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Properties

Label 25.25.2276114846...0625.3
Degree $25$
Signature $[25, 0]$
Discriminant $5^{68}\cdot 31^{20}$
Root discriminant $1242.48$
Ramified primes $5, 31$
Class number Not computed
Class group Not computed
Galois group $C_{25}$ (as 25T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3224340405664787501, -16035495984262633550, -5843218933392249350, 62488560763081705900, 33554962682608500875, -33914550662807657240, -29490586343949441125, -3547856441731801175, 2379329196310910025, 560815730021192375, -73928366383086995, -25735760122584450, 1108880965450200, 599195478192300, -8359365771625, -8121689002885, 29460319900, 66941602275, -38319875, -335148750, 0, 985025, 0, -1550, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 38319875*x^18 + 66941602275*x^17 + 29460319900*x^16 - 8121689002885*x^15 - 8359365771625*x^14 + 599195478192300*x^13 + 1108880965450200*x^12 - 25735760122584450*x^11 - 73928366383086995*x^10 + 560815730021192375*x^9 + 2379329196310910025*x^8 - 3547856441731801175*x^7 - 29490586343949441125*x^6 - 33914550662807657240*x^5 + 33554962682608500875*x^4 + 62488560763081705900*x^3 - 5843218933392249350*x^2 - 16035495984262633550*x + 3224340405664787501)
 
gp: K = bnfinit(x^25 - 1550*x^23 + 985025*x^21 - 335148750*x^19 - 38319875*x^18 + 66941602275*x^17 + 29460319900*x^16 - 8121689002885*x^15 - 8359365771625*x^14 + 599195478192300*x^13 + 1108880965450200*x^12 - 25735760122584450*x^11 - 73928366383086995*x^10 + 560815730021192375*x^9 + 2379329196310910025*x^8 - 3547856441731801175*x^7 - 29490586343949441125*x^6 - 33914550662807657240*x^5 + 33554962682608500875*x^4 + 62488560763081705900*x^3 - 5843218933392249350*x^2 - 16035495984262633550*x + 3224340405664787501, 1)
 

Normalized defining polynomial

\( x^{25} - 1550 x^{23} + 985025 x^{21} - 335148750 x^{19} - 38319875 x^{18} + 66941602275 x^{17} + 29460319900 x^{16} - 8121689002885 x^{15} - 8359365771625 x^{14} + 599195478192300 x^{13} + 1108880965450200 x^{12} - 25735760122584450 x^{11} - 73928366383086995 x^{10} + 560815730021192375 x^{9} + 2379329196310910025 x^{8} - 3547856441731801175 x^{7} - 29490586343949441125 x^{6} - 33914550662807657240 x^{5} + 33554962682608500875 x^{4} + 62488560763081705900 x^{3} - 5843218933392249350 x^{2} - 16035495984262633550 x + 3224340405664787501 \)

