Normalized defining polynomial
\( x^{20} - 6 x^{19} + 7 x^{18} + 22 x^{17} + 398 x^{16} - 2164 x^{15} + 7336 x^{14} - 15442 x^{13} + 99688 x^{12} - 300174 x^{11} + 1595106 x^{10} - 3565206 x^{9} + 17149330 x^{8} - 32155744 x^{7} + 155208869 x^{6} - 234163102 x^{5} + 1025221284 x^{4} - 1123862306 x^{3} + 4154165587 x^{2} - 2548342424 x + 7273808849 \)
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: | \(948576001354145151944312834959360000000000=2^{20}\cdot 5^{10}\cdot 11^{16}\cdot 17^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $125.56$ | 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: | $2, 5, 11, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(3740=2^{2}\cdot 5\cdot 11\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{3740}(1,·)$, $\chi_{3740}(2821,·)$, $\chi_{3740}(2379,·)$, $\chi_{3740}(2381,·)$, $\chi_{3740}(339,·)$, $\chi_{3740}(2039,·)$, $\chi_{3740}(3161,·)$, $\chi_{3740}(3059,·)$, $\chi_{3740}(2721,·)$, $\chi_{3740}(1699,·)$, $\chi_{3740}(1259,·)$, $\chi_{3740}(3501,·)$, $\chi_{3740}(1939,·)$, $\chi_{3740}(3061,·)$, $\chi_{3740}(3639,·)$, $\chi_{3740}(441,·)$, $\chi_{3740}(2619,·)$, $\chi_{3740}(1599,·)$, $\chi_{3740}(1021,·)$, $\chi_{3740}(1461,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{37} a^{15} + \frac{14}{37} a^{14} + \frac{13}{37} a^{13} + \frac{4}{37} a^{12} + \frac{15}{37} a^{11} - \frac{15}{37} a^{10} - \frac{18}{37} a^{9} + \frac{6}{37} a^{8} + \frac{14}{37} a^{7} + \frac{9}{37} a^{6} + \frac{9}{37} a^{5} + \frac{12}{37} a^{4} - \frac{16}{37} a^{3} - \frac{14}{37} a^{2} - \frac{14}{37} a - \frac{2}{37}$, $\frac{1}{37} a^{16} + \frac{2}{37} a^{14} + \frac{7}{37} a^{13} - \frac{4}{37} a^{12} - \frac{3}{37} a^{11} + \frac{7}{37} a^{10} - \frac{1}{37} a^{9} + \frac{4}{37} a^{8} - \frac{2}{37} a^{7} - \frac{6}{37} a^{6} - \frac{3}{37} a^{5} + \frac{1}{37} a^{4} - \frac{12}{37} a^{3} - \frac{3}{37} a^{2} + \frac{9}{37} a - \frac{9}{37}$, $\frac{1}{37} a^{17} + \frac{16}{37} a^{14} + \frac{7}{37} a^{13} - \frac{11}{37} a^{12} + \frac{14}{37} a^{11} - \frac{8}{37} a^{10} + \frac{3}{37} a^{9} - \frac{14}{37} a^{8} + \frac{3}{37} a^{7} + \frac{16}{37} a^{6} - \frac{17}{37} a^{5} + \frac{1}{37} a^{4} - \frac{8}{37} a^{3} - \frac{18}{37} a + \frac{4}{37}$, $\frac{1}{118885233560860870490259239} a^{18} - \frac{950822519265042529913483}{118885233560860870490259239} a^{17} - \frac{1117170719912119708134174}{118885233560860870490259239} a^{16} + \frac{30577622671423703151282}{118885233560860870490259239} a^{15} - \frac{19301042787626922494576896}{118885233560860870490259239} a^{14} + \frac{36528796005468723595710798}{118885233560860870490259239} a^{13} + \frac{1422638903851506751942996}{118885233560860870490259239} a^{12} + \frac{30630382205623487768781638}{118885233560860870490259239} a^{11} + \frac{8642503520020537888578575}{118885233560860870490259239} a^{10} + \frac{55909676282478760839992317}{118885233560860870490259239} a^{9} + \frac{47539193582860859463596732}{118885233560860870490259239} a^{8} + \frac{24941058924516117818713164}{118885233560860870490259239} a^{7} + \frac{37272286098225709649592402}{118885233560860870490259239} a^{6} - \frac{6952578194574844302897083}{118885233560860870490259239} a^{5} + \frac{39419909874868348911346601}{118885233560860870490259239} a^{4} - \frac{30247271741238035367877577}{118885233560860870490259239} a^{3} - \frac{1237257252755860096827253}{118885233560860870490259239} a^{2} - \frac{7375269576602910621132102}{118885233560860870490259239} a + \frac{5260636174463948579723311}{118885233560860870490259239}$, $\frac{1}{4213870604165472439381617710210289628039189322595147550511} a^{19} - \frac{2220849114262768440818236976230}{4213870604165472439381617710210289628039189322595147550511} a^{18} + \frac{40740989183237720859284110212732834766134240638776134839}{4213870604165472439381617710210289628039189322595147550511} a^{17} + \frac{55033885631773944446901904068680139558466233128853705324}{4213870604165472439381617710210289628039189322595147550511} a^{16} + \frac{41683558671694636591467166489021536658388840720664404201}{4213870604165472439381617710210289628039189322595147550511} a^{15} - \frac{2071813839317324480732397256897129887958328317160469884844}{4213870604165472439381617710210289628039189322595147550511} a^{14} - \frac{372333747110509802282238737210389838170898703475765038892}{4213870604165472439381617710210289628039189322595147550511} a^{13} - \frac{2020926617187943952970904300106383563011428091236411563122}{4213870604165472439381617710210289628039189322595147550511} a^{12} - \frac{34333885920545091592011691814752271563945768325202772003}{113888394707174930794097775951629449406464576286355339203} a^{11} + \frac{1249526917300938162468729141930970160493242536328133669610}{4213870604165472439381617710210289628039189322595147550511} a^{10} + \frac{1652535960063729919781641999749167284907722029988866964901}{4213870604165472439381617710210289628039189322595147550511} a^{9} - \frac{1336460659461507905453368295867309541712067067551136421997}{4213870604165472439381617710210289628039189322595147550511} a^{8} - \frac{318902965026530756357769625250529865044594322161572388083}{4213870604165472439381617710210289628039189322595147550511} a^{7} + \frac{1841511770047357615619236266441951136790420403477050221534}{4213870604165472439381617710210289628039189322595147550511} a^{6} + \frac{10636037202013515517945512415968251084429301585614503573}{4213870604165472439381617710210289628039189322595147550511} a^{5} - \frac{425225786950895260435062066362884488157655128686573785}{4260738730197646551447540657442153314498674744787813499} a^{4} - \frac{897120362306142219160863827138051924425407484805278308721}{4213870604165472439381617710210289628039189322595147550511} a^{3} - \frac{1365102926841378250882003053084804434175803687956250475355}{4213870604165472439381617710210289628039189322595147550511} a^{2} + \frac{888136792971536001283144795925591435594137341684195403495}{4213870604165472439381617710210289628039189322595147550511} a - \frac{1641143383559869362325512880886547999611964478458614936542}{4213870604165472439381617710210289628039189322595147550511}$
Class group and class number
$C_{10}\times C_{198710}$, which has order $1987100$ (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: | \( 3338983.6210111137 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{10}$ (as 20T3):
| An abelian group of order 20 |
| The 20 conjugacy class representatives for $C_2\times C_{10}$ |
| Character table for $C_2\times C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-5}, \sqrt{17})\), \(\Q(\zeta_{11})^+\), 10.0.685948419200000.1, 10.0.973948664640054400000.1, 10.10.304358957700017.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ | ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.11 | $x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
| 5 | Data not computed | ||||||
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 17 | Data not computed | ||||||