Normalized defining polynomial
\( x^{26} + 73 x^{24} + 2180 x^{22} + 35829 x^{20} + 363625 x^{18} + 2405673 x^{16} + 10625163 x^{14} + 31462021 x^{12} + 61715876 x^{10} + 77878153 x^{8} + 59854953 x^{6} + 25249803 x^{4} + 4581570 x^{2} + 85849 \)
Invariants
| Degree: | $26$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 13]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-234331179334410135333725229627752024799305672723267584=-\,2^{26}\cdot 79^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $112.90$ | 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: | $2, 79$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(316=2^{2}\cdot 79\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{316}(1,·)$, $\chi_{316}(259,·)$, $\chi_{316}(65,·)$, $\chi_{316}(141,·)$, $\chi_{316}(143,·)$, $\chi_{316}(275,·)$, $\chi_{316}(21,·)$, $\chi_{316}(87,·)$, $\chi_{316}(225,·)$, $\chi_{316}(89,·)$, $\chi_{316}(283,·)$, $\chi_{316}(159,·)$, $\chi_{316}(289,·)$, $\chi_{316}(67,·)$, $\chi_{316}(131,·)$, $\chi_{316}(101,·)$, $\chi_{316}(97,·)$, $\chi_{316}(299,·)$, $\chi_{316}(301,·)$, $\chi_{316}(255,·)$, $\chi_{316}(179,·)$, $\chi_{316}(245,·)$, $\chi_{316}(247,·)$, $\chi_{316}(223,·)$, $\chi_{316}(125,·)$, $\chi_{316}(117,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23} a^{16} + \frac{6}{23} a^{14} - \frac{2}{23} a^{12} - \frac{3}{23} a^{10} + \frac{8}{23} a^{8} + \frac{10}{23} a^{6} + \frac{11}{23} a^{4} - \frac{5}{23} a^{2} - \frac{10}{23}$, $\frac{1}{23} a^{17} + \frac{6}{23} a^{15} - \frac{2}{23} a^{13} - \frac{3}{23} a^{11} + \frac{8}{23} a^{9} + \frac{10}{23} a^{7} + \frac{11}{23} a^{5} - \frac{5}{23} a^{3} - \frac{10}{23} a$, $\frac{1}{23} a^{18} + \frac{8}{23} a^{14} + \frac{9}{23} a^{12} + \frac{3}{23} a^{10} + \frac{8}{23} a^{8} - \frac{3}{23} a^{6} - \frac{2}{23} a^{4} - \frac{3}{23} a^{2} - \frac{9}{23}$, $\frac{1}{23} a^{19} + \frac{8}{23} a^{15} + \frac{9}{23} a^{13} + \frac{3}{23} a^{11} + \frac{8}{23} a^{9} - \frac{3}{23} a^{7} - \frac{2}{23} a^{5} - \frac{3}{23} a^{3} - \frac{9}{23} a$, $\frac{1}{54487} a^{20} + \frac{969}{54487} a^{18} - \frac{706}{54487} a^{16} - \frac{15958}{54487} a^{14} + \frac{1757}{54487} a^{12} - \frac{17092}{54487} a^{10} + \frac{17125}{54487} a^{8} + \frac{23577}{54487} a^{6} + \frac{26660}{54487} a^{4} + \frac{18387}{54487} a^{2} + \frac{24846}{54487}$, $\frac{1}{54487} a^{21} + \frac{969}{54487} a^{19} - \frac{706}{54487} a^{17} - \frac{15958}{54487} a^{15} + \frac{1757}{54487} a^{13} - \frac{17092}{54487} a^{11} + \frac{17125}{54487} a^{9} + \frac{23577}{54487} a^{7} + \frac{26660}{54487} a^{5} + \frac{18387}{54487} a^{3} + \frac{24846}{54487} a$, $\frac{1}{1253201} a^{22} + \frac{11}{1253201} a^{20} - \frac{21681}{1253201} a^{18} + \frac{13653}{1253201} a^{16} + \frac{578031}{1253201} a^{14} - \frac{1414}{54487} a^{12} - \frac{468912}{1253201} a^{10} + \frac{577498}{1253201} a^{8} - \frac{582893}{1253201} a^{6} + \frac{94104}{1253201} a^{4} + \frac{248670}{1253201} a^{2} + \frac{434771}{1253201}$, $\frac{1}{1253201} a^{23} + \frac{11}{1253201} a^{21} - \frac{21681}{1253201} a^{19} + \frac{13653}{1253201} a^{17} + \frac{578031}{1253201} a^{15} - \frac{1414}{54487} a^{13} - \