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Properties

Label 26.0.33831769337...7767.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,7^{13}\cdot 79^{24}$
Root discriminant $149.35$
Ramified primes $7, 79$
Class number $8511191$ (GRH)
Class group $[8511191]$ (GRH)
Galois group $C_{26}$ (as 26T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1492440721, 2418267426, 2932101364, 2235176701, 1699916559, 1033869253, 625580801, 300082081, 187657668, 106509143, 48261148, 5630364, -988693, 1760065, 2148333, 37356, -503077, -154129, 64360, 32061, -2308, -3704, 18, 220, 8, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 11*x^25 + 8*x^24 + 220*x^23 + 18*x^22 - 3704*x^21 - 2308*x^20 + 32061*x^19 + 64360*x^18 - 154129*x^17 - 503077*x^16 + 37356*x^15 + 2148333*x^14 + 1760065*x^13 - 988693*x^12 + 5630364*x^11 + 48261148*x^10 + 106509143*x^9 + 187657668*x^8 + 300082081*x^7 + 625580801*x^6 + 1033869253*x^5 + 1699916559*x^4 + 2235176701*x^3 + 2932101364*x^2 + 2418267426*x + 1492440721)
 
gp: K = bnfinit(x^26 - 11*x^25 + 8*x^24 + 220*x^23 + 18*x^22 - 3704*x^21 - 2308*x^20 + 32061*x^19 + 64360*x^18 - 154129*x^17 - 503077*x^16 + 37356*x^15 + 2148333*x^14 + 1760065*x^13 - 988693*x^12 + 5630364*x^11 + 48261148*x^10 + 106509143*x^9 + 187657668*x^8 + 300082081*x^7 + 625580801*x^6 + 1033869253*x^5 + 1699916559*x^4 + 2235176701*x^3 + 2932101364*x^2 + 2418267426*x + 1492440721, 1)
 

Normalized defining polynomial

\( x^{26} - 11 x^{25} + 8 x^{24} + 220 x^{23} + 18 x^{22} - 3704 x^{21} - 2308 x^{20} + 32061 x^{19} + 64360 x^{18} - 154129 x^{17} - 503077 x^{16} + 37356 x^{15} + 2148333 x^{14} + 1760065 x^{13} - 988693 x^{12} + 5630364 x^{11} + 48261148 x^{10} + 106509143 x^{9} + 187657668 x^{8} + 300082081 x^{7} + 625580801 x^{6} + 1033869253 x^{5} + 1699916559 x^{4} + 2235176701 x^{3} + 2932101364 x^{2} + 2418267426 x + 1492440721 \)

