Normalized defining polynomial
\( x^{26} + 121 x^{24} + 6052 x^{22} + 163895 x^{20} + 2652975 x^{18} + 26945411 x^{16} + 177180824 x^{14} + 769764830 x^{12} + 2226843762 x^{10} + 4256806366 x^{8} + 5227349377 x^{6} + 3883036579 x^{4} + 1545040687 x^{2} + 245454889 \)
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: | \(-43781534893766029378911430108761129396314054312752152838144=-\,2^{26}\cdot 131^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $180.07$ | 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: | $2, 131$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(524=2^{2}\cdot 131\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{524}(1,·)$, $\chi_{524}(107,·)$, $\chi_{524}(325,·)$, $\chi_{524}(263,·)$, $\chi_{524}(375,·)$, $\chi_{524}(369,·)$, $\chi_{524}(45,·)$, $\chi_{524}(211,·)$, $\chi_{524}(215,·)$, $\chi_{524}(473,·)$, $\chi_{524}(477,·)$, $\chi_{524}(453,·)$, $\chi_{524}(99,·)$, $\chi_{524}(243,·)$, $\chi_{524}(39,·)$, $\chi_{524}(361,·)$, $\chi_{524}(193,·)$, $\chi_{524}(455,·)$, $\chi_{524}(301,·)$, $\chi_{524}(113,·)$, $\chi_{524}(307,·)$, $\chi_{524}(183,·)$, $\chi_{524}(505,·)$, $\chi_{524}(191,·)$, $\chi_{524}(445,·)$, $\chi_{524}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{89} a^{18} + \frac{4}{89} a^{16} + \frac{25}{89} a^{14} + \frac{4}{89} a^{12} + \frac{9}{89} a^{10} - \frac{40}{89} a^{8} - \frac{20}{89} a^{6} - \frac{20}{89} a^{4} - \frac{43}{89} a^{2} + \frac{4}{89}$, $\frac{1}{89} a^{19} + \frac{4}{89} a^{17} + \frac{25}{89} a^{15} + \frac{4}{89} a^{13} + \frac{9}{89} a^{11} - \frac{40}{89} a^{9} - \frac{20}{89} a^{7} - \frac{20}{89} a^{5} - \frac{43}{89} a^{3} + \frac{4}{89} a$, $\frac{1}{4717} a^{20} + \frac{12}{4717} a^{18} + \frac{947}{4717} a^{16} + \frac{115}{4717} a^{14} - \frac{2095}{4717} a^{12} + \frac{9}{89} a^{10} - \frac{1675}{4717} a^{8} + \frac{443}{4717} a^{6} - \frac{292}{4717} a^{4} + \frac{2330}{4717} a^{2} + \frac{1278}{4717}$, $\frac{1}{4717} a^{21} + \frac{12}{4717} a^{19} + \frac{947}{4717} a^{17} + \frac{115}{4717} a^{15} - \frac{2095}{4717} a^{13} + \frac{9}{89} a^{11} - \frac{1675}{4717} a^{9} + \frac{443}{4717} a^{7} - \frac{292}{4717} a^{5} + \frac{2330}{4717} a^{3} + \frac{1278}{4717} a$, $\frac{1}{134302975889} a^{22} + \frac{10725326}{134302975889} a^{20} - \frac{389869598}{134302975889} a^{18} - \frac{5125378554}{134302975889} a^{16} - \frac{27434544643}{134302975889} a^{14} - \frac{18111369913}{134302975889} a^{12} + \frac{27517959732}{134302975889} a^{10} + \frac{38379855929}{134302975889} a^{8} + \frac{34447188688}{134302975889} a^{6} + \frac{1798184570}{134302975889} a^{4} - \frac{19598688248}{134302975889} a^{2} - \frac{33406148756}{134302975889}$, $\frac{1}{134302975889} a^{23} + \frac{10725326}{134302975889} a^{21} - \frac{389869598}{134302975889} a^{19} - \frac{5125378554}{134302975889} a^{17} - \frac{27434544643}{134302975889} a^{15} - \frac{18111369913}{134302975889} a^{13} + \frac{27517959732}{134302975889} a^{11} + \frac{38379855929}{134302975889} a^{9} + \frac{34447188688}{134302975889} a^{7} + \frac{1798184570}{134302975889} a^{5} - \frac{19598688248}{134302975889} a^{3} - \frac{33406148756}{134302975889} a$, $\frac{1}{1425627471973463772986698023553} a^{24} + \frac{3232303857148555705}{1425627471973463772986698023553} a^{22} + \frac{19160753344996581799132333}{1425627471973463772986698023553} a^{20} + \frac{641710145148298818066283253}{1425627471