Normalized defining polynomial
\( x^{26} - x^{25} + 134 x^{24} - 135 x^{23} + 7272 x^{22} - 7408 x^{21} + 207314 x^{20} - 214859 x^{19} + 3365872 x^{18} - 3529266 x^{17} + 31452906 x^{16} - 33307875 x^{15} + 160481844 x^{14} - 201399519 x^{13} + 399638724 x^{12} - 1090191903 x^{11} + 1078517226 x^{10} - 5333780593 x^{9} + 7447220393 x^{8} - 10347834170 x^{7} + 17322572104 x^{6} - 3801869175 x^{5} + 50805450213 x^{4} - 94464627281 x^{3} + 97635221932 x^{2} + 113892513590 x + 60572712821 \)
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: | \(-441572122027487213320469174167608817426459689061923004183=-\,11^{13}\cdot 53^{25}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $150.89$ | 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: | $11, 53$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(583=11\cdot 53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{583}(1,·)$, $\chi_{583}(386,·)$, $\chi_{583}(131,·)$, $\chi_{583}(452,·)$, $\chi_{583}(197,·)$, $\chi_{583}(582,·)$, $\chi_{583}(329,·)$, $\chi_{583}(331,·)$, $\chi_{583}(461,·)$, $\chi_{583}(274,·)$, $\chi_{583}(483,·)$, $\chi_{583}(342,·)$, $\chi_{583}(89,·)$, $\chi_{583}(155,·)$, $\chi_{583}(540,·)$, $\chi_{583}(219,·)$, $\chi_{583}(100,·)$, $\chi_{583}(43,·)$, $\chi_{583}(428,·)$, $\chi_{583}(494,·)$, $\chi_{583}(241,·)$, $\chi_{583}(309,·)$, $\chi_{583}(364,·)$, $\chi_{583}(122,·)$, $\chi_{583}(252,·)$, $\chi_{583}(254,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{83} a^{21} + \frac{35}{83} a^{20} - \frac{25}{83} a^{19} - \frac{23}{83} a^{18} - \frac{16}{83} a^{17} + \frac{34}{83} a^{16} + \frac{25}{83} a^{15} - \frac{3}{83} a^{14} + \frac{18}{83} a^{13} - \frac{40}{83} a^{12} - \frac{41}{83} a^{11} + \frac{1}{83} a^{10} - \frac{9}{83} a^{9} - \frac{27}{83} a^{8} - \frac{37}{83} a^{7} - \frac{40}{83} a^{6} - \frac{36}{83} a^{5} + \frac{23}{83} a^{4} + \frac{30}{83} a^{3} - \frac{8}{83} a^{2} - \frac{16}{83} a + \frac{25}{83}$, $\frac{1}{83} a^{22} - \frac{5}{83} a^{20} + \frac{22}{83} a^{19} - \frac{41}{83} a^{18} + \frac{13}{83} a^{17} - \frac{3}{83} a^{16} + \frac{35}{83} a^{15} + \frac{40}{83} a^{14} - \frac{6}{83} a^{13} + \frac{31}{83} a^{12} + \frac{25}{83} a^{11} + \frac{39}{83} a^{10} + \frac{39}{83} a^{9} - \frac{5}{83} a^{8} + \frac{10}{83} a^{7} + \frac{36}{83} a^{6} + \frac{38}{83} a^{5} - \frac{28}{83} a^{4} + \frac{21}{83} a^{3} + \frac{15}{83} a^{2} + \frac{4}{83} a + \frac{38}{83}$, $\frac{1}{83} a^{23} + \frac{31}{83} a^{20} - \frac{19}{83} a^{18} + \frac{39}{83} a^{16} - \frac{1}{83} a^{15} - \frac{21}{83} a^{14} + \frac{38}{83} a^{13} - \frac{9}{83} a^{12} - \frac{39}{83} a^{10} + \frac{33}{83} a^{9} + \frac{41}{83} a^{8} + \frac{17}{83} a^{7} + \frac{4}{83} a^{6} + \frac{41}{83} a^{5} - \frac{30}{83} a^{4} - \frac{1}{83} a^{3} - \frac{36}{83} a^{2} + \frac{41}{83} a - \frac{41}{83}$, $\frac{1}{83} a^{24} - \frac{6}{83} a^{20} + \frac{9}{83} a^{19} - \frac{34}{83} a^{18} + \frac{37}{83} a^{17} + \frac{24}{83} a^{16} + \frac{34}{83} a^{15} - \frac{35}{83} a^{14} + \frac{14}{83} a^{13} - \frac{5}{83} a^{12} - \frac{13}{83} a^{11} + \frac{2}{83} a^{10} - \frac{12}{83} a^{9} + \frac{24}{83} a^{8} - \frac{11}{83} a^{7} + \frac{36}{83} a^{6} + \frac{7}{83} a^{5} + \frac{33}{83} a^{4} + \frac{30}{83} a^{3} + \frac{40}{83} a^{2} + \frac{40}{83} a - \frac{28}{83}$, $\frac{1}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{25} + \frac{34752968549301248550934565514761354308163172951089839327114183502727932161529690811205455901607860229214486047002592111801118760417878054630994}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{24} + \frac{24133760283797720672715410331905568739125180445845505005624253867053690855863515715790015232441333816234785249473149046125986894191412651898379}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{23} - \frac{26454231602019781981347371115137361807188168331216497845225670772156777391413307728907448370313144838164932149217294389469060589814484773263038}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{22} + \frac{26017891332502002708267314320914812504742705047544255582218521062439829866024475526179304262992117808730758707890747715754192838679056439450059}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{21} - \frac{1375243581965139216047300030587848240225048181690578350525081074124822153646257933298623102190912623766180383496302069515276791921608331604408482}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{20} + \frac{347239840973790193047536333011265137897529808389953339814481318648819068176713314717746045429514524255086415129757504185278789113887985761913462}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{19} + \frac{1435438142436425673000485098635361296852234486533482523105513614704