Normalized defining polynomial
\( x^{25} - 550 x^{23} + 124025 x^{21} - 14973750 x^{19} - 2102375 x^{18} + 1060144525 x^{17} + 573527900 x^{16} - 45626793135 x^{15} - 57745934125 x^{14} + 1198081082550 x^{13} + 2710469962200 x^{12} - 17657242921450 x^{11} - 65075477797495 x^{10} + 96351540709875 x^{9} + 755653228638275 x^{8} + 707942320831075 x^{7} - 2724949254850625 x^{6} - 8413751638067490 x^{5} - 10066386119106625 x^{4} - 6004767748400850 x^{3} - 1749838739064350 x^{2} - 210794145955800 x - 6151028734499 \)
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$$\chi_{1375}(1,·)$, $\chi_{1375}(966,·)$, $\chi_{1375}(906,·)$, $\chi_{1375}(971,·)$, $\chi_{1375}(141,·)$, $\chi_{1375}(1101,·)$, $\chi_{1375}(1336,·)$, $\chi_{1375}(146,·)$, $\chi_{1375}(276,·)$, $\chi_{1375}(1241,·)$, $\chi_{1375}(1181,·)$, $\chi_{1375}(1246,·)$, $\chi_{1375}(81,·)$, $\chi_{1375}(416,·)$, $\chi_{1375}(356,·)$, $\chi_{1375}(1061,·)$, $\chi_{1375}(551,·)$, $\chi_{1375}(421,·)$, $\chi_{1375}(236,·)$, $\chi_{1375}(786,·)$, $\chi_{1375}(691,·)$, $\chi_{1375}(631,·)$, $\chi_{1375}(696,·)$, $\chi_{1375}(826,·)$, $\chi_{1375}(511,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{11} a^{6}$, $\frac{1}{11} a^{7}$, $\frac{1}{11} a^{8}$, $\frac{1}{11} a^{9}$, $\frac{1}{121} a^{10}$, $\frac{1}{121} a^{11}$, $\frac{1}{121} a^{12}$, $\frac{1}{121} a^{13}$, $\frac{1}{121} a^{14}$, $\frac{1}{1331} a^{15}$, $\frac{1}{1331} a^{16}$, $\frac{1}{1331} a^{17}$, $\frac{1}{1331} a^{18}$, $\frac{1}{1331} a^{19}$, $\frac{1}{14641} a^{20}$, $\frac{1}{14641} a^{21}$, $\frac{1}{1478741} a^{22} - \frac{8}{1478741} a^{21} - \frac{30}{1478741} a^{20} - \frac{19}{134431} a^{19} - \frac{12}{134431} a^{18} + \frac{6}{134431} a^{17} - \frac{17}{134431} a^{16} + \frac{3}{134431} a^{15} + \frac{5}{12221} a^{14} - \frac{14}{12221} a^{13} - \frac{1}{12221} a^{12} + \frac{39}{12221} a^{11} + \frac{28}{12221} a^{10} + \frac{29}{1111} a^{9} + \frac{17}{1111} a^{8} + \frac{18}{1111} a^{7} - \frac{16}{1111} a^{6} + \frac{6}{1111} a^{5} - \frac{22}{101} a^{4} + \frac{18}{101} a^{3} + \frac{21}{101} a^{2} - \frac{11}{101} a$, $\frac{1}{232162337} a^{23} - \frac{15}{232162337} a^{22} - \frac{2297}{232162337} a^{21} + \frac{607}{232162337} a^{20} + \frac{2747}{21105667} a^{19} - \frac{5869}{21105667} a^{18} + \frac{1355}{21105667} a^{17} - \frac{7251}{21105667} a^{16} + \frac{3468}{21105667} a^{15} - \frac{4998}{1918697} a^{14} + \frac{1006}{1918697} a^{13} + \frac{2773}{1918697} a^{12} - \frac{6406}{1918697} a^{11} + \frac{123}{1918697} a^{10} + \frac{3551}{174427} a^{9} - \frac{65}{1727} a^{8} + \frac{3494}{174427} a^{7} + \frac{5673}{174427} a^{6} + \frac{322}{174427} a^{5} + \frac{5525}{15857} a^{4} - \frac{5660}{15857} a^{3} - \frac{6016}{15857} a^{2} + \frac{1794}{15857} a + \frac{8}{157}$, $\frac{1}{1897284930247498075654018658841511367155408708620935364935041334726207325974659508525223370317154330507621642909081637} a^{24} + \frac{3666376401457719259339023647384079522357383747586800749639681646561613410908434192925176992810625947526291986}{1897284930247498075654018658841511367155408708620935364935041334726207325974659508525223370317154330507621642909081637} a^{23} - \frac{118804688016944712430406759981398052347262325207828984769822495265776994327478313925074468082673875565977404366}{1897284930247498075654018658841511367155408708620935364935041334726207325974659508525223370317154330507621642909081637} a^{22} - \frac{33842554039629513745498362270717901441504539139651675398917076122684101725550971765345865322624857491640816041365}{1897284930247498075654018658841511367155408708620935364935041334726207325974659508525223370317154330507621642909081637} a^{21} + \frac{62780830742172761014504098540454100286909072655411605928873260037246747877699378710500366191090905740481207571547}{1897284930247498075654018658841511367155408708620935364935041334726207325974659508525223370317154330507621642909081637} a^{20} - \frac{1326988718143695567277340386629324491816537092967684994107836337840143103083533512346512643825764724689362659524}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{19} - \frac{12577151937812823234045825561422258814063352256429532333429803146458292182057515134383900583984622672681028103755}{172480448204318006877638059894682851559582609874630487721367394066018847815878137138656670028832211864329240264