↑

Properties

Label 26.0.33757598619...7664.1
Degree $26$
Signature $[0, 13]$
Discriminant $-\,2^{26}\cdot 157^{24}$
Root discriminant $212.82$
Ramified primes $2, 157$
Class number $55496389$ (GRH)
Class group $[13, 4268953]$ (GRH)
Galois group $C_{26}$ (as 26T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3769837201, 0, 31642042988, 0, 86626329100, 0, 101147766894, 0, 62077115096, 0, 22740238646, 0, 5340563011, 0, 835400600, 0, 88430622, 0, 6317962, 0, 297801, 0, 8786, 0, 145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^26 + 145*x^24 + 8786*x^22 + 297801*x^20 + 6317962*x^18 + 88430622*x^16 + 835400600*x^14 + 5340563011*x^12 + 22740238646*x^10 + 62077115096*x^8 + 101147766894*x^6 + 86626329100*x^4 + 31642042988*x^2 + 3769837201)
 
gp: K = bnfinit(x^26 + 145*x^24 + 8786*x^22 + 297801*x^20 + 6317962*x^18 + 88430622*x^16 + 835400600*x^14 + 5340563011*x^12 + 22740238646*x^10 + 62077115096*x^8 + 101147766894*x^6 + 86626329100*x^4 + 31642042988*x^2 + 3769837201, 1)
 

Normalized defining polynomial

\( x^{26} + 145 x^{24} + 8786 x^{22} + 297801 x^{20} + 6317962 x^{18} + 88430622 x^{16} + 835400600 x^{14} + 5340563011 x^{12} + 22740238646 x^{10} + 62077115096 x^{8} + 101147766894 x^{6} + 86626329100 x^{4} + 31642042988 x^{2} + 3769837201 \)

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:  \(-3375759861985195519612137229985370185177350891905754190577664=-\,2^{26}\cdot 157^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $212.82$
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:  $2, 157$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(628=2^{2}\cdot 157\)
Dirichlet character group:    $\lbrace$$\chi_{628}(1,·)$, $\chi_{628}(67,·)$, $\chi_{628}(487,·)$, $\chi_{628}(389,·)$, $\chi_{628}(601,·)$, $\chi_{628}(265,·)$, $\chi_{628}(75,·)$, $\chi_{628}(579,·)$, $\chi_{628}(203,·)$, $\chi_{628}(407,·)$, $\chi_{628}(485,·)$, $\chi_{628}(153,·)$, $\chi_{628}(413,·)$, $\chi_{628}(517,·)$, $\chi_{628}(353,·)$, $\chi_{628}(99,·)$, $\chi_{628}(101,·)$, $\chi_{628}(39,·)$, $\chi_{628}(171,·)$, $\chi_{628}(173,·)$, $\chi_{628}(93,·)$, $\chi_{628}(467,·)$, $\chi_{628}(415,·)$, $\chi_{628}(315,·)$, $\chi_{628}(287,·)$, $\chi_{628}(381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{1}{13} a^{9} + \frac{1}{13} a^{7} + \frac{1}{13} a^{5} + \frac{1}{13} a^{3} + \frac{1}{13} a$, $\frac{1}{13} a^{12} + \frac{1}{13} a^{10} + \frac{1}{13} a^{8} + \frac{1}{13} a^{6} + \frac{1}{13} a^{4} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{4}$, $\frac{1}{13} a^{17} - \frac{1}{13} a^{5}$, $\frac{1}{169} a^{18} + \frac{3}{169} a^{16} + \frac{5}{169} a^{14} + \frac{6}{169} a^{12} + \frac{45}{169} a^{10} + \frac{58}{169} a^{8} + \frac{31}{169} a^{6} - \frac{10}{169} a^{4} - \frac{25}{169} a^{2}$, $\frac{1}{169} a^{19} + \frac{3}{169} a^{17} + \frac{5}{169} a^{15} + \frac{6}{169} a^{13} + \frac{6}{169} a^{11} + \frac{19}{169} a^{9} - \frac{8}{169} a^{7} - \frac{49}{169} a^{5} - \frac{64}{169} a^{3} - \frac{3}{13} a$, $\frac{1}{169} a^{20} - \frac{4}{169} a^{16} + \frac{4}{169} a^{14} + \frac{1}{169} a^{12} + \frac{66}{169} a^{10} + \frac{40}{169} a^{6} - \frac{21}{169} a^{4} + \frac{36}{169} a^{2}$, $\frac{1}{169} a^{21} - \frac{4}{169} a^{17} + \frac{4}{169} a^{15} + \frac{1}{169} a^{13} + \frac{1}{169} a^{11} - \frac{5}{13} a^{9} - \frac{25}{169} a^{7} + \frac{83}{169} a^{5} - \frac{29}{169} a^{3} - \frac{5}{13} a$, $\frac{1}{687661} a^{22} + \frac{1470}{687661} a^{20} - \frac{1824}{687661} a^{18} + \frac{1624}{52897} a^{16} - \frac{18767}{687661} a^{14} + \frac{15797}{687661} a^{12} + \frac{5994}{687661} a^{10} - \frac{740}{687661} a^{8} + \frac{176598}{687661} a^{6} - \frac{110147}{687661} a^{4} - \frac{207639}{687661} a^{2} + \frac{1849}{4069}$, $\frac{1}{687661} a^{23} + \frac{1470}{687661} a^{21} - \frac{1824}{687661} a^{19} + \frac{1624}{52897} a^{17} - \frac{18767}{687661} a^{15} + \frac{15797}{687661} a^{13} + \frac{5994}{687661} a^{11} - \frac{740}{687661} a^{9} + \frac{176598}{687661} a^{7} - \frac{110147}{687661} a^{5} - \frac{207639}{687661} a^{3} + \frac{1849}{4069} a$, $\frac{1}{106520820451097081289736595536203738435809} a^{24} - \frac{59792274433508202672987655848343830}{106520820451097081289736595536203738435809} a^{22} + \frac{3562081969687284019904208807112745915}{8193909265469006253056661195092595264293} a^{20} + \frac{296866448756815252793769850334452846166}{106520820451097081289736595536203738435809} a^{18} - \frac{4761211711307889050932485874142700755}{253018575893342235842604739990982751629} a^{16} + \frac{1915638478494197467889421143242843248810}{106520820451097081289736595536203738435809} a^{14} + \frac{3401981517497397839945842506517883609674}{106520820451097081289736595536203738435809} a^{12} - \frac{7781467753824852443617692468168202952512}{106520820451097081289736595536203738435809} a^{10} - \frac{14371835166615280123794119494076300836398}{106520820451097081289736595536203738435809} a^{8} - \frac{2105264170727713453471728922092482070969}{8193909265469006253056661195092595264293} a^{6} + \frac{19357708554244623471927212566343472760498}{106520820451097081289736595536203738435809} a^{4} - \frac{53236476639576442567657422668268840001739}{106520820451097081289736595536203738435809} a^{2} - \frac{195444463129211467867279769485980408236}{630300712728385096388973938084045789561}$, $\frac{1}{503097834990531514931425940717490256632325907} a^{25} + \frac{64689703405208902780514799544724558612}{503097834990531514931425940717490256632325907} a^{23} - \frac{417402495460857997116895525543014262532}{2976910266216162810245123909570948264096603} a^{21} - \frac{589901447504471379575882233628460817589231}{503097834990531514931425940717490256632325907} a^{19} + \frac{34598586722188711207024437401562318422573}{1195006733944255379884622186977411535943767} a^{17} - \frac{16216286012015105378800342429464450745800687}{503097834990531514931425940717490256632325907} a^{15} + \frac{5853092622842781632288125734761590679944086}{503097834990531514931425940717490256632325907} a^{13} - \frac{12508061964520066691972129181930502530658492}{503097834990531514931425940717490256632325907} a^{11} - \frac{166330572074015712010891566157089107168518790}{503097834990531514931425940717490256632325907} a^{9} - \frac{13113497089588647316890030775012546315856942}{38699833460810116533186610824422327433255839} a^{7} + \frac{117852705215596486346273881902142010363214067}{503097834990531514931425940717490256632325907} a^{5} - \frac{45838486580158286312424039132300415192531087}{503097834990531514931425940717490256632325907} a^{3} + \frac{1297364466457041848484458006293275190499743}{2976910266216162810245123909570948264096603} a$

