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Properties

Label 25.25.1212078750...4001.1
Degree $25$
Signature $[25, 0]$
Discriminant $11^{20}\cdot 41^{20}$
Root discriminant $132.84$
Ramified primes $11, 41$
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![-359776, -3466480, 1956424, 45549604, 22993680, -185477445, -179794682, 260565807, 352085268, -119774447, -284820250, -11753214, 111486276, 23943273, -23379160, -7517715, 2743554, 1136264, -178142, -94439, 5884, 4358, -76, -104, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 104*x^23 - 76*x^22 + 4358*x^21 + 5884*x^20 - 94439*x^19 - 178142*x^18 + 1136264*x^17 + 2743554*x^16 - 7517715*x^15 - 23379160*x^14 + 23943273*x^13 + 111486276*x^12 - 11753214*x^11 - 284820250*x^10 - 119774447*x^9 + 352085268*x^8 + 260565807*x^7 - 179794682*x^6 - 185477445*x^5 + 22993680*x^4 + 45549604*x^3 + 1956424*x^2 - 3466480*x - 359776)
 
gp: K = bnfinit(x^25 - 104*x^23 - 76*x^22 + 4358*x^21 + 5884*x^20 - 94439*x^19 - 178142*x^18 + 1136264*x^17 + 2743554*x^16 - 7517715*x^15 - 23379160*x^14 + 23943273*x^13 + 111486276*x^12 - 11753214*x^11 - 284820250*x^10 - 119774447*x^9 + 352085268*x^8 + 260565807*x^7 - 179794682*x^6 - 185477445*x^5 + 22993680*x^4 + 45549604*x^3 + 1956424*x^2 - 3466480*x - 359776, 1)
 

Normalized defining polynomial

\( x^{25} - 104 x^{23} - 76 x^{22} + 4358 x^{21} + 5884 x^{20} - 94439 x^{19} - 178142 x^{18} + 1136264 x^{17} + 2743554 x^{16} - 7517715 x^{15} - 23379160 x^{14} + 23943273 x^{13} + 111486276 x^{12} - 11753214 x^{11} - 284820250 x^{10} - 119774447 x^{9} + 352085268 x^{8} + 260565807 x^{7} - 179794682 x^{6} - 185477445 x^{5} + 22993680 x^{4} + 45549604 x^{3} + 1956424 x^{2} - 3466480 x - 359776 \)

