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Properties

Label 25.25.3423342906...4401.1
Degree $25$
Signature $[25, 0]$
Discriminant $11^{20}\cdot 61^{20}$
Root discriminant $182.55$
Ramified primes $11, 61$
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![944416, -30059216, -346963496, -677788540, 1212591512, 3265983995, -1434566212, -5197520035, 721456442, 3871967469, -207652438, -1568386838, 45584040, 371551113, -7391668, -53660765, 739142, 4796520, -40058, -262485, 1042, 8430, -10, -144, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 144*x^23 - 10*x^22 + 8430*x^21 + 1042*x^20 - 262485*x^19 - 40058*x^18 + 4796520*x^17 + 739142*x^16 - 53660765*x^15 - 7391668*x^14 + 371551113*x^13 + 45584040*x^12 - 1568386838*x^11 - 207652438*x^10 + 3871967469*x^9 + 721456442*x^8 - 5197520035*x^7 - 1434566212*x^6 + 3265983995*x^5 + 1212591512*x^4 - 677788540*x^3 - 346963496*x^2 - 30059216*x + 944416)
 
gp: K = bnfinit(x^25 - 144*x^23 - 10*x^22 + 8430*x^21 + 1042*x^20 - 262485*x^19 - 40058*x^18 + 4796520*x^17 + 739142*x^16 - 53660765*x^15 - 7391668*x^14 + 371551113*x^13 + 45584040*x^12 - 1568386838*x^11 - 207652438*x^10 + 3871967469*x^9 + 721456442*x^8 - 5197520035*x^7 - 1434566212*x^6 + 3265983995*x^5 + 1212591512*x^4 - 677788540*x^3 - 346963496*x^2 - 30059216*x + 944416, 1)
 

Normalized defining polynomial

\( x^{25} - 144 x^{23} - 10 x^{22} + 8430 x^{21} + 1042 x^{20} - 262485 x^{19} - 40058 x^{18} + 4796520 x^{17} + 739142 x^{16} - 53660765 x^{15} - 7391668 x^{14} + 371551113 x^{13} + 45584040 x^{12} - 1568386838 x^{11} - 207652438 x^{10} + 3871967469 x^{9} + 721456442 x^{8} - 5197520035 x^{7} - 1434566212 x^{6} + 3265983995 x^{5} + 1212591512 x^{4} - 677788540 x^{3} - 346963496 x^{2} - 30059216 x + 944416 \)

