Properties

Label 26.0.115...543.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.154\times 10^{40}$
Root discriminant $34.74$
Ramified primes $23, 73$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 + 11*x^24 - 20*x^23 + 107*x^22 - 382*x^21 + 1248*x^20 - 3197*x^19 + 6521*x^18 - 10581*x^17 + 13912*x^16 - 15385*x^15 + 14827*x^14 - 15025*x^13 + 15241*x^12 - 12717*x^11 + 9243*x^10 - 4542*x^9 - 233*x^8 + 4189*x^7 - 7810*x^6 + 6629*x^5 - 1342*x^4 - 1780*x^3 + 1964*x^2 - 1216*x + 361)
 
gp: K = bnfinit(x^26 - x^25 + 11*x^24 - 20*x^23 + 107*x^22 - 382*x^21 + 1248*x^20 - 3197*x^19 + 6521*x^18 - 10581*x^17 + 13912*x^16 - 15385*x^15 + 14827*x^14 - 15025*x^13 + 15241*x^12 - 12717*x^11 + 9243*x^10 - 4542*x^9 - 233*x^8 + 4189*x^7 - 7810*x^6 + 6629*x^5 - 1342*x^4 - 1780*x^3 + 1964*x^2 - 1216*x + 361, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![361, -1216, 1964, -1780, -1342, 6629, -7810, 4189, -233, -4542, 9243, -12717, 15241, -15025, 14827, -15385, 13912, -10581, 6521, -3197, 1248, -382, 107, -20, 11, -1, 1]);
 

\( x^{26} - x^{25} + 11 x^{24} - 20 x^{23} + 107 x^{22} - 382 x^{21} + 1248 x^{20} - 3197 x^{19} + 6521 x^{18} - 10581 x^{17} + 13912 x^{16} - 15385 x^{15} + 14827 x^{14} - 15025 x^{13} + 15241 x^{12} - 12717 x^{11} + 9243 x^{10} - 4542 x^{9} - 233 x^{8} + 4189 x^{7} - 7810 x^{6} + 6629 x^{5} - 1342 x^{4} - 1780 x^{3} + 1964 x^{2} - 1216 x + 361 \)

