Normalized defining polynomial
\( x^{26} - 2 x^{25} - 21 x^{24} + 38 x^{23} + 294 x^{22} - 464 x^{21} - 1357 x^{20} + 1930 x^{19} + 9504 x^{18} - 8228 x^{17} + 25716 x^{16} - 28630 x^{15} + 286963 x^{14} - 378044 x^{13} + 2103312 x^{12} - 1678782 x^{11} + 11411452 x^{10} - 5574160 x^{9} + 50609745 x^{8} - 14301644 x^{7} + 160353701 x^{6} - 26227056 x^{5} + 361035384 x^{4} - 44819066 x^{3} + 515469360 x^{2} - 43859864 x + 329025401 \)
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: | \(-132675490325051365459874809737094631268103216670179328=-\,2^{39}\cdot 53^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $110.45$ | 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: | $2, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(424=2^{3}\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{424}(1,·)$, $\chi_{424}(259,·)$, $\chi_{424}(227,·)$, $\chi_{424}(225,·)$, $\chi_{424}(201,·)$, $\chi_{424}(395,·)$, $\chi_{424}(417,·)$, $\chi_{424}(81,·)$, $\chi_{424}(275,·)$, $\chi_{424}(203,·)$, $\chi_{424}(281,·)$, $\chi_{424}(153,·)$, $\chi_{424}(331,·)$, $\chi_{424}(155,·)$, $\chi_{424}(289,·)$, $\chi_{424}(99,·)$, $\chi_{424}(195,·)$, $\chi_{424}(97,·)$, $\chi_{424}(169,·)$, $\chi_{424}(107,·)$, $\chi_{424}(387,·)$, $\chi_{424}(49,·)$, $\chi_{424}(307,·)$, $\chi_{424}(89,·)$, $\chi_{424}(121,·)$, $\chi_{424}(187,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23} a^{22} - \frac{6}{23} a^{21} - \frac{4}{23} a^{20} - \frac{2}{23} a^{19} + \frac{1}{23} a^{18} - \frac{11}{23} a^{17} - \frac{9}{23} a^{16} + \frac{7}{23} a^{15} - \frac{6}{23} a^{14} + \frac{2}{23} a^{13} - \frac{10}{23} a^{12} - \frac{4}{23} a^{11} + \frac{2}{23} a^{10} + \frac{1}{23} a^{9} + \frac{7}{23} a^{8} - \frac{2}{23} a^{7} + \frac{7}{23} a^{6} - \frac{2}{23} a^{5} + \frac{8}{23} a^{4} - \frac{11}{23} a^{3} - \frac{1}{23} a^{2} + \frac{9}{23} a + \frac{6}{23}$, $\frac{1}{1909} a^{23} + \frac{17}{1909} a^{22} - \frac{648}{1909} a^{21} + \frac{504}{1909} a^{20} - \frac{183}{1909} a^{19} - \frac{471}{1909} a^{18} + \frac{796}{1909} a^{17} - \frac{131}{1909} a^{16} - \frac{167}{1909} a^{15} + \frac{416}{1909} a^{14} - \frac{493}{1909} a^{13} + \frac{19}{1909} a^{12} - \frac{412}{1909} a^{11} + \frac{93}{1909} a^{10} + \frac{260}{1909} a^{9} + \frac{113}{1909} a^{8} + \frac{53}{1909} a^{7} - \frac{163}{1909} a^{6} + \frac{77}{1909} a^{5} + \frac{35}{1909} a^{4} + \frac{712}{1909} a^{3} - \frac{865}{1909} a^{2} + \frac{834}{1909} a + \frac{8}{83}$, $\frac{1}{46556235749} a^{24} - \frac{2538442}{46556235749} a^{23} + \frac{667381408}{46556235749} a^{22} - \frac{13730156511}{46556235749} a^{21} - \frac{15492865346}{46556235749} a^{20} + \frac{5592677835}{46556235749} a^{19} - \frac{148864578}{435105007} a^{18} + \frac{15578016692}{46556235749} a^{17} - \frac{22108418631}{46556235749} a^{16} + \frac{4943024733}{46556235749} a^{15} + \frac{8075717252}{46556235749} a^{14} - \frac{17464510569}{46556235749} a^{13} + \frac{6765095360}{46556235749} a^{12} + \frac{21109855762}{46556235749} a^{11} - \frac{206278182}{46556235749} a^{10} + \frac{12511024429}{46556235749} a^{9} - \frac{3536278384}{46556235749} a^{8} - \frac{6025674974}{46556235749} a^{7} - \frac{8847845368}{46556235749} a^{6} - \frac{1099413561}{46556235749} a^{5} - \frac{12396291691}{46556235749} a^{4} - \frac{2957496828}{46556235749} a^{3} - \frac{12654458003}{46556235749} a^{2} + \frac{626375174}{46556235749} a - \frac{4908460323}{46556235749}$, $\frac{1}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{25} - \frac{7386026046021206031517033855561120689428915068201572712746074507318968262098089115}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{24} - \frac{1258547209664925155183547322964679369373372432912241965206293592179564681409696647752532611}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{23} + \frac{162404100459974776433719875419894676284536176347720416152191399014912883927336185170543546534}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{22} - \frac{11836532053750489767635680007972974086396763549502697920704322243847733318413798204230677922304}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{21} + \frac{1522501258780488500328957674185626