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

Class group and class number

$C_{13}\times C_{4268953}$, which has order $55496389$ (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:  \( -\frac{3453937810176431842324}{3308700176843220638364245187659} a^{25} - \frac{480128120190588989170507}{3308700176843220638364245187659} a^{23} - \frac{2111766065315231616290615}{254515398218709279874172706743} a^{21} - \frac{861801974554805568236383303}{3308700176843220638364245187659} a^{19} - \frac{39261447139104115264208748}{7859145313166794865473266479} a^{17} - \frac{202441016265507829013163293224}{3308700176843220638364245187659} a^{15} - \frac{1599060069500146769278402748908}{3308700176843220638364245187659} a^{13} - \frac{8003331143846220131054136244513}{3308700176843220638364245187659} a^{11} - \frac{24126290036857196509944964238003}{3308700176843220638364245187659} a^{9} - \frac{3023744864929009222293111227923}{254515398218709279874172706743} a^{7} - \frac{26357660572698679134019679412727}{3308700176843220638364245187659} a^{5} + \frac{1606169599550550988386049093170}{3308700176843220638364245187659} a^{3} + \frac{48771091076762291483583223310}{19578107555285329221090208211} a \) (order $4$)
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:  \( 5466968796671.363 \) (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

\(\Q(\sqrt{-1}) \), 13.13.224282727500720205065439601.1

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 R $26$ ${\href{/LocalNumberField/5.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{26}$ ${\href{/LocalNumberField/17.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/29.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/37.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/LocalNumberField/53.13.0.1}{13} }^{2}$ $26$

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
2Data not computed
$157$157.13.12.1$x^{13} - 157$$13$$1$$12$$C_{13}$$[\ ]_{13}$
157.13.12.1$x^{13} - 157$$13$$1$$12$$C_{13}$$[\ ]_{13}$