746398283996305949497497428203159106018608439473090416382403953207086211556578}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{18} - \frac{659667732685956229586610994789571436829644900909963854265210179430114701737995647422020082073761318830693183599427138700643196013889510142863218}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{17} + \frac{2677997684654437074173008948246138739662363690241528568589920334301724376087306195903990314903084250660212453385196055164464323149366147366690476}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{16} + \frac{1092517955244117455362018548322316003463308930716291832532306598108379435759104869665632792496307917207420391906866844100570342439039329436579006}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{15} + \frac{515373266565922182885132029447636366721027741561181247120208261086168927340527406190026658709298435050252407672374457363963477188456577668652740}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{14} + \frac{1826087674103044950353888648359236796716537736173435354709567580879122982204182267279896641916688142248567864297433590429847933796038353472991069}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{13} - \frac{2343436373900203598153959135789468217796004426976540896011785853730851936442925804607760030615404635107773475814359853249008101672280764579991402}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{12} + \frac{666160056689759524429979838556292679189275383789977175902679051841850446436789656474546174730233536069969065384972222963434414285264185667206742}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{11} + \frac{2585081611366841573236217381750696327034524958012795162631478120521669257003853884757599995971079082343069610358394410208023297776930037136508694}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{10} - \frac{866681504739802526013764775789637002580989058099014093211824632627929217685236252237366200324832907838339672202914301103865437838368468616029138}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{9} - \frac{1121295321047977033285566522175966227595526794799874920744206295099884014929451405977183880509098442203089244495611726927454956841630634787576672}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{8} - \frac{6926936845667627432865390902809924505463736500741833998770333415073701223598904012172369095997136742656773975786976022403270610743244047582487}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{7} - \frac{660014265456129828966074288731266472739085337249894273442402056461911947603957238634478530673053116111293707435502472935147030821496109366336701}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{6} + \frac{1609210068643467699739820116453714518405434441631515864384791503007517367966337140641296807362865199454383314410103321134677138886151043725196230}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{5} + \frac{3386854993380685395191487496860581809558538622115247757870236085241313231533771808751473331394736672095771958819342504440468221752925095593991207}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{4} - \frac{536010958427888098950652018306044153973881023057482702733828099802084926674426051391985190835626821390626663449054636403744381243537726329767982}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{3} + \frac{650291192966544293409953016299400116830635699273710688058716314378315197168031393717771457820524129213646609237412406596206872246413171473147673}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a^{2} + \frac{1823047927048225878553911057135356915684034650340204614223800718217678697148339984399696995023381098128425139441006366951694703678168245219092590}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357} a + \frac{882055364417855677931767562841509871605683171682278391043103924901851301417512318098558529688697941013493463440582606771619255062606618104629122}{7030193456406996204237124258076637439793817295985563129522324478681554275036708001142221555451727212528444900116373622782495277419591081115859357}$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | 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
| \(\Q(\sqrt{-583}) \), 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 | ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ | $26$ | $26$ | $26$ | R | $26$ | $26$ | ${\href{/LocalNumberField/19.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}$ | $26$ | $26$ | ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ | ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ | $26$ | ${\href{/LocalNumberField/47.13.0.1}{13} }^{2}$ | 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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 53 | Data not computed | ||||||