Normalized defining polynomial
\( x^{28} - x^{27} - 57 x^{26} + 57 x^{25} + 1364 x^{24} - 1364 x^{23} - 17950 x^{22} + 18037 x^{21} + 142913 x^{20} - 145523 x^{19} - 713109 x^{18} + 743298 x^{17} + 2232740 x^{16} - 2405812 x^{15} - 4269553 x^{14} + 4796744 x^{13} + 4718011 x^{12} - 5558257 x^{11} - 2806909 x^{10} + 3434904 x^{9} + 899378 x^{8} - 1037302 x^{7} - 179219 x^{6} + 128266 x^{5} + 22273 x^{4} - 2466 x^{3} - 463 x^{2} - x + 1 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[28, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1455848000512373226044338588471370773272037506103515625=5^{21}\cdot 29^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $85.98$ | 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: | $5, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(145=5\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{145}(64,·)$, $\chi_{145}(1,·)$, $\chi_{145}(3,·)$, $\chi_{145}(4,·)$, $\chi_{145}(133,·)$, $\chi_{145}(129,·)$, $\chi_{145}(136,·)$, $\chi_{145}(9,·)$, $\chi_{145}(111,·)$, $\chi_{145}(12,·)$, $\chi_{145}(141,·)$, $\chi_{145}(142,·)$, $\chi_{145}(144,·)$, $\chi_{145}(81,·)$, $\chi_{145}(98,·)$, $\chi_{145}(27,·)$, $\chi_{145}(97,·)$, $\chi_{145}(34,·)$, $\chi_{145}(36,·)$, $\chi_{145}(37,·)$, $\chi_{145}(16,·)$, $\chi_{145}(43,·)$, $\chi_{145}(108,·)$, $\chi_{145}(109,·)$, $\chi_{145}(47,·)$, $\chi_{145}(48,·)$, $\chi_{145}(118,·)$, $\chi_{145}(102,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{157} a^{25} + \frac{70}{157} a^{24} + \frac{36}{157} a^{23} + \frac{11}{157} a^{22} + \frac{54}{157} a^{21} - \frac{15}{157} a^{20} + \frac{29}{157} a^{19} - \frac{37}{157} a^{18} + \frac{65}{157} a^{17} - \frac{74}{157} a^{16} - \frac{72}{157} a^{15} + \frac{13}{157} a^{14} + \frac{35}{157} a^{13} - \frac{61}{157} a^{12} + \frac{14}{157} a^{11} - \frac{45}{157} a^{10} - \frac{35}{157} a^{9} - \frac{75}{157} a^{8} + \frac{11}{157} a^{7} + \frac{21}{157} a^{6} - \frac{13}{157} a^{5} - \frac{78}{157} a^{4} - \frac{59}{157} a^{3} - \frac{38}{157} a^{2} + \frac{60}{157} a - \frac{61}{157}$, $\frac{1}{1183937} a^{26} + \frac{2075}{1183937} a^{25} + \frac{430208}{1183937} a^{24} - \frac{281687}{1183937} a^{23} - \frac{13687}{1183937} a^{22} + \frac{503267}{1183937} a^{21} - \frac{31930}{1183937} a^{20} - \frac{546185}{1183937} a^{19} - \frac{311818}{1183937} a^{18} + \frac{9361}{1183937} a^{17} - \frac{147814}{1183937} a^{16} + \frac{547709}{1183937} a^{15} - \frac{300617}{1183937} a^{14} + \frac{490560}{1183937} a^{13} + \frac{82437}{1183937} a^{12} + \frac{55029}{1183937} a^{11} + \frac{246034}{1183937} a^{10} + \frac{295874}{1183937} a^{9} - \frac{104049}{1183937} a^{8} + \frac{248627}{1183937} a^{7} + \frac{217304}{1183937} a^{6} + \frac{210770}{1183937} a^{5} + \frac{453496}{1183937} a^{4} - \frac{155071}{1183937} a^{3} - \frac{85236}{1183937} a^{2} + \frac{549320}{1183937} a + \frac{259519}{1183937}$, $\frac{1}{151018107920685246151444695105403205187952185789869} a^{27} - \frac{19142917337198780813700771473417123894736029}{151018107920685246151444695105403205187952185789869} a^{26} - \frac{297360878556903402965426842538704621658223743282}{151018107920685246151444695105403205187952185789869} a^{25} + \frac{74769883972877078765947915779475543806177218931383}{151018107920685246151444695105403205187952185789869} a^{24} - \frac{61505676941465902061322698692856636201445460096781}{151018107920685246151444695105403205187952185789869} a^{23} - \frac{10242017373504998087691356802812649246659343735443}{151018107920685246151444695105403