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-11543464978247129822340588525652022819543\)\(\medspace = -\,23^{13}\cdot 73^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.74$
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:  $23, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{23} a^{14} + \frac{8}{23} a^{13} - \frac{9}{23} a^{12} + \frac{1}{23} a^{11} + \frac{7}{23} a^{10} + \frac{7}{23} a^{9} - \frac{9}{23} a^{8} + \frac{10}{23} a^{7} + \frac{9}{23} a^{6} - \frac{11}{23} a^{5} + \frac{7}{23} a^{4} - \frac{11}{23} a^{3} + \frac{10}{23} a^{2} - \frac{5}{23} a + \frac{8}{23}$, $\frac{1}{23} a^{15} - \frac{4}{23} a^{13} + \frac{4}{23} a^{12} - \frac{1}{23} a^{11} - \frac{3}{23} a^{10} + \frac{4}{23} a^{9} - \frac{10}{23} a^{8} - \frac{2}{23} a^{7} + \frac{9}{23} a^{6} + \frac{3}{23} a^{5} + \frac{2}{23} a^{4} + \frac{6}{23} a^{3} + \frac{7}{23} a^{2} + \frac{2}{23} a + \frac{5}{23}$, $\frac{1}{23} a^{16} - \frac{10}{23} a^{13} + \frac{9}{23} a^{12} + \frac{1}{23} a^{11} + \frac{9}{23} a^{10} - \frac{5}{23} a^{9} + \frac{8}{23} a^{8} + \frac{3}{23} a^{7} - \frac{7}{23} a^{6} + \frac{4}{23} a^{5} + \frac{11}{23} a^{4} + \frac{9}{23} a^{3} - \frac{4}{23} a^{2} + \frac{8}{23} a + \frac{9}{23}$, $\frac{1}{23} a^{17} - \frac{3}{23} a^{13} + \frac{3}{23} a^{12} - \frac{4}{23} a^{11} - \frac{4}{23} a^{10} + \frac{9}{23} a^{9} + \frac{5}{23} a^{8} + \frac{1}{23} a^{7} + \frac{2}{23} a^{6} - \frac{7}{23} a^{5} + \frac{10}{23} a^{4} + \frac{1}{23} a^{3} - \frac{7}{23} a^{2} + \frac{5}{23} a + \frac{11}{23}$, $\frac{1}{23} a^{18} + \frac{4}{23} a^{13} - \frac{8}{23} a^{12} - \frac{1}{23} a^{11} + \frac{7}{23} a^{10} + \frac{3}{23} a^{9} - \frac{3}{23} a^{8} + \frac{9}{23} a^{7} - \frac{3}{23} a^{6} - \frac{1}{23} a^{4} + \frac{6}{23} a^{3} - \frac{11}{23} a^{2} - \frac{4}{23} a + \frac{1}{23}$, $\frac{1}{23} a^{19} + \frac{6}{23} a^{13} - \frac{11}{23} a^{12} + \frac{3}{23} a^{11} - \frac{2}{23} a^{10} - \frac{8}{23} a^{9} - \frac{1}{23} a^{8} + \frac{3}{23} a^{7} + \frac{10}{23} a^{6} - \frac{3}{23} a^{5} + \frac{1}{23} a^{4} + \frac{10}{23} a^{3} + \frac{2}{23} a^{2} - \frac{2}{23} a - \frac{9}{23}$, $\frac{1}{529} a^{20} + \frac{7}{529} a^{19} - \frac{5}{529} a^{18} - \frac{6}{529} a^{17} - \frac{3}{529} a^{16} + \frac{5}{529} a^{15} + \frac{9}{529} a^{14} + \frac{132}{529} a^{13} + \frac{75}{529} a^{12} + \frac{89}{529} a^{11} + \frac{199}{529} a^{10} - \frac{93}{529} a^{9} + \frac{156}{529} a^{8} + \frac{129}{529} a^{7} - \frac{228}{529} a^{6} - \frac{54}{529} a^{5} - \frac{109}{529} a^{4} - \frac{155}{529} a^{3} + \frac{209}{529} a^{2} + \frac{214}{529} a + \frac{256}{529}$, $\frac{1}{529} a^{21} - \frac{8}{529} a^{19} + \frac{6}{529} a^{18} - \frac{7}{529} a^{17} + \frac{3}{529} a^{16} - \frac{3}{529} a^{15} + \frac{117}{529} a^{13} + \frac{139}{529} a^{12} - \frac{194}{529} a^{11} - \frac{198}{529} a^{10} + \frac{209}{529} a^{9} + \frac{95}{529} a^{8} + \frac{65}{529} a^{7} + \frac{139}{529} a^{6} + \frac{131}{529} a^{5} + \frac{56}{529} a^{4} + \frac{144}{529} a^{3} + \frac{39}{529} a^{2} - \frac{9}{23} a - \frac{205}{529}$, $\frac{1}{529} a^{22} - \frac{7}{529} a^{19} - \frac{1}{529} a^{18} + \frac{1}{529} a^{17} - \frac{4}{529} a^{16} - \frac{6}{529} a^{15} + \frac{5}{529} a^{14} - \frac{162}{529} a^{13} - \frac{31}{529} a^{12} - \frac{38}{529} a^{11} + \frac{76}{529} a^{10} - \frac{74}{529} a^{9} + \frac{71}{529} a^{8} - \frac{255}{529} a^{7} + \frac{101}{529} a^{6} - \frac{100}{529} a^{5} + \frac{77}{529} a^{4} - \frac{143}{529} a^{3} - \frac{168}{529} a^{2} + \frac{58}{529} a + \frac{139}{529}$, $\frac{1}{529} a^{23} + \frac{2}{529} a^{19} - \frac{11}{529} a^{18} - \frac{4}{529} a^{16} - \frac{6}{529} a^{15} - \frac{7}{529} a^{14} + \frac{203}{529} a^{13} + \frac{142}{529} a^{12} - \frac{14}{529} a^{11} + \frac{261}{529} a^{10} + \frac{87}{529} a^{9} - \frac{198}{529} a^{8} + \frac{84}{529} a^{7} + \frac{236}{529} a^{6} + \frac{44}{529} a^{5} - \frac{239}{529} a^{4} + \frac{35}{529} a^{3} - \frac{227}{529} a^{2} - \frac{88}{529} a - \frac{255}{529}$, $\frac{1}{371887} a^{24} + \frac{216}{371887} a^{23} - \frac{351}{371887} a^{22} - \frac{168}{371887} a^{21} - \frac{211}{371887} a^{20} + \frac{4295}{371887} a^{19} - \frac{1784}{371887} a^{18} + \frac{144}{371887} a^{17} - \frac{7565}{371887} a^{16} - \frac{1713}{371887} a^{15} - \frac{4751}{371887} a^{14} + \frac{14532}{371887} a^{13} + \frac{184625}{371887} a^{12} + \frac{183370}{371887} a^{11} + \frac{39892}{371887} a^{10} - \frac{131436}{371887} a^{9} - \frac{5227}{19573} a^{8} - \frac{131348}{371887} a^{7} - \frac{27069}{371887} a^{6} + \frac{255}{529} a^{5} + \frac{3554}{10051} a^{4} + \frac{127138}{371887} a^{3} - \frac{31860}{371887} a^{2} - \frac{100613}{371887} a + \frac{4752}{19573}$, $\frac{1}{26246140626632223463666233231769} a^{25} + \frac{20446516544110977174217662}{26246140626632223463666233231769} a^{24} - \frac{23092320612528386575055610016}{26246140626632223463666233231769} a^{23} - \frac{10136455399687508144608133950}{26246140626632223463666233231769} a^{22} - \frac{7654403302297737062728329424}{26246140626632223463666233231769} a^{21} + \frac{20844826164334799095530874278}{26246140626632223463666233231769} a^{20} - \frac{61990989030219173236457897480}{26246140626632223463666233231769} a^{19} - \frac{156569336092383559948906845721}{26246140626632223463666233231769} a^{18} + \frac{314428674723340934446859782515}{26246140626632223463666233231769} a^{17} - \frac{191078224490006908437453440230}{26246140626632223463666233231769} a^{16} - \frac{425765236532090714585708302308}{26246140626632223463666233231769} a^{15} - \frac{115475186482657142864514828}{18922956471977089735880485387} a^{14} + \frac{12030205455565394334303626438511}{26246140626632223463666233231769} a^{13} + \frac{1553642215842557659678745105654}{26246140626632223463666233231769} a^{12} + \frac{2010012279269296807722228362405}{26246140626632223463666233231769} a^{11} + \frac{8140448592390318030466806471887}{26246140626632223463666233231769} a^{10} - \frac{10302928415331611487646636142452}{26246140626632223463666233231769} a^{9} - \frac{9967999585358655912045431264442}{26246140626632223463666233231769} a^{8} - \frac{7345244725777486931299021527193}{26246140626632223463666233231769} a^{7} - \frac{11845544881337668022125264552019}{26246140626632223463666233231769} a^{6} - \frac{204273272725187288779234633334}{709355152071141174693681979237} a^{5} + \frac{1294548877285837778873234636785}{26246140626632223463666233231769} a^{4} - \frac{4640450309898442766979782970406}{26246140626632223463666233231769} a^{3} + \frac{308664349651543098193828115863}{709355152071141174693681979237} a^{2} - \frac{10722471501580284960216974276120}{26246140626632223463666233231769} a - \frac{558685795556497198169622985785}{1381375822454327550719275433251}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
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:  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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1140159040.1752596 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 1140159040.1752596 \cdot 3}{2\sqrt{11543464978247129822340588525652022819543}}\approx 0.378640345672729$ (assuming GRH)

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

\(\Q(\sqrt{-23}) \), 13.1.22402896724819285921.1

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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/3.13.0.1}{13} }^{2}$ $26$ $26$ $26$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $26$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$
73.2.1.1$x^{2} - 73$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.73.2t1.a.a$1$ $ 73 $ \(\Q(\sqrt{73}) \) $C_2$ (as 2T1) $1$ $1$
1.1679.2t1.a.a$1$ $ 23 \cdot 73 $ \(\Q(\sqrt{-1679}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.23.2t1.a.a$1$ $ 23 $ \(\Q(\sqrt{-23}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.1679.26t3.a.d$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.1679.13t2.a.b$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.13t2.a.d$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.26t3.a.b$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.1679.13t2.a.f$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.26t3.a.c$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.1679.13t2.a.e$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.13t2.a.a$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.13t2.a.c$2$ $ 23 \cdot 73 $ 13.1.22402896724819285921.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.1679.26t3.a.a$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.1679.26t3.a.f$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.1679.26t3.a.e$2$ $ 23 \cdot 73 $ 26.0.11543464978247129822340588525652022819543.1 $D_{26}$ (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 *.