frac{468912}{1253201} a^{11} + \frac{577498}{1253201} a^{9} - \frac{582893}{1253201} a^{7} + \frac{94104}{1253201} a^{5} + \frac{248670}{1253201} a^{3} + \frac{434771}{1253201} a$, $\frac{1}{36312097996795594591} a^{24} - \frac{3042854934540}{36312097996795594591} a^{22} + \frac{163102868478168}{36312097996795594591} a^{20} - \frac{729910204257191736}{36312097996795594591} a^{18} - \frac{80847591801097298}{36312097996795594591} a^{16} - \frac{9661084337237949398}{36312097996795594591} a^{14} - \frac{17285195517999639779}{36312097996795594591} a^{12} - \frac{1479230434318096355}{36312097996795594591} a^{10} + \frac{12683277970346318026}{36312097996795594591} a^{8} - \frac{7448301868357136890}{36312097996795594591} a^{6} + \frac{10032743142454686592}{36312097996795594591} a^{4} + \frac{538331259201357278}{1578786869425895417} a^{2} - \frac{2720085983921497350}{36312097996795594591}$, $\frac{1}{10639444713061109215163} a^{25} - \frac{2321081094405820}{10639444713061109215163} a^{23} + \frac{65966413391469129}{10639444713061109215163} a^{21} - \frac{139867634661935348719}{10639444713061109215163} a^{19} - \frac{96188045872268062084}{10639444713061109215163} a^{17} - \frac{1616146772196135558938}{10639444713061109215163} a^{15} - \frac{4424692997232316545979}{10639444713061109215163} a^{13} - \frac{4685677053184660126906}{10639444713061109215163} a^{11} - \frac{641821309451440109558}{10639444713061109215163} a^{9} - \frac{4348651077032782709266}{10639444713061109215163} a^{7} - \frac{5134304159957988487522}{10639444713061109215163} a^{5} + \frac{95761049294509888738}{462584552741787357181} a^{3} + \frac{2233637208239285062162}{10639444713061109215163} a$
Class group and class number
$C_{189047}$, which has order $189047$ (assuming GRH)
Unit group
| Rank: | $12$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1971930520834328}{462584552741787357181} a^{25} + \frac{140923162736181882}{462584552741787357181} a^{23} + \frac{4079661439500119245}{462584552741787357181} a^{21} + \frac{64194256870339635003}{462584552741787357181} a^{19} + \frac{612991798622570530525}{462584552741787357181} a^{17} + \frac{3718263039526908026100}{462584552741787357181} a^{15} + \frac{14451882346560990127351}{462584552741787357181} a^{13} + \frac{35086534019966094031366}{462584552741787357181} a^{11} + \frac{49061702742471158935382}{462584552741787357181} a^{9} + \frac{30259970969665189480089}{462584552741787357181} a^{7} - \frac{5415753680691374056530}{462584552741787357181} a^{5} - \frac{604858056457361772475}{20112371858338580747} a^{3} - \frac{3477616094881911388040}{462584552741787357181} a \) (order $4$) | 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: | \( 57529828940.82975 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 26 |
| The 26 conjugacy class representatives for $C_{26}$ |
| Character table for $C_{26}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 13.13.59091511031674153381441.1 |
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 | $26$ | ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ | $26$ |
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 | Data not computed | ||||||
| 79 | Data not computed | ||||||