205187952185789869} a^{22} + \frac{68359350857166428363219121658928348113231939012332}{151018107920685246151444695105403205187952185789869} a^{21} - \frac{42879175700810282139597482545565902327410934301969}{151018107920685246151444695105403205187952185789869} a^{20} - \frac{28483286242966432913069596530742253992462004723033}{151018107920685246151444695105403205187952185789869} a^{19} - \frac{22687015065026354076443790806326726586937488945314}{151018107920685246151444695105403205187952185789869} a^{18} + \frac{71696074675090914460738329057658031000355278341615}{151018107920685246151444695105403205187952185789869} a^{17} - \frac{23786258123491260040795635434231114816834413340762}{151018107920685246151444695105403205187952185789869} a^{16} + \frac{6891949727470326764910053306047505182007328170217}{151018107920685246151444695105403205187952185789869} a^{15} + \frac{46495861285975869774082615299482388715069033467113}{151018107920685246151444695105403205187952185789869} a^{14} + \frac{4424988345014596614914679140265840913915229318603}{151018107920685246151444695105403205187952185789869} a^{13} + \frac{2430596463356426876498823119449312863759872799102}{151018107920685246151444695105403205187952185789869} a^{12} + \frac{5336256357983556685213935687579461976705801979100}{151018107920685246151444695105403205187952185789869} a^{11} - \frac{73299331340417596955920796460974677411177803735627}{151018107920685246151444695105403205187952185789869} a^{10} - \frac{74471474815717722541444826942599572394517082285989}{151018107920685246151444695105403205187952185789869} a^{9} + \frac{64110302365050864267818431394905404677192253853194}{151018107920685246151444695105403205187952185789869} a^{8} + \frac{34161031805672193539393097503599384193086615848513}{151018107920685246151444695105403205187952185789869} a^{7} + \frac{19658555655255776314619108341619560342684199952031}{151018107920685246151444695105403205187952185789869} a^{6} - \frac{40900427929324387063172308450673330340211488622077}{151018107920685246151444695105403205187952185789869} a^{5} + \frac{49373115880590285993797217660235206376749861998636}{151018107920685246151444695105403205187952185789869} a^{4} - \frac{827713066075859079369504869856982362036294635238}{151018107920685246151444695105403205187952185789869} a^{3} - \frac{23752340555382770772207582943083521811097756748086}{151018107920685246151444695105403205187952185789869} a^{2} + \frac{50197769489815389920597060436821169886356893060502}{151018107920685246151444695105403205187952185789869} a + \frac{57010110340987785814250541775913103748770762064402}{151018107920685246151444695105403205187952185789869}$
Class group and class number
$C_{4}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $27$ | 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: | \( 238735871646007500 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 28 |
| The 28 conjugacy class representatives for $C_{28}$ |
| Character table for $C_{28}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{145}) \), 4.4.3048625.2, 7.7.594823321.1, 14.14.801611618199890796015625.1 |
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 | ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/3.7.0.1}{7} }^{4}$ | R | $28$ | $28$ | $28$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ | $28$ | $28$ | R | $28$ | ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ | ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ | $28$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| 29 | Data not computed | ||||||