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:  \(342334290668031200335701860762422121140804207127700374401=11^{20}\cdot 61^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.55$
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:  $11, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(671=11\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{671}(192,·)$, $\chi_{671}(1,·)$, $\chi_{671}(386,·)$, $\chi_{671}(70,·)$, $\chi_{671}(9,·)$, $\chi_{671}(522,·)$, $\chi_{671}(203,·)$, $\chi_{671}(400,·)$, $\chi_{671}(81,·)$, $\chi_{671}(339,·)$, $\chi_{671}(20,·)$, $\chi_{671}(278,·)$, $\chi_{671}(34,·)$, $\chi_{671}(485,·)$, $\chi_{671}(489,·)$, $\chi_{671}(619,·)$, $\chi_{671}(302,·)$, $\chi_{671}(367,·)$, $\chi_{671}(306,·)$, $\chi_{671}(180,·)$, $\chi_{671}(245,·)$, $\chi_{671}(630,·)$, $\chi_{671}(375,·)$, $\chi_{671}(58,·)$, $\chi_{671}(119,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{58} a^{20} - \frac{1}{29} a^{19} + \frac{1}{29} a^{18} + \frac{1}{29} a^{17} + \frac{2}{29} a^{15} + \frac{1}{29} a^{14} - \frac{5}{29} a^{13} - \frac{1}{29} a^{12} - \frac{3}{58} a^{11} - \frac{13}{29} a^{10} - \frac{9}{58} a^{9} - \frac{3}{29} a^{8} - \frac{1}{58} a^{7} - \frac{11}{58} a^{6} - \frac{14}{29} a^{5} - \frac{13}{58} a^{4} - \frac{7}{29} a^{3} - \frac{5}{29} a^{2} + \frac{1}{58} a + \frac{12}{29}$, $\frac{1}{58} a^{21} - \frac{1}{29} a^{19} + \frac{3}{29} a^{18} + \frac{2}{29} a^{17} + \frac{2}{29} a^{16} + \frac{5}{29} a^{15} - \frac{3}{29} a^{14} + \frac{7}{58} a^{13} - \frac{7}{58} a^{12} - \frac{3}{58} a^{11} + \frac{13}{29} a^{10} - \frac{12}{29} a^{9} - \frac{13}{58} a^{8} + \frac{8}{29} a^{7} - \frac{21}{58} a^{6} - \frac{11}{58} a^{5} - \frac{11}{58} a^{4} + \frac{10}{29} a^{3} + \frac{5}{29} a^{2} - \frac{3}{58} a - \frac{5}{29}$, $\frac{1}{116} a^{22} + \frac{1}{58} a^{19} + \frac{2}{29} a^{18} + \frac{2}{29} a^{17} - \frac{19}{116} a^{16} + \frac{1}{58} a^{15} - \frac{9}{58} a^{14} + \frac{1}{58} a^{13} - \frac{7}{116} a^{12} + \frac{5}{29} a^{11} + \frac{11}{116} a^{10} - \frac{1}{58} a^{9} + \frac{1}{29} a^{8} - \frac{13}{29} a^{7} + \frac{25}{116} a^{6} + \frac{5}{29} a^{5} - \frac{35}{116} a^{4} + \frac{10}{29} a^{3} + \frac{35}{116} a^{2} - \frac{2}{29} a + \frac{12}{29}$, $\frac{1}{81896} a^{23} - \frac{153}{40948} a^{22} - \frac{125}{20474} a^{21} - \frac{197}{40948} a^{20} - \frac{8019}{40948} a^{19} + \frac{1147}{40948} a^{18} - \frac{13205}{81896} a^{17} + \frac{925}{20474} a^{16} - \frac{1286}{10237} a^{15} - \frac{7973}{40948} a^{14} + \frac{10223}{81896} a^{13} + \frac{849}{40948} a^{12} + \frac{11485}{81896} a^{11} - \frac{2601}{40948} a^{10} - \frac{2321}{40948} a^{9} + \frac{16705}{40948} a^{8} + \frac{27125}{81896} a^{7} + \frac{7041}{20474} a^{6} - \frac{16943}{81896} a^{5} + \frac{781}{40948} a^{4} - \frac{30001}{81896} a^{3} - \frac{1761}{40948} a^{2} - \frac{2899}{20474} a + \frac{3111}{10237}$, $\frac{1}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{24} - \frac{10604033023349227758225857406153134217896393811622930320461291325208159452113929494647169206}{8056553096150228666849330026105027580918224177996995065090481879499601503462011907277738338814489} a^{23} + \frac{6157176280830031392518278778189909727484008644061221501909987842809221504320189212481059642229}{32226212384600914667397320104420110323672896711987980260361927517998406013848047629110953355257956} a^{22} - \frac{298822598625306298439092882847696328581002175239821141082065928519307352333665802487856680597637}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{21} + \frac{454444034037514504188855356253706680088539061548882447439065923675210237290854849649861451781875}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{20} + \frac{2728885357660677075408134108096420234201146821503838535324536267503071871229771251324426663627121}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{19} + \frac{25142660700456733723787214623904628593052397008142136337033305323306636997813642685540813130850867}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{18} + \frac{10593531197024066724911395279058394324189959964311993086330795711987110347233726237380985783700867}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{17} - \frac{769748154668871317531269579010142993745567861124801587158339124677028921731423345744933572353615}{32226212384600914667397320104420110323672896711987980260361927517998406013848047629110953355257956} a^{16} + \frac{13583684120582391952182940781748873646259210401746643485118635869030496301282343389822004283953467}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{15} + \frac{30690377852556376453833925396258471936990937612529485516429927809827565523486908496415709193401179}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{14} + \frac{618008420582557932380225554694747661936717631035642943441457456011887856354605851005468684764605}{32226212384600914667397320104420110323672896711987980260361927517998406013848047629110953355257956} a^{13} + \frac{29128207497641810052564597007212356888410797867419304532809737865169917273852048753257912345450221}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{12} + \frac{3903346237869054923095735452889523301396382843393157895212997723240522310516587432103201318581139}{16113106192300457333698660052210055161836448355993990130180963758999203006924023814555476677628978} a^{11} - \frac{7176596656801072368709634751677443917024918196757936533592498442201612778552421332746905472568677}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{10} + \frac{27374012501202856259229626251168689904154553821043976555863126172081801182195264220296193451450573}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{9} + \frac{58774509210693018235170634816318535288437151825119429231869701113037050013276549984891294599817845}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{8} - \frac{16191314411368021799315396589011230473316230578663402495569039702580199800770897859528672925358379}{64452424769201829334794640208840220647345793423975960520723855035996812027696095258221906710515912} a^{7} + \frac{13053210978362794182586822204589688971428601355694770472891214345645728972269377808443755642970465}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{6} - \frac{727908382106825861266625023950544831959580409539164860684377683606575213119600596991817356306161}{32226212384600914667397320104420110323672896711987980260361927517998406013848047629110953355257956} a^{5} + \frac{42898333645452166349715967756809191674986714344643971312822775281511007381660057751013053963892279}{128904849538403658669589280417680441294691586847951921041447710071993624055392190516443813421031824} a^{4} - \frac{243080368301522318610611057971579723662173518135763137392232378435632759666053978805765028535611}{555624351458636459782712415593450177994360288137723797592447026172386310583587028088119885435482} a^{3} + \frac{2474097001713258931046220471242281748798595896154213333206549767420559151110801368743879435551578}{8056553096150228666849330026105027580918224177996995065090481879499601503462011907277738338814489} a^{2} + \frac{5177320828592313052345563896374597025169420017574425720164894066788431159840653095815396138631913}{16113106192300457333698660052210055161836448355993990130180963758999203006924023814555476677628978} a - \frac{172451073786028498626822291647866790440428364192555746882113918094763401322209683440413568552636}{8056553096150228666849330026105027580918224177996995065090481879499601503462011907277738338814489}$

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:  \( 281978220069726200000 \) (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

5.5.13845841.1, 5.5.202716958081.3, 5.5.202716958081.1, \(\Q(\zeta_{11})^+\), 5.5.202716958081.4, 5.5.202716958081.2

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
61Data not computed