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:  \(121207875070942751916201424921290281964762434202234001=11^{20}\cdot 41^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.84$
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:  $11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(451=11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{451}(256,·)$, $\chi_{451}(1,·)$, $\chi_{451}(324,·)$, $\chi_{451}(133,·)$, $\chi_{451}(262,·)$, $\chi_{451}(201,·)$, $\chi_{451}(119,·)$, $\chi_{451}(141,·)$, $\chi_{451}(78,·)$, $\chi_{451}(16,·)$, $\chi_{451}(344,·)$, $\chi_{451}(346,·)$, $\chi_{451}(411,·)$, $\chi_{451}(92,·)$, $\chi_{451}(221,·)$, $\chi_{451}(223,·)$, $\chi_{451}(379,·)$, $\chi_{451}(100,·)$, $\chi_{451}(37,·)$, $\chi_{451}(42,·)$, $\chi_{451}(174,·)$, $\chi_{451}(180,·)$, $\chi_{451}(247,·)$, $\chi_{451}(59,·)$, $\chi_{451}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{15} + \frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{16} + \frac{1}{6} a^{14} + \frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{19} - \frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{18} a^{20} - \frac{1}{18} a^{19} + \frac{1}{18} a^{17} + \frac{1}{18} a^{16} - \frac{1}{6} a^{14} + \frac{1}{18} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{10} + \frac{1}{9} a^{9} - \frac{1}{2} a^{8} + \frac{1}{9} a^{7} - \frac{5}{18} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{6} a^{3} + \frac{7}{18} a^{2} - \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{18} a^{21} - \frac{1}{18} a^{19} + \frac{1}{18} a^{18} - \frac{1}{18} a^{17} + \frac{1}{18} a^{16} - \frac{1}{9} a^{14} + \frac{1}{18} a^{13} + \frac{1}{6} a^{11} + \frac{5}{18} a^{10} - \frac{2}{9} a^{9} - \frac{7}{18} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{9} a^{3} + \frac{7}{18} a - \frac{2}{9}$, $\frac{1}{36} a^{22} + \frac{1}{18} a^{18} + \frac{1}{18} a^{17} + \frac{1}{36} a^{16} - \frac{1}{18} a^{15} - \frac{1}{18} a^{14} - \frac{2}{9} a^{13} - \frac{1}{12} a^{12} + \frac{2}{9} a^{11} - \frac{13}{36} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{36} a^{6} - \frac{1}{3} a^{5} - \frac{17}{36} a^{4} - \frac{1}{36} a^{2} - \frac{1}{18} a + \frac{2}{9}$, $\frac{1}{72} a^{23} + \frac{1}{36} a^{19} - \frac{1}{18} a^{18} + \frac{1}{72} a^{17} - \frac{1}{36} a^{16} + \frac{1}{18} a^{15} + \frac{5}{36} a^{14} - \frac{5}{24} a^{13} - \frac{1}{18} a^{12} - \frac{7}{72} a^{11} + \frac{7}{18} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{23}{72} a^{7} - \frac{1}{2} a^{6} - \frac{5}{72} a^{5} + \frac{1}{12} a^{4} - \frac{25}{72} a^{3} + \frac{7}{18} a^{2} - \frac{7}{18} a + \frac{1}{3}$, $\frac{1}{4645054387503589812323170137426705538267157804061555148142457105139645239887856} a^{24} - \frac{859354812071012804359247599724671235467937442517168449536891069045371596563}{193543932812649575513465422392779397427798241835898131172602379380818551661994} a^{23} - \frac{38786899632211496885306513181478292830273067623148411690420065419059898863}{580631798437948726540396267178338192283394725507694393517807138142455654985982} a^{22} + \frac{8239299915306429296353236419055810352423720443091081562964572327400042638725}{1161263596875897453080792534356676384566789451015388787035614276284911309971964} a^{21} - \frac{23272401210683355761005581399286899972833354990041047291977651520933493708205}{2322527193751794906161585068713352769133578902030777574071228552569822619943928} a^{20} - \frac{73599837319932613928157706101795369873948461861639190786435123182892411077129}{1161263596875897453080792534356676384566789451015388787035614276284911309971964} a^{19} - \frac{4980104221904733911841651523217650969703330680297179502047377466399423407167}{516117154167065534702574459714078393140795311562395016460273011682182804431984} a^{18} - \frac{107025797679644420639275043009777697393760638817064038471994886460438400840671}{2322527193751794906161585068713352769133578902030777574071228552569822619943928} a^{17} - \frac{16258394601701430459743692428989076937248209777438152244196964449837795467456}{290315899218974363270198133589169096141697362753847196758903569071227827492991} a^{16} - \frac{28117657008738771425342982226088734984092085007506327729172124410075292264493}{774175731250598302053861689571117589711192967343592524690409517523274206647976} a^{15} - \frac{368668965828531779600530650638127952384098655191870922053674145723752605744009}{1548351462501196604107723379142235179422385934687185049380819035046548413295952} a^{14} - \frac{52915124658312778934423549275170949044395014277274158125536062013432147069190}{290315899218974363270198133589169096141697362753847196758903569071227827492991} a^{13} - \frac{537147684229684405195775537708321543347120936510610641150122861195214564444447}{4645054387503589812323170137426705538267157804061555148142457105139645239887856} a^{12} - \frac{199341850082789679804712248472515790670097973606558846704715339473565393035265}{1161263596875897453080792534356676384566789451015388787035614276284911309971964} a^{11} + \frac{45766713965901180998989120774502146206079737073294096748981176519682512611803}{774175731250598302053861689571117589711192967343592524690409517523274206647976} a^{10} + \frac{304621019263311166427207743572007556900884305489814828186627954972449982812911}{2322527193751794906161585068713352769133578902030777574071228552569822619943928} a^{9} + \frac{537327502320482445025527774188445767899808232295105995384016757333230292089667}{1548351462501196604107723379142235179422385934687185049380819035046548413295952} a^{8} - \frac{29795000357785207119727538051429034591038014402785520569174860525750672193963}{129029288541766383675643614928519598285198827890598754115068252920545701107996} a^{7} - \frac{1509951389481125960324349845868878606180021405382501738666728166383324722324457}{4645054387503589812323170137426705538267157804061555148142457105139645239887856} a^{6} + \frac{268954024093274166791684224851886071730205812238386709068328546874707398384001}{774175731250598302053861689571117589711192967343592524690409517523274206647976} a^{5} - \frac{10817922464143947867116574530547239719398476895969820947299969294306620830365}{4645054387503589812323170137426705538267157804061555148142457105139645239887856} a^{4} + \frac{61783581881944563891428235586305199323643306982972702286848807440496039084051}{290315899218974363270198133589169096141697362753847196758903569071227827492991} a^{3} - \frac{142714134497341868478994537392419292828811986574662281096247692334856891600213}{387087865625299151026930844785558794855596483671796262345204758761637103323988} a^{2} + \frac{67057845983509539274634223519453637992560847437559610129393263537856130265113}{193543932812649575513465422392779397427798241835898131172602379380818551661994} a + \frac{13408880696265461458397778067753367282251580203348786092936270080044821778307}{96771966406324787756732711196389698713899120917949065586301189690409275830997}$

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:  \( 3134200286519037400 \) (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

5.5.41371966801.1, \(\Q(\zeta_{11})^+\), 5.5.41371966801.2, 5.5.41371966801.4, 5.5.41371966801.3, 5.5.2825761.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
11Data not computed
41Data not computed