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Properties

Label 25.25.3797235377...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 191^{20}$
Root discriminant $877.36$
Ramified primes $5, 191$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group $C_5^2$ (as 25T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11923703816995, -24154429512175, -30832950658125, 108147977960025, -45711997402870, -101516189962054, 120222850235380, -17703547775880, -45208123167080, 30848179881080, -3489773984829, -4492862723775, 2381625278510, -427122481340, -24561224315, 22400877853, -2670442740, -234426780, 76430385, -2353665, -796816, 61290, 3510, -420, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 420*x^23 + 3510*x^22 + 61290*x^21 - 796816*x^20 - 2353665*x^19 + 76430385*x^18 - 234426780*x^17 - 2670442740*x^16 + 22400877853*x^15 - 24561224315*x^14 - 427122481340*x^13 + 2381625278510*x^12 - 4492862723775*x^11 - 3489773984829*x^10 + 30848179881080*x^9 - 45208123167080*x^8 - 17703547775880*x^7 + 120222850235380*x^6 - 101516189962054*x^5 - 45711997402870*x^4 + 108147977960025*x^3 - 30832950658125*x^2 - 24154429512175*x + 11923703816995)
 
gp: K = bnfinit(x^25 - 5*x^24 - 420*x^23 + 3510*x^22 + 61290*x^21 - 796816*x^20 - 2353665*x^19 + 76430385*x^18 - 234426780*x^17 - 2670442740*x^16 + 22400877853*x^15 - 24561224315*x^14 - 427122481340*x^13 + 2381625278510*x^12 - 4492862723775*x^11 - 3489773984829*x^10 + 30848179881080*x^9 - 45208123167080*x^8 - 17703547775880*x^7 + 120222850235380*x^6 - 101516189962054*x^5 - 45711997402870*x^4 + 108147977960025*x^3 - 30832950658125*x^2 - 24154429512175*x + 11923703816995, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 420 x^{23} + 3510 x^{22} + 61290 x^{21} - 796816 x^{20} - 2353665 x^{19} + 76430385 x^{18} - 234426780 x^{17} - 2670442740 x^{16} + 22400877853 x^{15} - 24561224315 x^{14} - 427122481340 x^{13} + 2381625278510 x^{12} - 4492862723775 x^{11} - 3489773984829 x^{10} + 30848179881080 x^{9} - 45208123167080 x^{8} - 17703547775880 x^{7} + 120222850235380 x^{6} - 101516189962054 x^{5} - 45711997402870 x^{4} + 108147977960025 x^{3} - 30832950658125 x^{2} - 24154429512175 x + 11923703816995 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(37972353775599550581612761468462973090283165229266160167753696441650390625=5^{40}\cdot 191^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $877.36$
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:  $5, 191$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4775=5^{2}\cdot 191\)
Dirichlet character group:    $\lbrace$$\chi_{4775}(1,·)$, $\chi_{4775}(4241,·)$, $\chi_{4775}(3286,·)$, $\chi_{4775}(4311,·)$, $\chi_{4775}(4251,·)$, $\chi_{4775}(3356,·)$, $\chi_{4775}(2341,·)$, $\chi_{4775}(3296,·)$, $\chi_{4775}(2401,·)$, $\chi_{4775}(4386,·)$, $\chi_{4775}(2331,·)$, $\chi_{4775}(421,·)$, $\chi_{4775}(1446,·)$, $\chi_{4775}(3431,·)$, $\chi_{4775}(1386,·)$, $\chi_{4775}(491,·)$, $\chi_{4775}(2476,·)$, $\chi_{4775}(3821,·)$, $\chi_{4775}(431,·)$, $\chi_{4775}(1376,·)$, $\chi_{4775}(2866,·)$, $\chi_{4775}(1521,·)$, $\chi_{4775}(566,·)$, $\chi_{4775}(1911,·)$, $\chi_{4775}(956,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{3850} a^{20} + \frac{41}{770} a^{19} + \frac{5}{77} a^{18} - \frac{173}{770} a^{17} + \frac{5}{154} a^{16} + \frac{57}{350} a^{15} + \frac{92}{385} a^{14} - \frac{15}{77} a^{13} - \frac{16}{77} a^{12} - \frac{2}{385} a^{11} - \frac{843}{1925} a^{10} - \frac{79}{385} a^{9} - \frac{317}{770} a^{8} + \frac{67}{770} a^{7} - \frac{2}{35} a^{6} + \frac{349}{1925} a^{5} - \frac{166}{385} a^{4} - \frac{65}{154} a^{3} - \frac{20}{77} a^{2} - \frac{23}{77} a + \frac{181}{385}$, $\frac{1}{3850} a^{21} + \frac{23}{154} a^{19} - \frac{2}{55} a^{18} + \frac{1}{11} a^{17} + \frac{27}{3850} a^{16} - \frac{113}{770} a^{15} - \frac{2}{11} a^{14} + \frac{5}{22} a^{13} + \frac{71}{770} a^{12} + \frac{489}{3850} a^{11} + \frac{53}{770} a^{10} - \frac{267}{770} a^{9} + \frac{186}{385} a^{8} - \frac{152}{385} a^{7} + \frac{1523}{3850} a^{6} + \frac{31}{77} a^{5} - \frac{5}{154} a^{4} - \frac{18}{77} a^{3} + \frac{69}{154} a^{2} + \frac{157}{770} a + \frac{19}{154}$, $\frac{1}{38500} a^{22} + \frac{1}{9625} a^{21} - \frac{1}{38500} a^{20} + \frac{49}{550} a^{19} + \frac{3}{700} a^{18} + \frac{353}{2750} a^{17} + \frac{17}{250} a^{16} + \frac{196}{1375} a^{15} + \frac{454}{1925} a^{14} + \frac{47}{275} a^{13} + \frac{8409}{38500} a^{12} + \frac{147}{1375} a^{11} - \frac{10889}{38500} a^{10} - \frac{522}{1925} a^{9} - \frac{641}{1925} a^{8} + \frac{3077}{9625} a^{7} - \frac{2022}{9625} a^{6} + \frac{1251}{19250} a^{5} + \frac{1151}{3850} a^{4} + \frac{87}{770} a^{3} - \frac{1499}{3850} a^{2} + \frac{96}{275} a + \frac{153}{7700}$, $\frac{1}{8239000} a^{23} + \frac{69}{8239000} a^{22} + \frac{369}{8239000} a^{21} + \frac{3}{1647800} a^{20} - \frac{141677}{1647800} a^{19} - \frac{1953733}{8239000} a^{18} + \frac{532799}{4119500} a^{17} + \frac{669189}{4119500} a^{16} + \frac{7569}{164780} a^{15} - \frac{119547}{823900} a^{14} + \frac{888859}{8239000} a^{13} - \frac{1022999}{8239000} a^{12} - \frac{527309}{8239000} a^{11} + \frac{55911}{329560} a^{10} - \frac{37601}{411950} a^{9} + \frac{512076}{1029875} a^{8} + \frac{24169}{294250} a^{7} + \frac{18789}{1029875} a^{6} - \frac{12304}{205975} a^{5} - \frac{7767}{16478} a^{4} + \frac{106951}{823900} a^{3} + \frac{35519}{74900} a^{2} + \frac{60213}{149800} a + \frac{52599}{329560}$, $\frac{1}{3915239378840541286577256528211830632290504371779020432100634762535088751177737780337999294503009621378553234595440987385150078000} a^{24} + \frac{86922651390039012218841094752386280587527054787543022198348483230593053340394259967241047793050032332725530998509894582199}{1957619689420270643288628264105915316145252185889510216050317381267544375588868890168999647251504810689276617297720493692575039000} a^{23} + \frac{2879303724927899943577742051357200875332341022095564027069758248170190417522907437684191747404075719695082806340081800692759}{279659955631467234755518323443702188020750312269930030864331054466792053655552698595571378178786401527039516756817213384653577000} a^{22} - \frac{6185733012950712829887640865615331490997360673104051360265478167711818398414389254819287957618848648251643264290037887261514}{48940492235506766082215706602647882903631304647237755401257934531688609389721722254224991181287620267231915432443012342314375975} a^{21} - \frac{106697147018238888412887096598633671404343964150252500311482787876065831296476325495979571197297915793110036661205553817090533}{1957619689420270643288628264105915316145252185889510216050317381267544375588868890168999647251504810689276617297720493692575039000} a^{20} + \frac{466325866810364138229599293690342943667893666012685374305809284522218469976741302816147468929939171733474265905523964361538483691}{1957619689420270643288628264105915316145252185889510216050317381267544375588868890168999647251504810689276617297720493692575039000} a^{19} + \frac{477228453467872183422909728185648121915654464824282904464103160577026043531112252271068349074103053075273440739601735318799924581}{3915239378840541286577256528211830632290504371779020432100634762535088751177737780337999294503009621378553234595440987385150078000} a^{18} - \frac{123645708642790936515103332858012251158207506646604590342238944028409366705435063144514032032290825022501312773057789916610217057}{978809844710135321644314132052957658072626092944755108025158690633772187794434445084499823625752405344638308648860246846287519500} a^{17} + \frac{109596847589097053224308729664921936163044351076223167474004155551799610411634041594020813095586641995757180226673909374841436}{9788098447101353216443141320529576580726260929447551080251586906337721877944344450844998236257524053446383086488602468462875195} a^{16} + \frac{150977053481341264233097743495179481145093229532749532196483026180172590434554345425395497444698535408485834129020545795068572797}{978809844710135321644314132052957658072626092944755108025158690633772187794434445084499823625752405344638308648860246846287519500} a^{15} + \frac{45401104314163844607582433672736168960963531437976849734825475549161631924377492649757935099325547343057319232618960266617284819}{355930852621867389688841502564711875662773124707183675645512251139553522834339798212545390409364511034413930417767362489559098000} a^{14} + \frac{3131804447777123373997258762974452952093967925582450102158527649182358414835145930864167630095634924666600940365494692460623867}{22245678288866711855552593910294492228923320294198979727844515696222095177146237388