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:  \(227611484625260901780055732479538161718013444811958834179677069187164306640625=5^{68}\cdot 31^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1242.48$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3875=5^{3}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3875}(1,·)$, $\chi_{3875}(66,·)$, $\chi_{3875}(3846,·)$, $\chi_{3875}(776,·)$, $\chi_{3875}(841,·)$, $\chi_{3875}(1551,·)$, $\chi_{3875}(1616,·)$, $\chi_{3875}(2326,·)$, $\chi_{3875}(2391,·)$, $\chi_{3875}(411,·)$, $\chi_{3875}(3101,·)$, $\chi_{3875}(3166,·)$, $\chi_{3875}(481,·)$, $\chi_{3875}(1186,·)$, $\chi_{3875}(1256,·)$, $\chi_{3875}(1961,·)$, $\chi_{3875}(746,·)$, $\chi_{3875}(2031,·)$, $\chi_{3875}(2736,·)$, $\chi_{3875}(1521,·)$, $\chi_{3875}(2806,·)$, $\chi_{3875}(3511,·)$, $\chi_{3875}(2296,·)$, $\chi_{3875}(3581,·)$, $\chi_{3875}(3071,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{31} a^{5}$, $\frac{1}{31} a^{6}$, $\frac{1}{31} a^{7}$, $\frac{1}{31} a^{8}$, $\frac{1}{31} a^{9}$, $\frac{1}{961} a^{10}$, $\frac{1}{961} a^{11}$, $\frac{1}{961} a^{12}$, $\frac{1}{961} a^{13}$, $\frac{1}{961} a^{14}$, $\frac{1}{29791} a^{15}$, $\frac{1}{29791} a^{16}$, $\frac{1}{29791} a^{17}$, $\frac{1}{29791} a^{18}$, $\frac{1}{29791} a^{19}$, $\frac{1}{923521} a^{20}$, $\frac{1}{923521} a^{21}$, $\frac{1}{923521} a^{22}$, $\frac{1}{923521} a^{23}$, $\frac{1}{14905520996932641866350042689123068525604136910992557363016058416295180816085136509489979389849927391687544688174826613911243020006152529855078866051363555906736901713197834103822529} a^{24} + \frac{2626393098154747583959023807613734519349198263193795261296149035611572572869604862649430228951496178572868461968365470074024059063506688547942601178109295255500902547757478014}{14905520996932641866350042689123068525604136910992557363016058416295180816085136509489979389849927391687544688174826613911243020006152529855078866051363555906736901713197834103822529} a^{23} - \frac{4269982206150728779203624818115128279329709132997242006373440757474495442856604396996699507645764885765562585434708340318903811253121060865831690599499471639437131344697084888}{14905520996932641866350042689123068525604136910992557363016058416295180816085136509489979389849927391687544688174826613911243020006152529855078866051363555906736901713197834103822529} a^{22} - \frac{989674996451892750650169835870688672939742422130815873948807729238628190098805080858500913364328634448289533188080588564951165308403194161189724549795891491929846692134226254}{14905520996932641866350042689123068525604136910992557363016058416295180816085136509489979389849927391687544688174826613911243020006152529855078866051363555906736901713197834103822529} a^{21} - \frac{2950158332985906321590169526166788848848484572808127349113183765704239518033028995439999050877050680417509424964400510456557282119658200402891556999206165518396510153455646413}{14905520996932641866350042689123068525604136910992557363016058416295180816085136509489979389849927391687544688174826613911243020006152529855078866051363555906736901713197834103822529} a^{20} + \frac{5356776184761984491082929246437616906967310942789375599300366234923611194437886854029715331342037748222063324326984834939253493396565199208679245946293756888316078994996288862}{480823257965569092462904602874937694374326997128792173000518013428876800518875371273870302898384754570565957683058923029394936129230726769518673098431082448604416184296704325929759} a^{19} + \frac{4737012994836423769958529570247266227829155586128741566558951069964268578600010836631235152440960803769178116779288581369826716649215111713682857254180709421757530824030022821}{480823257965569092462904602874937694374326997128792173000518013428876800518875371273870302898384754570565957683058923029394936129230726769518673098431082448604416184296704325929759} a^{18} + \frac{1475521470184657463206432299396600382021080879971070001441072191819097046192578337271079558679206541754903448837799454122137823003654189309052955553410557247568542688109384153}{480823257965569092462904602874937694374326997128792173000518013428876800518875371273870302898384754570565957683058923029394936129230726769518673098431082448604416184296704325929759} a^{17} - \frac{5189284766878744659465962932274967720258854436989551778120591130765604554385340108462403519014145374643612247605107458192774042718218779489157168100456256007305008974610821151}{480823257965569092462904602874937694374326997128792173000518013428876800518875371273870302898384754570565957683058923029394936129230726769518673098431082448604416184296704325929759} a^{16} + \frac{4901939429998769943123626859759764452628405228172595294275011372536049525179513876084502335645322893997934521050173097242818903011359277635278722290833333959088270929915118742}{480823257965569092462904602874937694374326997128792173000518013428876800518875371273870302898384754570565957683058923029394936129230726769518673098431082448604416184296704325929759} a^{15} - \frac{6464188536557953399532811660244345893673939758389818414834940591286254042117130535397462962794900821483441505114388966174667597147904898608491234601258694677570435606610668485}{15510427676308680402029180737901215947558935391251360419371548820286348403834689395931300093496282405502127667195449129980481810620346024823183003175196208019497296267635623417089} a^{14} - \frac{4956312990404488578024044476785438617485966515332267408276734850773094225075014119553785435580221187132293591075409175315206379191057227127329267928307385243324416692680161956}{15