461967} a^{18} - \frac{33968387188858075849477423679337848713330887326153980455171733140417557466376293202641880028512072108264087366432}{172480448204318006877638059894682851559582609874630487721367394066018847815878137138656670028832211864329240264461967} a^{17} + \frac{9181067720600665420274678537219064192428552812633068678363177649998969385850148054142803625574086241208064115793}{172480448204318006877638059894682851559582609874630487721367394066018847815878137138656670028832211864329240264461967} a^{16} - \frac{53153385249247609106798288915112311816129206299820297321677932818833021804553358058931884933171083964117185476959}{172480448204318006877638059894682851559582609874630487721367394066018847815878137138656670028832211864329240264461967} a^{15} + \frac{64731352599757944654388002758952460026071743284188611033108128576311262745735633210520868870080776042838112184180}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{14} + \frac{33520890049312162336105918643794380630338570398348414787825669855825225554969603158159537480516291275213238366560}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{13} - \frac{31216606417535382544421581610266695217149723375488531302266819263094785624687024531565148838114053192706229360051}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{12} + \frac{58376282656215687827697057269513899122788796686262582591670001497954594849299287471309958741749543465392289395160}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{11} + \frac{48876401213182254458685422915322686511520802026455443706211448593936724488880698149063976183644552604895096901718}{15680040745847091534330732717698441050871146352239135247397035824183531619625285194423333638984746533120840024041997} a^{10} + \frac{40857519821783977032216218601245499409006339193913001468988241413520460184912566713559099750355121193217681592136}{1425458249622462866757339337972585550079195122930830477036094165834866510875025926765757603544067866647349093094727} a^{9} - \frac{54781913979385707027251693825281933905971612932159755335873354848277799918019564794495841709652636653745007707245}{1425458249622462866757339337972585550079195122930830477036094165834866510875025926765757603544067866647349093094727} a^{8} + \frac{7765209509590603181962636512330314683675868753509350284248661188178848165896747898873355118924787555870863750644}{1425458249622462866757339337972585550079195122930830477036094165834866510875025926765757603544067866647349093094727} a^{7} + \frac{41253090448510291131312732878944654391814097656915136542539808184140998424520292455646561992457124324351529316453}{1425458249622462866757339337972585550079195122930830477036094165834866510875025926765757603544067866647349093094727} a^{6} + \frac{2397888646990139954500395967187848972697108778637073606539472542682074903097143907891749937850941093344152084077}{1425458249622462866757339337972585550079195122930830477036094165834866510875025926765757603544067866647349093094727} a^{5} + \frac{60384562541874084528251205558988672346311209509274515455730957916708590068852944288459703965744961390766407500910}{129587113602042078796121757997507777279926829357348225185099469621351500988638720615068873049460715149759008463157} a^{4} - \frac{1824274438904564560572371988110904830759996783427992863430006549206065593084762354175282841889175141683492432226}{129587113602042078796121757997507777279926829357348225185099469621351500988638720615068873049460715149759008463157} a^{3} + \frac{41497722736399350452516603814660190195138313146014859739303534612861273774662557719559878909645920675172852343154}{129587113602042078796121757997507777279926829357348225185099469621351500988638720615068873049460715149759008463157} a^{2} + \frac{18037529692059536678717397263350784228463565048664328406044219416390519477717787521719413987166969021356736747039}{129587113602042078796121757997507777279926829357348225185099469621351500988638720615068873049460715149759008463157} a + \frac{590443356054208817340339817789299719025590587073009404632239001720765272268224421669460333177412752775999879248}{1283040728733089889070512455420869081979473557993546784010885837835163376125135847673949238113472427225336717457}$
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: | \( 39114029004193790000000000 \) (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 | ||||||