284086900585281939650870651110460155597443625} a^{13} - \frac{37207327430974238701880400688612940688595649980129563211830513916395762422554304076855290931603343682594903510730311152727168836}{244702461177533830411078533013239414518156523236188777006289672658443046948608611271124955906438101336159577162215061711571879875} a^{12} - \frac{4029470275837752105496790067789077271550306573973608747725883549162612239628638557472331307166124682839893328689371374621334753}{35593085262186738968884150256471187566277312470718367564551225113955352283433979821254539040936451103441393041776736248955909800} a^{11} + \frac{1397203167338267320306800684006705071393052823015266166525809062621130584514935142107705414443525796002839614160278165875316276931}{3915239378840541286577256528211830632290504371779020432100634762535088751177737780337999294503009621378553234595440987385150078000} a^{10} - \frac{22004260187797186967635556501361313858739237164546260962846429077477262031766769162865403462717991667324780735573909862521773627}{69914988907866808688879580860925547005187578067482507716082763616698013413888174648892844544696600381759879189204303346163394250} a^{9} - \frac{9730811006600446885175317122668820598622191107156362467712220999321984836104232519797704587663965079877486539835902980449130606}{22245678288866711855552593910294492228923320294198979727844515696222095177146237388284086900585281939650870651110460155597443625} a^{8} - \frac{50017984400266403228881213009489703951758410424223079797198820101573587763467635243647649806378050237936618881726615942408059939}{489404922355067660822157066026478829036313046472377554012579345316886093897217222542249911812876202672319154324430123423143759750} a^{7} + \frac{1580591327748737257709713576895686363109743509298334387687231987838153689944512908842852096966282057337588914540564627661408857}{97880984471013532164431413205295765807262609294475510802515869063377218779443444508449982362575240534463830864886024684628751950} a^{6} - \frac{288762200745467249104767082388274743121875833916484908798899630867964405450215099828078994845476161780365618483413532181469243727}{978809844710135321644314132052957658072626092944755108025158690633772187794434445084499823625752405344638308648860246846287519500} a^{5} - \frac{93051984845057272300043162944184454552025795247280345454827755753205838402591915105915199286407301219420636100931981739668920333}{391523937884054128657725652821183063229050437177902043210063476253508875117773778033799929450300962137855323459544098738515007800} a^{4} + \frac{41741307108680230272164329470265765560149129349041857794973101884040570139200388555335020774238303376006581210472327864781953399}{195761968942027064328862826410591531614525218588951021605031738126754437558886889016899964725150481068927661729772049369257503900} a^{3} + \frac{322050207532432069274316474434581456836749136860846530753736049243661925591952752497546632957169320643948734216002177130428830177}{783047875768108257315451305642366126458100874355804086420126952507017750235547556067599858900601924275710646919088197477030015600} a^{2} - \frac{6644728380509971039475969943749008835208578880947662877102962188609702622158425008919420714325370624472509587877489472302014251}{15660957515362165146309026112847322529162017487116081728402539050140355004710951121351997178012038485514212938381763949540600312} a - \frac{152542426966809059211932290909205585773177626256784933288705008444964265245293751009211359160841978621285781232216401224276367297}{783047875768108257315451305642366126458100874355804086420126952507017750235547556067599858900601924275710646919088197477030015600}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}$, which has order $11$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.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:  \( 14690097512775467000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2$ (as 25T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 25
The 25 conjugacy class representatives for $C_5^2$
Character table for $C_5^2$ is not computed

Intermediate fields

5.5.390625.1, 5.5.519868500390625.1, 5.5.1330863361.1, 5.5.519868500390625.3, 5.5.519868500390625.2, 5.5.519868500390625.4

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
191Data not computed