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Properties

Label 25.25.4628037546...0625.1
Degree $25$
Signature $[25, 0]$
Discriminant $5^{40}\cdot 61^{20}$
Root discriminant $352.06$
Ramified primes $5, 61$
Class number $1$ (GRH)
Class group Trivial (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![-9926699, 171719495, 104189315, -3196341625, -1993468850, 17044842134, 17746925340, -17431228350, -18676720520, 9144688380, 8503890713, -2953591475, -2105459870, 609685640, 308241235, -80350625, -27473160, 6679000, 1481655, -341415, -46346, 10200, 760, -160, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 5*x^24 - 160*x^23 + 760*x^22 + 10200*x^21 - 46346*x^20 - 341415*x^19 + 1481655*x^18 + 6679000*x^17 - 27473160*x^16 - 80350625*x^15 + 308241235*x^14 + 609685640*x^13 - 2105459870*x^12 - 2953591475*x^11 + 8503890713*x^10 + 9144688380*x^9 - 18676720520*x^8 - 17431228350*x^7 + 17746925340*x^6 + 17044842134*x^5 - 1993468850*x^4 - 3196341625*x^3 + 104189315*x^2 + 171719495*x - 9926699)
 
gp: K = bnfinit(x^25 - 5*x^24 - 160*x^23 + 760*x^22 + 10200*x^21 - 46346*x^20 - 341415*x^19 + 1481655*x^18 + 6679000*x^17 - 27473160*x^16 - 80350625*x^15 + 308241235*x^14 + 609685640*x^13 - 2105459870*x^12 - 2953591475*x^11 + 8503890713*x^10 + 9144688380*x^9 - 18676720520*x^8 - 17431228350*x^7 + 17746925340*x^6 + 17044842134*x^5 - 1993468850*x^4 - 3196341625*x^3 + 104189315*x^2 + 171719495*x - 9926699, 1)
 

Normalized defining polynomial

\( x^{25} - 5 x^{24} - 160 x^{23} + 760 x^{22} + 10200 x^{21} - 46346 x^{20} - 341415 x^{19} + 1481655 x^{18} + 6679000 x^{17} - 27473160 x^{16} - 80350625 x^{15} + 308241235 x^{14} + 609685640 x^{13} - 2105459870 x^{12} - 2953591475 x^{11} + 8503890713 x^{10} + 9144688380 x^{9} - 18676720520 x^{8} - 17431228350 x^{7} + 17746925340 x^{6} + 17044842134 x^{5} - 1993468850 x^{4} - 3196341625 x^{3} + 104189315 x^{2} + 171719495 x - 9926699 \)