614509002090672258552037839763560853889357617761042782311774}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{20} - \frac{790061223974532424265010146562790306088741846405345572853979477980730237481027177359571223768}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{19} + \frac{2572164320568525883219907987670490235309524967146826729978921801164788058193039929953943716661}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{18} - \frac{9420045256592187533237837360299557998117195007606355020971064343753833118389628357552623256977}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{17} + \frac{8394626497690498429982740501472984452558740824251987221313504067055765051747847658635623693309}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{16} + \frac{2442009292531111339076305675344443952207092201175514397835609291896568396328577372522278245024}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{15} - \frac{5178060822789391009633210886560068498043170271140477399358778742295724854971837653376774036067}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{14} + \frac{71938359395363536546432707187547121898916473253442646256047333426495794364290213784626946051}{1087587499475675411478382144535034779318029808638082593286147243178387831064912321176167862049} a^{13} + \frac{9354685705571132542360069626121242493035542456999872856410579365577961588494330510002042909192}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{12} + \frac{9453661707631789052559239169262958522390791209684750992948198918946012752798412464035254262014}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{11} - \frac{3017493161384291179057026605735934541172060822448551636973786222944795268077039676192748408057}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{10} + \frac{9229102135420641833243220532513943502375914252598083737921944062180503255593745426746342627856}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{9} + \frac{4646467006884179223088696649883401618942099383198594193216666072368010331774616485223006705278}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{8} + \frac{1240055974841981596156033040979182078254426055010502133784992213322283247676615507211838136941}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{7} + \frac{56323326209279128667934355425186329068017148380990351360463815814249804527095244157779819005}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{6} + \frac{3495901820205908383944910474204448704718925246921230682712397889584477549887337800057378213997}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{5} - \frac{9103289878659954396212894184162505418270419897806196651740947571299502833356962458273204120615}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{4} + \frac{5169547567145645778760932439894516139685270065338675091320284634371349573826130934087253628664}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{3} - \frac{9252275935686366607044370656860264625159649092011130049369271653276086393740265606922533838111}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a^{2} - \frac{3188901939960221492931401086355614879103234186289153623359187460327409460140654606238342259083}{25014512487940534464002789324305799924314685598675899645581386593102920114492983387051860827127} a - \frac{233947197656452521826735166461354488249932958684457593198471806086964352515468936056007727566}{1087587499475675411478382144535034779318029808638082593286147243178387831064912321176167862049}$
Class group and class number
$C_{1064051}$, which has order $1064051$ (assuming GRH)
Unit group
| Rank: | $12$ | 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: | \( 5382739421.971964 \) (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
| \(\Q(\sqrt{-2}) \), 13.13.491258904256726154641.1 |
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 | ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ | $26$ | $26$ | ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | $26$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ | $26$ | R | ${\href{/LocalNumberField/59.13.0.1}{13} }^{2}$ |
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 | Data not computed | ||||||
| 53 | Data not computed | ||||||