Normalized defining polynomial
\( x^{20} - 8 x^{19} + 12 x^{18} + 80 x^{17} - 290 x^{16} - 183 x^{15} + 2154 x^{14} - 1376 x^{13} - 8255 x^{12} + 12715 x^{11} + 11704 x^{10} - 36967 x^{9} + 3675 x^{8} + 48318 x^{7} - 3411 x^{6} - 61608 x^{5} - 23582 x^{4} + 88692 x^{3} + 7239 x^{2} - 48735 x + 9025 \)
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: | \(5834302965409512291058019417402368=2^{10}\cdot 19^{10}\cdot 43^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $48.79$ | 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: | $2, 19, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{3}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{3}{10} a^{5} - \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} - \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{260} a^{17} - \frac{9}{260} a^{16} + \frac{51}{130} a^{15} + \frac{23}{130} a^{14} - \frac{79}{260} a^{13} - \frac{2}{5} a^{12} - \frac{83}{260} a^{11} - \frac{22}{65} a^{10} - \frac{61}{130} a^{9} - \frac{7}{26} a^{8} - \frac{1}{13} a^{7} + \frac{81}{260} a^{6} + \frac{14}{65} a^{5} + \frac{7}{20} a^{4} + \frac{123}{260} a^{3} + \frac{12}{65} a^{2} - \frac{27}{65} a + \frac{7}{52}$, $\frac{1}{21508760} a^{18} + \frac{1489}{827260} a^{17} - \frac{5517}{21508760} a^{16} - \frac{1437767}{5377190} a^{15} + \frac{1691835}{4301752} a^{14} - \frac{9961389}{21508760} a^{13} + \frac{769083}{3072680} a^{12} + \frac{2023335}{4301752} a^{11} + \frac{2703439}{10754380} a^{10} + \frac{1331587}{2688595} a^{9} - \frac{2137}{43540} a^{8} + \frac{9622709}{21508760} a^{7} - \frac{2260669}{21508760} a^{6} + \frac{931023}{4301752} a^{5} - \frac{1488453}{5377190} a^{4} + \frac{8066953}{21508760} a^{3} + \frac{916638}{2688595} a^{2} - \frac{258347}{1132040} a - \frac{19301}{226408}$, $\frac{1}{1125566756730229891509498911681200} a^{19} + \frac{11224160666074156447728723}{1125566756730229891509498911681200} a^{18} - \frac{2711860632325300552957859095}{3463282328400707358490765882096} a^{17} - \frac{5448768150469136276298698585933}{225113351346045978301899782336240} a^{16} + \frac{83820535104523692243074429516759}{225113351346045978301899782336240} a^{15} + \frac{14721058118051319687463170550123}{80397625480730706536392779405800} a^{14} + \frac{39010018427459781485289164464557}{140695844591278736438687363960150} a^{13} + \frac{2367357890442364917701388609173}{28139168918255747287737472792030} a^{12} + \frac{12769236468487432869854105424197}{225113351346045978301899782336240} a^{11} + \frac{11214084096678144356049120733339}{112556675673022989150949891168120} a^{10} - \frac{164078499871507892680563090276143}{562783378365114945754749455840600} a^{9} + \frac{385106545728159995477246322723487}{1125566756730229891509498911681200} a^{8} + \frac{1004440207658486040127555819171}{10049703185091338317049097425725} a^{7} + \frac{3805816017742593477643407374391}{22511335134604597830189978233624} a^{6} - \frac{134082478661567281281363283687801}{1125566756730229891509498911681200} a^{5} - \frac{193361177743100533538838655046219}{1125566756730229891509498911681200} a^{4} + \frac{6741152248975157750840221134883}{13561045261809998692885529056400} a^{3} - \frac{6623822065727762720577772137613}{86582058210017683962269147052400} a^{2} - \frac{1417074744313473597487530438553}{2962017780869026030288155030740} a - \frac{314680462233888802340057305417}{2369614224695220824230524024592}$
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: | \( 3648112275.37 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 10240 |
| The 160 conjugacy class representatives for t20n423 are not computed |
| Character table for t20n423 is not computed |
Intermediate fields
| \(\Q(\sqrt{817}) \), 5.5.667489.1 x5, 10.10.364007458703857.1 |
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 | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.10.12 | $x^{10} - 11 x^{8} + 54 x^{6} - 10 x^{4} + 9 x^{2} - 11$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
| $19$ | 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $43$ | 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.4.3.1 | $x^{4} + 387$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |