Normalized defining polynomial
\( x^{25} - 550 x^{23} + 124025 x^{21} - 14973750 x^{19} - 1073875 x^{18} + 1061415025 x^{17} + 292953100 x^{16} - 45696670635 x^{15} - 29496124625 x^{14} + 1195775125050 x^{13} + 1388600089800 x^{12} - 18216821941450 x^{11} - 32760226572755 x^{10} + 141336159619875 x^{9} + 373032643425475 x^{8} - 331978561284425 x^{7} - 1686835306391125 x^{6} - 1031805037461990 x^{5} + 1080219244517875 x^{4} + 1100312373931650 x^{3} - 5183853220150 x^{2} - 156124659276300 x - 26153667939751 \)
Invariants
| Degree: | $25$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[25, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(227936564389216769066972959924266550757465665810741484165191650390625=5^{68}\cdot 11^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $542.39$ | 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: | $5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1375=5^{3}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1375}(256,·)$, $\chi_{1375}(1,·)$, $\chi_{1375}(1346,·)$, $\chi_{1375}(246,·)$, $\chi_{1375}(521,·)$, $\chi_{1375}(1356,·)$, $\chi_{1375}(1101,·)$, $\chi_{1375}(911,·)$, $\chi_{1375}(16,·)$, $\chi_{1375}(531,·)$, $\chi_{1375}(276,·)$, $\chi_{1375}(86,·)$, $\chi_{1375}(796,·)$, $\chi_{1375}(361,·)$, $\chi_{1375}(1186,·)$, $\chi_{1375}(291,·)$, $\chi_{1375}(806,·)$, $\chi_{1375}(551,·)$, $\chi_{1375}(1116,·)$, $\chi_{1375}(1071,·)$, $\chi_{1375}(566,·)$, $\chi_{1375}(841,·)$, $\chi_{1375}(1081,·)$, $\chi_{1375}(826,·)$, $\chi_{1375}(636,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{11} a^{9}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{121} a^{12}$, $\frac{1}{121} a^{13}$, $\frac{1}{121} a^{14}$, $\frac{1}{1331} a^{15}$, $\frac{1}{1331} a^{16}$, $\frac{1}{1331} a^{17}$, $\frac{1}{1331} a^{18}$, $\frac{1}{1331} a^{19}$, $\frac{1}{14641} a^{20}$, $\frac{1}{14641} a^{21}$, $\frac{1}{2181509} a^{22} - \frac{70}{2181509} a^{21} - \frac{3}{198319} a^{20} + \frac{47}{198319} a^{19} + \frac{61}{198319} a^{18} + \frac{40}{198319} a^{17} + \frac{52}{198319} a^{16} - \frac{15}{198319} a^{15} - \frac{6}{1639} a^{14} - \frac{56}{18029} a^{13} + \frac{25}{18029} a^{12} + \frac{65}{18029} a^{11} + \frac{56}{18029} a^{10} + \frac{50}{1639} a^{9} - \frac{58}{1639} a^{8} - \frac{8}{1639} a^{7} - \frac{3}{1639} a^{6} - \frac{69}{1639} a^{5} - \frac{40}{149} a^{4} - \frac{18}{149} a^{3} - \frac{4}{149} a^{2} - \frac{46}{149} a + \frac{39}{149}$, $\frac{1}{497604384409} a^{23} + \frac{80860}{497604384409} a^{22} - \frac{16372571}{497604384409} a^{21} + \frac{4238361}{497604384409} a^{20} - \frac{14715290}{45236762219} a^{19} - \frac{16039599}{45236762219} a^{18} - \frac{4057639}{45236762219} a^{17} - \frac{7929642}{45236762219} a^{16} + \frac{10093828}{45236762219} a^{15} + \frac{15807922}{4112432929} a^{14} + \frac{355643}{373857539} a^{13} - \frac{10767739}{4112432929} a^{12} - \frac{14597767}{4112432929} a^{11} - \frac{14825748}{4112432929} a^{10} - \frac{16045910}{373857539} a^{9} + \frac{16536317}{373857539} a^{8} - \frac{11155072}{373857539} a^{7} + \frac{4407133}{373857539} a^{6} + \frac{451516}{33987049} a^{5} - \frac{303408}{33987049} a^{4} + \frac{8547265}{33987049} a^{3} + \frac{16000823}{33987049} a^{2} - \frac{2947494}{33987049} a - \frac{7453722}{33987049}$, $\frac{1}{45409998588175398900468871735826665973423644184176503984407230891410371793775951014359279929024929606349653873608313385610646844091} a^{24} + \frac{39124878360373503698022694911322690759760331629265157545752008177424483954552237653828619507991968587113654215894235967}{45409998588175398900468871735826665973423644184176503984407230891410371793775951014359279929024929606349653873608313385610646844091} a^{23} + \frac{7353435615092882167591276257358223777943799148845874227165815642403909740510452074641686811515921273545816868870651345379613}{45409998588175398900468871735826665973423644184176503984407230891410371793775951014359279929024929606349653873608313385610646844091} a^{22} + \frac{952299724703822675700845564517079710691592487352506397939098697164400288135319192697781334332870379212448050841569548515955059}{45409998588175398900468871735826665973423644184176503984407230891410371793775951014359279929024929606349653873608313385610646844091} a^{21} + \frac{410119414701038517885208349240640079520610402640632412146812646681552968215504509785418652405466772544596394955058906821867}{45409998588175398900468871735826665973423644184176503984407230891410371793775951014359279929024929606349653873608313385610646844091} a^{20} + \frac{1016224959929627343790563953758335042495456254708422755004065115145184419465140136641755827697864566520777399717181797574635568}{4128181689834127172769897430529696906674876744016045816764293717400942890343268274032661811729539055122695806691664853237331531281} a^{19} - \frac{301755333755987298906944984962087605329990972369443415921189459889771815759909700573726354499299084564039338626292736958118176}{4128181689834127172769897430529696906674876744016045816764293717400942890343268274032661811729539055122695806691664853237331531281} a^{18} - \frac{1193273479440407863584481362350298420132887968092615665153924481390531113618355291363325323742567252886769493629458122551475342}{4128181689834127172769897430529696906674876744016045816764293717400942890343268274032661811729539055122695806691664853237331531281} a^{