Normalized defining polynomial
\( x^{20} - 4 x^{19} + 34 x^{18} - 132 x^{17} + 709 x^{16} - 2396 x^{15} + 8264 x^{14} - 24012 x^{13} + 37584 x^{12} - 68008 x^{11} + 3274 x^{10} + 280784 x^{9} + 361817 x^{8} - 3064120 x^{7} + 10758886 x^{6} - 28153856 x^{5} + 55154491 x^{4} - 84942664 x^{3} + 107321392 x^{2} - 92496540 x + 37311425 \)
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: | \(142272327122261506049033808572607102976=2^{30}\cdot 19^{10}\cdot 43^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $80.85$ | 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 Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{30} a^{15} - \frac{1}{15} a^{14} + \frac{1}{30} a^{13} + \frac{7}{30} a^{12} + \frac{1}{15} a^{11} - \frac{1}{10} a^{10} - \frac{11}{30} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{7} + \frac{1}{30} a^{6} - \frac{3}{10} a^{5} + \frac{11}{30} a^{4} - \frac{1}{15} a^{3} - \frac{1}{10} a^{2} - \frac{2}{5} a - \frac{1}{3}$, $\frac{1}{570} a^{16} + \frac{7}{570} a^{15} - \frac{47}{570} a^{14} - \frac{119}{570} a^{13} - \frac{5}{114} a^{12} - \frac{2}{19} a^{11} - \frac{34}{285} a^{10} - \frac{27}{190} a^{9} + \frac{27}{190} a^{8} - \frac{173}{570} a^{7} - \frac{6}{19} a^{6} - \frac{47}{114} a^{5} + \frac{13}{30} a^{4} + \frac{14}{95} a^{3} - \frac{83}{190} a^{2} - \frac{89}{285} a - \frac{8}{19}$, $\frac{1}{331170} a^{17} - \frac{1}{15770} a^{16} - \frac{1107}{110390} a^{15} - \frac{488}{2905} a^{14} - \frac{31663}{165585} a^{13} - \frac{24133}{165585} a^{12} + \frac{26521}{331170} a^{11} + \frac{31606}{165585} a^{10} + \frac{44559}{110390} a^{9} - \frac{13679}{33117} a^{8} - \frac{60943}{331170} a^{7} - \frac{12619}{47310} a^{6} + \frac{4441}{23655} a^{5} + \frac{15653}{33117} a^{4} - \frac{3558}{11039} a^{3} + \frac{24293}{331170} a^{2} + \frac{15340}{33117} a + \frac{642}{11039}$, $\frac{1}{4305210} a^{18} - \frac{1}{2152605} a^{17} + \frac{542}{717535} a^{16} - \frac{1227}{143507} a^{15} + \frac{853}{25935} a^{14} - \frac{16184}{307515} a^{13} - \frac{727}{615030} a^{12} - \frac{20430}{143507} a^{11} + \frac{215044}{2152605} a^{10} + \frac{74687}{2152605} a^{9} + \frac{32822}{143507} a^{8} - \frac{17476}{113295} a^{7} + \frac{280567}{615030} a^{6} - \frac{50941}{2152605} a^{5} + \frac{1973171}{4305210} a^{4} - \frac{824969}{2152605} a^{3} - \frac{148207}{717535} a^{2} - \frac{201626}{430521} a - \frac{54039}{287014}$, $\frac{1}{313413624091831606182113633713917173359287104584857834523710} a^{19} + \frac{1885770196773180542361678711150574929265221473938383}{20894241606122107078807575580927811557285806972323855634914} a^{18} - \frac{102553511460951968675634605605785230769476337948751211}{104471208030610535394037877904639057786429034861619278174570} a^{17} - \frac{20165197482232396663594790853014033671335305664839256019}{44773374870261658026016233387702453337041014940693976360530} a^{16} - \frac{3614546176655089027069329564968908064248211318762087464}{574017626541816128538669658816698119705654037701204825135} a^{15} + \frac{1214155074752706236130906433630791303915904213646025378482}{52235604015305267697018938952319528893214517430809639087285} a^{14} + \frac{1205368628296390526832246686687143403143955449311371793199}{10447120803061053539403787790463905778642903486161927817457} a^{13} - \frac{35164336208794971516745630145417629458797624857057727488}{211480178199616468408983558511415096733662013889917567155} a^{12} - \frac{22641708022213520616874245081217500079044766938000448456519}{104471208030610535394037877904639057786429034861619278174570} a^{11} + \frac{136968571358402164798723138325553639137597754123825796552}{2749242316595014089316786260648396257537606180568928373015} a^{10} - \frac{1183995704443312769827427690468530252765647528484242162248}{8247726949785042267950358781945188772612818541706785119045} a^{9} - \frac{28350269252573935641924484682809965535343149940977231059727}{313413624091831606182113633713917173359287104584857834523710} a^{8} + \frac{29663446962656897820358503415471178398258083133550311113697}{313413624091831606182113633713917173359287104584857834523710} a^{7} - \frac{119268594544530344741455144250974656957435322456320645786913}{313413624091831606182113633713917173359287104584857834523710} a^{6} - \frac{201361681884512642347217612707142491152607473309516925336}{22386687435130829013008116693851226668520507470346988180265} a^{5} - \frac{142408944539521618633878010512566616993529930493530379544149}{313413624091831606182113633713917173359287104584857834523710} a^{4} - \frac{3247669404159538303335481979855403541858534320890665064992}{8247726949785042267950358781945188772612818541706785119045} a^{3} + \frac{3883309321065235898763363246289141205553586537516383563031}{14924458290087219342005411129234151112347004980231325453510} a^{2} - \frac{94913536282442910876478162358880440880083316536130699601847}{313413624091831606182113633713917173359287104584857834523710} a - \frac{4428327922341205621887932614734712400067995615951830560386}{10447120803061053539403787790463905778642903486161927817457}$
Class group and class number
$C_{62}\times C_{496}$, which has order $30752$ (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: | \( 4747366.99665 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 20 |
| The 8 conjugacy class representatives for $D_{10}$ |
| Character table for $D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{817}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1634}) \), \(\Q(\sqrt{-2}, \sqrt{817})\), 5.5.667489.1 x5, 10.10.364007458703857.1, 10.0.14599506005884928.1 x5, 10.0.11927796406807986176.1 x5 |
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 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ | R | ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ | R | ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.15.9 | $x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.9 | $x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
| $19$ | 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 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $43$ | 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 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 43.2.1.2 | $x^{2} + 387$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |