Properties

Label 26.2.713...893.1
Degree $26$
Signature $[2, 12]$
Discriminant $7.139\times 10^{41}$
Root discriminant $40.72$
Ramified primes $13, 191$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513)
 
gp: K = bnfinit(x^26 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11716513, -9412671, 41639217, 16397553, -74591007, -12442481, 74301119, 7281003, -48757198, -3804296, 22802242, 1620832, -7863783, -593805, 2044391, 177750, -407978, -39168, 63242, 5987, -7532, -602, 656, 36, -37, -1, 1]);
 

\( x^{26} - x^{25} - 37 x^{24} + 36 x^{23} + 656 x^{22} - 602 x^{21} - 7532 x^{20} + 5987 x^{19} + 63242 x^{18} - 39168 x^{17} - 407978 x^{16} + 177750 x^{15} + 2044391 x^{14} - 593805 x^{13} - 7863783 x^{12} + 1620832 x^{11} + 22802242 x^{10} - 3804296 x^{9} - 48757198 x^{8} + 7281003 x^{7} + 74301119 x^{6} - 12442481 x^{5} - 74591007 x^{4} + 16397553 x^{3} + 41639217 x^{2} - 9412671 x - 11716513 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(713943735055873357083710772983186869616893\)\(\medspace = 13^{13}\cdot 191^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $40.72$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 191$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{7} a^{21} + \frac{2}{7} a^{20} - \frac{2}{7} a^{19} - \frac{1}{7} a^{18} + \frac{3}{7} a^{17} - \frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7}$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{20} + \frac{3}{7} a^{19} - \frac{2}{7} a^{18} + \frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{2}{7} a^{15} - \frac{2}{7} a^{14} + \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{23} + \frac{1}{7} a^{20} + \frac{2}{7} a^{18} + \frac{3}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} + \frac{3}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{407030407} a^{24} + \frac{26500680}{407030407} a^{23} - \frac{26746959}{407030407} a^{22} - \frac{16747158}{407030407} a^{21} - \frac{809895}{3602039} a^{20} - \frac{168841075}{407030407} a^{19} + \frac{121204474}{407030407} a^{18} + \frac{2485405}{8306743} a^{17} - \frac{72052441}{407030407} a^{16} - \frac{152047013}{407030407} a^{15} + \frac{61556484}{407030407} a^{14} + \frac{4123975}{58147201} a^{13} + \frac{16250571}{407030407} a^{12} + \frac{179770040}{407030407} a^{11} - \frac{414085}{8306743} a^{10} - \frac{23961327}{407030407} a^{9} - \frac{94350002}{407030407} a^{8} - \frac{2417505}{407030407} a^{7} + \frac{41840655}{407030407} a^{6} + \frac{21041534}{407030407} a^{5} + \frac{5485625}{21422653} a^{4} - \frac{85395049}{407030407} a^{3} - \frac{73695659}{407030407} a^{2} + \frac{26168305}{407030407} a - \frac{17597504}{407030407}$, $\frac{1}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{25} + \frac{2817902550531034922440596344724118625648516882205332745727943828816057507851}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{24} + \frac{25633179683937210952200413519014468635540031485672101444897935286563806296882838900}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{23} + \frac{285188829684754932185736697198495335886856683284196967275062599671412895042222108250}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{22} - \frac{1528944139521844170390685906279603248340666650545833364067668312814649557297489612}{33650476793020304990329736110825152454052789558807519697514445883430365590369913451} a^{21} + \frac{240271694835107178741605553828903905427934370647819591304960433994277108556797299890}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{20} - \frac{27160438997221409640914807351379884154858358699600731965088024239841130015226210674}{91337008438197970688037855157953985232429000231048982036110638826453849459575479367} a^{19} + \frac{196597662592663935607412397912609323751996643731476029114867328367660936756082149132}{406864855770154596701259536612704116035365546483763647251765572954203511229018044453} a^{18} - \frac{1225836950162667252198973362115337557400889629564884924586909968655157276916995268635}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{17} - \frac{111507807809919078537153231334440439687401118693625384511681444631979637306947065899}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{16} + \frac{1120238464320768430023543122132617228149679595278101156802977899088080943477670414656}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{15} + \frac{2191687245068664655474871902446777596249228543514073727261284509057779019884548540368}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{14} + \frac{188058963851948476218220194693495684343871835099381481859061501333764092329568457946}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{13} + \frac{111482647893157219231457639087075652191861541709712454976666882403285815532575729979}{639359059067385794816264986105677896627003001617342874252774471785176946217028355569} a^{12} + \frac{129633899200774173357995098702706190040437652830157608874181175241218438300069260022}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{11} - \frac{1367654813433467918224250935390690015795292785499024959384840687208284550380828178867}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{10} + \frac{2017268097943887318798001482822784911153281156023722418937116900002631183304424658775}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{9} - \frac{1427983676775956461361093700774150873232955778315643038511212195513690778554054169010}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{8} - \frac{1910242237873322387430769400140709413490819144258311134576346566017197365996989197612}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{7} - \frac{440842511816791535095401996371200254386108415409145172765345223996820882756314172973}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{6} + \frac{1704640218688968812449021278266973425767774868395587637628587264117715908959257219996}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{5} - \frac{2232896232029545305418650273464719051343814777827982383786483716058969796388364810982}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{4} - \frac{2089368399433922149756832433185955799843556156338278487526370626383839155456408886108}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{3} + \frac{510946582554466015771480386600272033650080704854328993474863782288222764158669481486}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983} a^{2} + \frac{6740537480707759232441605735749622760543786006925400955923025642966773484861626006}{39606313393554872245255353121590666162734699215233629378490454004391492243532729991} a + \frac{1211557338016559476502033276566541352600596217290438386907870571220906801993495147779}{4475513413471700563713854902739745276389021011321400119769421302496238623519198488983}$

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

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$D_{26}$ (as 26T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{13}) \), 13.1.48551226272641.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 26 sibling: Deg 26

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ R ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/23.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $26$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
$191$$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 2$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 382$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
1.2483.2t1.a.a$1$ $ 13 \cdot 191 $ \(\Q(\sqrt{-2483}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.32279.26t3.a.d$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.e$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.32279.26t3.a.f$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.a$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.32279.26t3.a.c$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.b$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.