17} + \frac{1457391522357928387968964883656319168237819620682028641036688856143525231744317837939527893441831506232572637578234024999566419}{4128181689834127172769897430529696906674876744016045816764293717400942890343268274032661811729539055122695806691664853237331531281} a^{16} - \frac{835396425200297841490982118296078267290270485492696376938929955184578895359643076800658171624828021097849183542114118058347380}{4128181689834127172769897430529696906674876744016045816764293717400942890343268274032661811729539055122695806691664853237331531281} a^{15} + \frac{1450437686793729943846238488638919431781523578005658088391858049934030687210933885834376392450246268617710921314137107079086965}{375289244530375197524536130048154264243170613092367801524026701581903899122115297639332891975412641374790527881060441203393775571} a^{14} - \frac{689602037126966704906563475389727303035250994493588429865345983985623019512461996868013970608315207014152111120964111600117202}{375289244530375197524536130048154264243170613092367801524026701581903899122115297639332891975412641374790527881060441203393775571} a^{13} + \frac{1062024773306528217403066198794400758595468553580434424059727778776185424684078287427961653577520783772747781199489790371617656}{375289244530375197524536130048154264243170613092367801524026701581903899122115297639332891975412641374790527881060441203393775571} a^{12} - \frac{81381716127255140161939809263687008914229682409220847244282974050327976932398900247190057989607690728613352198215791540431342}{375289244530375197524536130048154264243170613092367801524026701581903899122115297639332891975412641374790527881060441203393775571} a^{11} - \frac{1354739902459747139918286270536218130991574909918023739788093065806664942922484601188746694773760200303729106694841993665990567}{375289244530375197524536130048154264243170613092367801524026701581903899122115297639332891975412641374790527881060441203393775571} a^{10} + \frac{994569593068194693563340235219090592891588684307265101605301584173366161209807059232472370348546931088325458482086569599834154}{34117204048215927047685102731650387658470055735669800138547881961991263556555936149030262906855694670435502534641858291217615961} a^{9} - \frac{1149621471901603521764954565471440338825147724759586494284703369654540696246185731103478107266250819044064671615281977454022922}{34117204048215927047685102731650387658470055735669800138547881961991263556555936149030262906855694670435502534641858291217615961} a^{8} - \frac{1466913716429320678622117074962561318061355280501395114316115987185772563418991949146441259564026121898104308196426901918360316}{34117204048215927047685102731650387658470055735669800138547881961991263556555936149030262906855694670435502534641858291217615961} a^{7} + \frac{246469194288521618886125388409839010842057621141156881035536516952610512649538456829629030590974421851928098362457180534333236}{34117204048215927047685102731650387658470055735669800138547881961991263556555936149030262906855694670435502534641858291217615961} a^{6} + \frac{353685019743745192095303267245087677127866048226278772878920616452023651988551352630173097819375517595207956976581026309172811}{34117204048215927047685102731650387658470055735669800138547881961991263556555936149030262906855694670435502534641858291217615961} a^{5} - \frac{1213984408791631854521396734892216885999642606448365232050321184210777480446116966792414775963817918715178915999918698870524316}{3101564004383266095244100248331853423497277794151800012595261996544660323323266922639114809714154060948682048603805299201601451} a^{4} + \frac{1027487082516135677563231333347373291825247534458470025425851588002476036172098892900804529369386979285785601038999927776349461}{3101564004383266095244100248331853423497277794151800012595261996544660323323266922639114809714154060948682048603805299201601451} a^{3} + \frac{865149875600061416686799374402576140272747537624963814774462943302161828666464663525523576539669098614170368942494046073368553}{3101564004383266095244100248331853423497277794151800012595261996544660323323266922639114809714154060948682048603805299201601451} a^{2} - \frac{205805663006692157872084037057794531490562188253270362687891268054266630941795112207434719950085954863471765467263057541603555}{3101564004383266095244100248331853423497277794151800012595261996544660323323266922639114809714154060948682048603805299201601451} a - \frac{740862660567497367684041958556615580047440455158019050215146801432426993730415085560293093583354696563661042955534685944554652}{3101564004383266095244100248331853423497277794151800012595261996544660323323266922639114809714154060948682048603805299201601451}$
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
| Rank: | $24$ | 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: | \( 26679700051561640000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 25 |
| The 25 conjugacy class representatives for $C_{25}$ |
| Character table for $C_{25}$ is not computed |
Intermediate fields
| 5.5.390625.1 |
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 | $25$ | $25$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ | R | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | $25$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ | $25$ | $25$ | $25$ |
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 | ||||||
| 11 | Data not computed | ||||||