973463772986698023553} a^{18} + \frac{651901284260532630500674247742}{1425627471973463772986698023553} a^{16} - \frac{449011912628794719127063073436}{1425627471973463772986698023553} a^{14} + \frac{570711273754199691229943577705}{1425627471973463772986698023553} a^{12} + \frac{270067130765909534196540130985}{1425627471973463772986698023553} a^{10} + \frac{505509646503780085160219381636}{1425627471973463772986698023553} a^{8} - \frac{10430589350023176928267601432}{23370942163499406114536033173} a^{6} + \frac{224943672364730922848617230800}{1425627471973463772986698023553} a^{4} + \frac{365288603414150036918805164008}{1425627471973463772986698023553} a^{2} - \frac{362838502491920755215904647001}{1425627471973463772986698023553}$, $\frac{1}{22335305603408256931382597935004851} a^{25} + \frac{14853631709297502702528}{22335305603408256931382597935004851} a^{23} - \frac{84304307388778750744978965423}{22335305603408256931382597935004851} a^{21} - \frac{117027914949720802899339071317823}{22335305603408256931382597935004851} a^{19} + \frac{3781251986773911619968088982333129}{22335305603408256931382597935004851} a^{17} + \frac{1569356315592721731045988020180533}{22335305603408256931382597935004851} a^{15} - \frac{2728930294941109408045656968332000}{22335305603408256931382597935004851} a^{13} + \frac{8363491559858265546098822885707982}{22335305603408256931382597935004851} a^{11} + \frac{9757647898321579465879534103743420}{22335305603408256931382597935004851} a^{9} + \frac{96223681648919440292227408799125}{366152550875545195596436031721391} a^{7} + \frac{9412468735322836975233016717851900}{22335305603408256931382597935004851} a^{5} + \frac{273727903723511091280438036624297}{22335305603408256931382597935004851} a^{3} - \frac{2716563850157974871867864611858652}{22335305603408256931382597935004851} a$
Class group and class number
$C_{3}\times C_{3}\times C_{1620687}$, which has order $14586183$ (assuming GRH)
Unit group
| Rank: | $12$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{18742303651753981654}{5268580110437032626801293} a^{25} - \frac{2232702633907726946086}{5268580110437032626801293} a^{23} - \frac{109246311084833357631699}{5268580110437032626801293} a^{21} - \frac{2867217040610101824896010}{5268580110437032626801293} a^{19} - \frac{44357186238362705592131616}{5268580110437032626801293} a^{17} - \frac{422073619611879790622887207}{5268580110437032626801293} a^{15} - \frac{2532387269702891181856705091}{5268580110437032626801293} a^{13} - \frac{9704055717554598099271574919}{5268580110437032626801293} a^{11} - \frac{23673835294114779851195236185}{5268580110437032626801293} a^{9} - \frac{35836841376669367226742312082}{5268580110437032626801293} a^{7} - \frac{31694352450296996928587506755}{5268580110437032626801293} a^{5} - \frac{14463258486750102097348665865}{5268580110437032626801293} a^{3} - \frac{2543925526041780354828639082}{5268580110437032626801293} a \) (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: | \( 1197545162478.713 \) (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
| \(\Q(\sqrt{-1}) \), 13.13.25542038069936263923006961.1 |
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 | $26$ | ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/13.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/53.1.0.1}{1} }^{26}$ | $26$ |
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 | ||||||
| 131 | Data not computed | ||||||