18} + 10 x^{17} - 23 x^{16} + 32 x^{15} - 8 x^{14} - 23 x^{13} - 33 x^{12} + 89 x^{11} - 83 x^{10} + 41 x^{9} + 51 x^{8} - 2 x^{7} - 88 x^{6} + 34 x^{5} - 6 x^{4} - 16 x^{3} + 23 x^{2} + 5 x - 1 \)
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[6, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1667682586167898216778752=-\,2^{10}\cdot 11^{16}\cdot 23^{2}\cdot 67\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.26$ | 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: | $2, 11, 23, 67$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{27597185381273113} a^{19} - \frac{85822798780397}{253185186984157} a^{18} - \frac{3750654903669878}{27597185381273113} a^{17} + \frac{8165918092043807}{27597185381273113} a^{16} + \frac{13398059899219832}{27597185381273113} a^{15} + \frac{10733879212264172}{27597185381273113} a^{14} + \frac{5726089296000150}{27597185381273113} a^{13} - \frac{1963030233616063}{27597185381273113} a^{12} - \frac{6559261103652253}{27597185381273113} a^{11} - \frac{12603897430315720}{27597185381273113} a^{10} - \frac{3127031155424141}{27597185381273113} a^{9} + \frac{1139278928524169}{27597185381273113} a^{8} + \frac{9339113940230670}{27597185381273113} a^{7} + \frac{10387951924250922}{27597185381273113} a^{6} + \frac{9479691558079964}{27597185381273113} a^{5} + \frac{12117996663922885}{27597185381273113} a^{4} - \frac{7535268927163754}{27597185381273113} a^{3} - \frac{7792586376930855}{27597185381273113} a^{2} + \frac{5565611206314670}{27597185381273113} a - \frac{4666661620828409}{27597185381273113}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 17148.8921562 \) (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
| \(\Q(\zeta_{11})^+\), 10.4.4930254263.1 |
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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ | $20$ | $20$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | $20$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.0.1 | $x^{10} - x^{3} + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |
| 2.10.10.2 | $x^{10} - 5 x^{8} + 10 x^{6} - 2 x^{4} - 11 x^{2} + 39$ | $2$ | $5$ | $10$ | $C_2^4 : C_5$ | $[2, 2, 2, 2]^{5}$ | |
| 11 | Data not computed | ||||||
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 67 | Data not computed | ||||||