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:  \(4628037546532860960147412376577678523972281254827976226806640625=5^{40}\cdot 61^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $352.06$
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:  $5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1525=5^{2}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{1525}(1,·)$, $\chi_{1525}(386,·)$, $\chi_{1525}(131,·)$, $\chi_{1525}(1156,·)$, $\chi_{1525}(1221,·)$, $\chi_{1525}(1351,·)$, $\chi_{1525}(461,·)$, $\chi_{1525}(81,·)$, $\chi_{1525}(851,·)$, $\chi_{1525}(916,·)$, $\chi_{1525}(1301,·)$, $\chi_{1525}(1046,·)$, $\chi_{1525}(156,·)$, $\chi_{1525}(1376,·)$, $\chi_{1525}(546,·)$, $\chi_{1525}(611,·)$, $\chi_{1525}(996,·)$, $\chi_{1525}(741,·)$, $\chi_{1525}(1071,·)$, $\chi_{1525}(241,·)$, $\chi_{1525}(306,·)$, $\chi_{1525}(691,·)$, $\chi_{1525}(436,·)$, $\chi_{1525}(1461,·)$, $\chi_{1525}(766,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{6}$, $\frac{1}{6} a^{16} - \frac{1}{6} a^{14} + \frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{14} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{18} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{19} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{1218} a^{20} - \frac{89}{1218} a^{19} - \frac{16}{609} a^{18} - \frac{17}{406} a^{17} - \frac{4}{609} a^{16} - \frac{33}{406} a^{15} - \frac{2}{609} a^{14} - \frac{52}{609} a^{13} - \frac{125}{1218} a^{12} + \frac{13}{58} a^{11} + \frac{89}{406} a^{10} + \frac{178}{609} a^{9} - \frac{89}{203} a^{8} - \frac{41}{87} a^{7} + \frac{373}{1218} a^{6} - \frac{260}{609} a^{5} + \frac{443}{1218} a^{4} + \frac{349}{1218} a^{3} - \frac{439}{1218} a^{2} - \frac{146}{609} a + \frac{307}{1218}$, $\frac{1}{1218} a^{21} - \frac{6}{203} a^{19} - \frac{19}{406} a^{18} - \frac{27}{406} a^{17} + \frac{1}{1218} a^{16} - \frac{43}{609} a^{15} + \frac{149}{1218} a^{14} - \frac{41}{203} a^{13} - \frac{148}{609} a^{12} + \frac{34}{203} a^{11} + \frac{55}{406} a^{10} - \frac{8}{87} a^{9} - \frac{32}{203} a^{8} + \frac{40}{203} a^{7} + \frac{200}{609} a^{6} + \frac{149}{406} a^{5} + \frac{197}{609} a^{4} + \frac{125}{406} a^{3} + \frac{19}{1218} a^{2} - \frac{103}{1218} a - \frac{41}{609}$, $\frac{1}{246036} a^{22} + \frac{1}{41006} a^{21} + \frac{1}{8484} a^{20} + \frac{7121}{123018} a^{19} + \frac{1331}{82012} a^{18} - \frac{7787}{123018} a^{17} + \frac{865}{61509} a^{16} - \frac{3020}{61509} a^{15} + \frac{12886}{61509} a^{14} - \frac{2539}{61509} a^{13} + \frac{59323}{246036} a^{12} + \frac{3071}{123018} a^{11} - \frac{23831}{82012} a^{10} + \frac{706}{20503} a^{9} - \frac{24713}{123018} a^{8} - \frac{52651}{123018} a^{7} - \frac{4509}{20503} a^{6} + \frac{8537}{20503} a^{5} - \frac{39043}{123018} a^{4} - \frac{31777}{123018} a^{3} + \frac{23293}{123018} a^{2} - \frac{2452}{20503} a + \frac{26111}{82012}$, $\frac{1}{492072} a^{23} - \frac{1}{492072} a^{22} + \frac{9}{23432} a^{21} - \frac{1}{4872} a^{20} - \frac{11229}{164024} a^{19} - \frac{12619}{492072} a^{18} + \frac{6357}{82012} a^{17} + \frac{4502}{61509} a^{16} + \frac{685}{246036} a^{15} + \frac{21187}{123018} a^{14} - \frac{47951}{492072} a^{13} + \frac{27403}{492072} a^{12} + \frac{3077}{492072} a^{11} - \frac{11553}{164024} a^{10} + \frac{82793}{246036} a^{9} - \frac{39745}{246036} a^{8} + \frac{5183}{35148} a^{7} - \frac{121283}{246036} a^{6} - \frac{947}{2929} a^{5} + \frac{122243}{246036} a^{4} + \frac{10167}{82012} a^{3} - \frac{53279}{123018} a^{2} + \frac{28915}{164024} a - \frac{63395}{164024}$, $\frac{1}{106953034143941568370431318247135167363918716095903818303532971982709977380254449570877853744} a^{24} - \frac{1450104737643814078866281926936466404541488500349041604654684637802553231720076628541}{2228188211332116007717319130148649320081639918664662881323603582973124528755301032726621953} a^{23} + \frac{735218872503712444479021002685120010331879640067447202375154084194068147734816663644}{954937804856621146164565341492278280034988536570569806281544392702767655180843299739980837} a^{22} - \frac{239078827618717103294069766467080955521783021814840320478450599303055341127244136612633}{636625203237747430776376894328185520023325691047046537521029595135178436787228866493320558} a^{21} + \frac{5287904572664530628771444746725975375386689582835918095513277594484379057169842962719131}{13369129267992696046303914780891895920489839511987977287941621497838747172531806196359731718} a^{20} - \frac{101515011894246568296711845684642400039901872558133694272896341133560421004234784530903383}{17825505690656928061738553041189194560653119349317303050588828663784996230042408261812975624} a^{19} + \frac{395204952956920298381254783302973483461097466337430399938918166569632413563409218822392687}{106953034143941568370431318247135167363918716095903818303532971982709977380254449570877853744} a^{18} + \frac{98804821521610299801285934318543878487658468658045613455942612233650503752206166229029577}{2546500812950989723105507577312742080093302764188186150084118380540713747148915465973282232} a^{17} - \frac{3280159243414791786598839033991886556036520207142795686474866470983712402499787201250079}{118049706560641907693632801597279434176510724167664258613171050753543021391009326237172024} a^{16} + \frac{1095589069627116906082157202528266871468467989221123476812745015624655210242319619706250245}{53476517071970784185215659123567583681959358047951909151766485991354988690127224785438926872} a^{15} + \frac{411414244465948301450304677066482564960321956653147207747683075164986552244197834358572333}{3688035660135916150704528215418454047031679865375993734604585240783102668284636192099236336} a^{14} - \frac{194401261338388464025821244364987246970117777239116704466681893636591842327230458429379825}{1909875609713242292329130682984556560069977073141139612563088785405535310361686599479961674} a^{13} - \frac{936874764696873214067105509132419503179587999945574123487331091524873954501564627537431771}{4456376422664232015434638260297298640163279837329325762647207165946249057510602065453243906} a^{12} - \frac{141014222161798654875995480334515433242472691062377096964175634746157540684693289009637433}{17825505690656928061738553041189194560653119349317303050588828663784996230042408261812975624} a^{11} - \frac{18970869723453099495651983429009557276985803478496022648537250629309044952921963248577487473}{106953034143941568370431318247135167363918716095903818303532971982709977380254449570877853744} a^{10} - \frac{6491409928277834428906360956160156076359891580124753780973274546317893231779876327659041307}{26738258535985392092607829561783791840979679023975954575883242995677494345063612392719463436} a^{9} + \frac{1245361274915811696414975911640880556784548602306931433316341507093819054362694934905666980}{6684564633996348023151957390445947960244919755993988643970810748919373586265903098179865859} a^{8} - \frac{2796220103166185960635496712662256819554590319740383174145413739430052989451095744479611895}{13369129267992696046303914780891895920489839511987977287941621497838747172531806196359731718} a^{7} + \frac{9384140164320692433785411408092386320252821966403126924136005206260574656417548682849546137}{53476517071970784185215659123567583681959358047951909151766485991354988690127224785438926872} a^{6} - \frac{7721668706613363484100302838155868165265962200096560415067293732424299246516807467210711843}{17825505690656928061738553041189194560653119349317303050588828663784996230042408261812975624} a^{5} + \frac{8927102489150759987250955527393671216341496117716056403699372417633337637267228336136735025}{26738258535985392092607829561783791840979679023975954575883242995677494345063612392719463436} a^{4} + \frac{2275607789654890167022076664357443446835553762417793079516245503161911069529969109172681937}{53476517071970784185215659123567583681959358047951909151766485991354988690127224785438926872} a^{3} - \frac{42496591602580828090939287944054400784847225165294468170217361161097676728114632274808594591}{106953034143941568370431318247135167363918716095903818303532971982709977380254449570877853744} a^{2} - \frac{8048676421617981336650132487351348751706733463982405066929779510499628314098379766259749}{65857779645284216976866575275329536554137140453142745260796165013983976219368503430343506} a + \frac{12947794045614164137850339769111020673943787703417150976030966575462127709969155904317760261}{35651011381313856123477106082378389121306238698634606101177657327569992460084816523625951248}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 775226479104812800000000 \) (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

5.5.5408531640625.1, 5.5.5408531640625.4, 5.5.13845841.1, 5.5.5408531640625.3, 5.5.390625.1, 5.5.5408531640625.2

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
5Data not computed
61Data not computed