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

Invariants

Degree:  $26$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 13]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-338317693370822771423393222562696496647525569958371057767=-\,7^{13}\cdot 79^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $149.35$
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:  $7, 79$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(553=7\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{553}(512,·)$, $\chi_{553}(1,·)$, $\chi_{553}(258,·)$, $\chi_{553}(225,·)$, $\chi_{553}(8,·)$, $\chi_{553}(204,·)$, $\chi_{553}(141,·)$, $\chi_{553}(526,·)$, $\chi_{553}(337,·)$, $\chi_{553}(146,·)$, $\chi_{553}(405,·)$, $\chi_{553}(22,·)$, $\chi_{553}(538,·)$, $\chi_{553}(475,·)$, $\chi_{553}(223,·)$, $\chi_{553}(97,·)$, $\chi_{553}(482,·)$, $\chi_{553}(484,·)$, $\chi_{553}(64,·)$, $\chi_{553}(302,·)$, $\chi_{553}(176,·)$, $\chi_{553}(433,·)$, $\chi_{553}(496,·)$, $\chi_{553}(125,·)$, $\chi_{553}(62,·)$, $\chi_{553}(447,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{23} a^{15} - \frac{2}{23} a^{14} - \frac{2}{23} a^{13} - \frac{6}{23} a^{12} - \frac{5}{23} a^{11} + \frac{3}{23} a^{10} - \frac{2}{23} a^{9} - \frac{4}{23} a^{8} + \frac{8}{23} a^{7} + \frac{10}{23} a^{6} - \frac{4}{23} a^{5} - \frac{11}{23} a^{4} + \frac{2}{23} a^{3} + \frac{10}{23} a^{2} + \frac{2}{23} a$, $\frac{1}{23} a^{16} - \frac{6}{23} a^{14} - \frac{10}{23} a^{13} + \frac{6}{23} a^{12} - \frac{7}{23} a^{11} + \frac{4}{23} a^{10} - \frac{8}{23} a^{9} + \frac{3}{23} a^{7} - \frac{7}{23} a^{6} + \frac{4}{23} a^{5} + \frac{3}{23} a^{4} - \frac{9}{23} a^{3} - \frac{1}{23} a^{2} + \frac{4}{23} a$, $\frac{1}{23} a^{17} + \frac{1}{23} a^{14} - \frac{6}{23} a^{13} + \frac{3}{23} a^{12} - \frac{3}{23} a^{11} + \frac{10}{23} a^{10} + \frac{11}{23} a^{9} + \frac{2}{23} a^{8} - \frac{5}{23} a^{7} - \frac{5}{23} a^{6} + \frac{2}{23} a^{5} - \frac{6}{23} a^{4} + \frac{11}{23} a^{3} - \frac{5}{23} a^{2} - \frac{11}{23} a$, $\frac{1}{23} a^{18} - \frac{4}{23} a^{14} + \frac{5}{23} a^{13} + \frac{3}{23} a^{12} - \frac{8}{23} a^{11} + \frac{8}{23} a^{10} + \frac{4}{23} a^{9} - \frac{1}{23} a^{8} + \frac{10}{23} a^{7} - \frac{8}{23} a^{6} - \frac{2}{23} a^{5} - \frac{1}{23} a^{4} - \frac{7}{23} a^{3} + \frac{2}{23} a^{2} - \frac{2}{23} a$, $\frac{1}{23} a^{19} - \frac{3}{23} a^{14} - \frac{5}{23} a^{13} - \frac{9}{23} a^{12} + \frac{11}{23} a^{11} - \frac{7}{23} a^{10} - \frac{9}{23} a^{9} - \frac{6}{23} a^{8} + \frac{1}{23} a^{7} - \frac{8}{23} a^{6} + \frac{6}{23} a^{5} - \frac{5}{23} a^{4} + \frac{10}{23} a^{3} - \frac{8}{23} a^{2} + \frac{8}{23} a$, $\frac{1}{23} a^{20} - \frac{11}{23} a^{14} + \frac{8}{23} a^{13} - \frac{7}{23} a^{12} + \frac{1}{23} a^{11} + \frac{11}{23} a^{9} - \frac{11}{23} a^{8} - \frac{7}{23} a^{7} - \frac{10}{23} a^{6} + \frac{6}{23} a^{5} - \frac{2}{23} a^{3} - \frac{8}{23} a^{2} + \frac{6}{23} a$, $\frac{1}{529} a^{21} + \frac{8}{529} a^{20} + \frac{1}{529} a^{19} - \frac{9}{529} a^{18} + \frac{5}{529} a^{17} - \frac{7}{529} a^{16} + \frac{5}{529} a^{15} - \frac{262}{529} a^{14} - \frac{261}{529} a^{13} - \frac{237}{529} a^{12} - \frac{24}{529} a^{11} - \frac{182}{529} a^{10} - \frac{27}{529} a^{9} + \frac{84}{529} a^{8} - \frac{165}{529} a^{7} - \frac{125}{529} a^{6} + \frac{13}{529} a^{5} + \frac{166}{529} a^{4} + \frac{38}{529} a^{3} + \frac{242}{529} a^{2} + \frac{9}{23} a$, $\frac{1}{529} a^{22} + \frac{6}{529} a^{20} + \frac{6}{529} a^{19} + \frac{8}{529} a^{18} - \frac{1}{529} a^{17} - \frac{8}{529} a^{16} - \frac{3}{529} a^{15} + \frac{87}{529} a^{14} + \frac{172}{529} a^{13} - \frac{37}{529} a^{12} + \frac{263}{529} a^{11} + \frac{210}{529} a^{10} - \frac{22}{529} a^{9} - \frac{124}{529} a^{8} - \frac{116}{529} a^{7} + \frac{231}{529} a^{6} - \frac{99}{529} a^{5} + \frac{182}{529} a^{4} + \frac{122}{529} a^{3} + \frac{226}{529} a^{2} - \frac{2}{23} a$, $\frac{1}{529} a^{23} + \frac{4}{529} a^{20} + \frac{2}{529} a^{19} + \frac{7}{529} a^{18} + \frac{8}{529} a^{17} - \frac{7}{529} a^{16} + \frac{11}{529} a^{15} + \frac{249}{529} a^{14} - \frac{173}{529} a^{13} - \frac{224}{529} a^{12} + \frac{124}{529} a^{11} - \frac{218}{529} a^{10} - \frac{261}{529} a^{9} + \frac{254}{529} a^{8} + \frac{232}{529} a^{7} + \frac{191}{529} a^{6} + \frac{35}{529} a^{5} - \frac{9}{23} a^{4} - \frac{2}{529} a^{3} + \frac{43}{529} a^{2} - \frac{3}{23} a$, $\frac{1}{28823623} a^{24} + \frac{22021}{28823623} a^{23} + \frac{18345}{28823623} a^{22} + \frac{16920}{28823623} a^{21} + \frac{441906}{28823623} a^{20} + \frac{556331}{28823623} a^{19} + \frac{210380}{28823623} a^{18} + \frac{137171}{28823623} a^{17} - \frac{96227}{28823623} a^{16} - \frac{575433}{28823623} a^{15} - \frac{2958}{28823623} a^{14} - \frac{12547399}{28823623} a^{13} - \frac{3846494}{28823623} a^{12} + \frac{10541683}{28823623} a^{11} + \frac{403067}{28823623} a^{10} + \frac{8821288}{28823623} a^{9} - \frac{4396675}{28823623} a^{8} + \frac{489294}{28823623} a^{7} - \frac{106738}{279841} a^{6} - \frac{10842518}{28823623} a^{5} - \frac{12333564}{28823623} a^{4} - \frac{2600531}{28823623} a^{3} + \frac{172119}{1253201} a^{2} + \frac{803}{54487} a - \frac{216}{2369}$, $\frac{1}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{25} + \frac{122675322882705031293623893442790386864283627987101323347007863512765535923200}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{24} + \frac{29394271347768050929606096409022120662514645795821068910928823966818974921970098332}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{23} - \frac{14304651727006441593200614706011519684212108430984276028163701244565487123660447101}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{22} + \frac{11104644828247924417567644654760685747113006554572694491877011420713571303241992825}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{21} + \frac{714382384108847451653047363154791453350676514078449453617748766190232864743278344042}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{20} + \frac{285456867896907052674232960408062852555276152082729409037302674657348189058237385533}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{19} - \frac{628644015693817610032886940515923613289878162571131010836790265890725512048715374567}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{18} + \frac{311212179282171788338123837254996375969753960995090881611404760612824171921332774460}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{17} + \frac{77402924717148713352498113055948130609541335498520085609181557632929561853982126199}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{16} - \frac{6322053379620138198270221454467114696793124027715582871940347837530842720223444936}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{15} - \frac{4434612243104579312341857894505637661408819971931663199148897035718367244537160307389}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{14} + \frac{4096072259702528178351688425974391097650028572382448872259491607768156619663201338503}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{13} - \frac{2332359104320264709492857236903721663613403911884381553388507343263812496838598806302}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{12} + \frac{908280012252542560992789234129858651518234426428931924178196546731276136004917272349}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{11} - \frac{16480803881334984897829901111403008462420434082826490729660586148654327010719784530953}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{10} - \frac{1238754198326914887046181317124239473191579939832715933492684188058087411747262562358}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{9} - \frac{10874833932686257009765622991318391620992493815212390803353100279843890596459833009728}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{8} - \frac{1057305487545591912388460207448092696408016879713325204688778643337752940565890966596}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{7} + \frac{14240421242147633756127944174270162824066036286110154801903538312392896659519606357385}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{6} + \frac{7499990997356650552325859177598336791288105856014311211524132781868180236496125803999}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{5} - \frac{15306940547791327787445096744693771720553013121848410761666114633502661869315327698997}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{4} + \frac{10036377864480467382340729369943842849526929035070000426424554244772724248899321400740}{33060617907924322736897751978549703444882842206209506294454017776575845255104016226391} a^{3} - \frac{700788919739435658825135917758096051770127829719709621344118622345219150625528904283}{1437418169909753162473815303415204497603601835052587230193652946807645445874087662017} a^{2} - \frac{18277168228525296662411400687353241522365341328337693460725544832165039130221231247}{62496442169989267933644143626748021634939210219677705660593606382941106342351637479} a - \frac{702995640183707369018582830936393102452295954116755288699409727890862939176509705}{2717236616086489910158441027249913984127791748681639376547548103606135058363114673}$

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

Class group and class number

$C_{8511191}$, which has order $8511191$ (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:  $12$
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:  \( 57529828940.82975 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{26}$ (as 26T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 26
The 26 conjugacy class representatives for $C_{26}$
Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 13.13.59091511031674153381441.1

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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ $26$ R ${\href{/LocalNumberField/11.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/43.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ $26$

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
7Data not computed
$79$79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$
79.13.12.1$x^{13} - 79$$13$$1$$12$$C_{13}$$[\ ]_{13}$