510427676308680402029180737901215947558935391251360419371548820286348403834689395931300093496282405502127667195449129980481810620346024823183003175196208019497296267635623417089} a^{13} + \frac{3229639261194317915671003912715366692479141388476590470396090956493228838653212208763834811046377901401915047254660014269839958180656380201255852019113062383833143414134781976}{15510427676308680402029180737901215947558935391251360419371548820286348403834689395931300093496282405502127667195449129980481810620346024823183003175196208019497296267635623417089} a^{12} + \frac{6744261716907198008277257204281372814080097784008366831249134763049245887035805662342497336440258260038228849547388709369576473509677039517060181956236369083214266181093332756}{15510427676308680402029180737901215947558935391251360419371548820286348403834689395931300093496282405502127667195449129980481810620346024823183003175196208019497296267635623417089} a^{11} + \frac{229362382077005063756823938709085686303319797886270373540792855291388251707309311647800891820532835213766835821701826308426657933339758704152694900895463965363031092789520069}{15510427676308680402029180737901215947558935391251360419371548820286348403834689395931300093496282405502127667195449129980481810620346024823183003175196208019497296267635623417089} a^{10} + \frac{1091400340388503721736354940853949886797687777966760078378555629149837955101002062137041322190137448963296591743408355706185843694615778172091562599441287533518810571792044478}{500336376655118722646102604448426320888997915846818078044243510331817690446280303094558067532138142112971860232111262257434897116785355639457516231457942194177332137665665271519} a^{9} - \frac{7592394628588615630010864824509553285243801210634440768400054990494223628299273267662814059797334468914674843609685008972024316282233480768621929887314804088407989096742490121}{500336376655118722646102604448426320888997915846818078044243510331817690446280303094558067532138142112971860232111262257434897116785355639457516231457942194177332137665665271519} a^{8} - \frac{3630561693219611706918518756739474700333703607170749585692382709844516684219043958163391666697989052153664589949991519702085071502723659598526962030743843326765220382233397052}{500336376655118722646102604448426320888997915846818078044243510331817690446280303094558067532138142112971860232111262257434897116785355639457516231457942194177332137665665271519} a^{7} + \frac{6482742046482986148659136599693464795394821716470642266317038791247502406468252729284511997782812138411528004957976353326612661670992639308249596533375060992043840550415573617}{500336376655118722646102604448426320888997915846818078044243510331817690446280303094558067532138142112971860232111262257434897116785355639457516231457942194177332137665665271519} a^{6} - \frac{4600373787363247427852835232870972212140876182302392307735890715894756568545323538103012456300197841839305331050404190410305300793767076267401388943893540161897552183478774520}{500336376655118722646102604448426320888997915846818078044243510331817690446280303094558067532138142112971860232111262257434897116785355639457516231457942194177332137665665271519} a^{5} - \frac{5656210327876931232034030387066704859866731414893279548280587964214203074893885318184749454684178464928352762075620605144970930925195231420484166993669851872271669768661222094}{16139883117907055569229116272529881318999932769252196065943339042961860982138074293372840888133488455257156781681008459917254745702753407724436007466385232070236520569860170049} a^{4} + \frac{1015564314492132699409971347473709881647182725010279803118762164143541891245577572737074784909146544956676222307359667937637189516715891566980075134298990423549717935854565248}{16139883117907055569229116272529881318999932769252196065943339042961860982138074293372840888133488455257156781681008459917254745702753407724436007466385232070236520569860170049} a^{3} - \frac{1333343838636480863969331453268048303872589578301668180646429898614064635999859957742328824500551926131544082740032134472314973674912939407419417726042214295471778674705222949}{16139883117907055569229116272529881318999932769252196065943339042961860982138074293372840888133488455257156781681008459917254745702753407724436007466385232070236520569860170049} a^{2} - \frac{7888109099963689640673476688632963218284469922131356579997750132364164234934220257026352082924998617249927553260496746558236334364623089376885196193964479413022075454320984150}{16139883117907055569229116272529881318999932769252196065943339042961860982138074293372840888133488455257156781681008459917254745702753407724436007466385232070236520569860170049} a - \frac{7537032451031539821144348894873340640952459210428584246586659687470538662652843077333286967953467827686946572772013807791565894050643204527525928612860239138345772117878064087}{16139883117907055569229116272529881318999932769252196065943339042961860982138074293372840888133488455257156781681008459917254745702753407724436007466385232070236520569860170049}$

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

Class group and class number

Not computed

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:  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

$C_{25}$ (as 25T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ $25$ $25$ $25$ $25$ $25$ $25$ R $25$ $25$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ $25$ $25$ $25$

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

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
5Data not computed
31Data not computed