Properties

Label 26.0.105...671.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.058\times 10^{44}$
Root discriminant $49.35$
Ramified primes $31, 113$
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 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675)
 
gp: K = bnfinit(x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![61675, 96160, 178609, 240404, -88862, 186178, -276734, -235197, 369592, -985605, 1103470, -1048444, 813670, -450753, 263098, -97147, 59870, -22984, 17106, -6943, 3738, -1233, 437, -114, 29, -4, 1]);
 

\(x^{26} - 4 x^{25} + 29 x^{24} - 114 x^{23} + 437 x^{22} - 1233 x^{21} + 3738 x^{20} - 6943 x^{19} + 17106 x^{18} - 22984 x^{17} + 59870 x^{16} - 97147 x^{15} + 263098 x^{14} - 450753 x^{13} + 813670 x^{12} - 1048444 x^{11} + 1103470 x^{10} - 985605 x^{9} + 369592 x^{8} - 235197 x^{7} - 276734 x^{6} + 186178 x^{5} - 88862 x^{4} + 240404 x^{3} + 178609 x^{2} + 96160 x + 61675\)  Toggle raw display

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:  \(-105838418476275527898387851073846090273368671\)\(\medspace = -\,31^{13}\cdot 113^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $49.35$
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:  $31, 113$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{25} a^{19} - \frac{2}{25} a^{18} - \frac{1}{25} a^{16} - \frac{1}{25} a^{14} - \frac{6}{25} a^{12} + \frac{6}{25} a^{11} + \frac{7}{25} a^{10} - \frac{6}{25} a^{9} - \frac{12}{25} a^{8} - \frac{1}{25} a^{7} - \frac{2}{25} a^{6} + \frac{4}{25} a^{5} - \frac{12}{25} a^{4} + \frac{2}{25} a^{3} + \frac{7}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{20} + \frac{1}{25} a^{18} - \frac{1}{25} a^{17} - \frac{2}{25} a^{16} - \frac{1}{25} a^{15} - \frac{2}{25} a^{14} - \frac{1}{25} a^{13} - \frac{11}{25} a^{12} - \frac{1}{25} a^{11} + \frac{3}{25} a^{10} + \frac{1}{25} a^{9} + \frac{1}{5} a^{8} + \frac{11}{25} a^{7} + \frac{1}{5} a^{6} + \frac{6}{25} a^{5} - \frac{12}{25} a^{4} - \frac{4}{25} a^{3} - \frac{1}{25} a^{2}$, $\frac{1}{775} a^{21} - \frac{7}{775} a^{20} - \frac{4}{775} a^{19} - \frac{58}{775} a^{18} - \frac{6}{155} a^{17} + \frac{3}{775} a^{16} + \frac{38}{775} a^{14} + \frac{71}{775} a^{13} - \frac{164}{775} a^{12} - \frac{11}{155} a^{11} - \frac{57}{155} a^{10} + \frac{233}{775} a^{9} + \frac{221}{775} a^{8} + \frac{213}{775} a^{7} - \frac{144}{775} a^{6} + \frac{101}{775} a^{5} + \frac{21}{155} a^{4} + \frac{202}{775} a^{3} + \frac{282}{775} a^{2} + \frac{21}{155} a + \frac{14}{31}$, $\frac{1}{775} a^{22} + \frac{9}{775} a^{20} + \frac{7}{775} a^{19} + \frac{12}{155} a^{18} + \frac{41}{775} a^{17} - \frac{41}{775} a^{16} - \frac{24}{775} a^{15} - \frac{7}{155} a^{14} - \frac{39}{775} a^{13} - \frac{118}{775} a^{12} - \frac{174}{775} a^{11} - \frac{6}{31} a^{10} + \frac{271}{775} a^{9} + \frac{334}{775} a^{8} - \frac{234}{775} a^{7} - \frac{318}{775} a^{6} - \frac{304}{775} a^{5} + \frac{162}{775} a^{4} + \frac{239}{775} a^{3} + \frac{33}{775} a^{2} + \frac{1}{5} a + \frac{5}{31}$, $\frac{1}{275125} a^{23} - \frac{3}{8875} a^{22} + \frac{164}{275125} a^{21} - \frac{4116}{275125} a^{20} + \frac{2509}{275125} a^{19} - \frac{1269}{55025} a^{18} - \frac{5063}{275125} a^{17} - \frac{179}{275125} a^{16} + \frac{14938}{275125} a^{15} + \frac{8858}{275125} a^{14} - \frac{4159}{55025} a^{13} + \frac{13466}{275125} a^{12} + \frac{29858}{275125} a^{11} - \frac{25087}{275125} a^{10} - \frac{4112}{55025} a^{9} + \frac{65579}{275125} a^{8} - \frac{44462}{275125} a^{7} - \frac{9708}{55025} a^{6} + \frac{120473}{275125} a^{5} + \frac{642}{2201} a^{4} + \frac{14496}{55025} a^{3} - \frac{1503}{8875} a^{2} + \frac{23926}{55025} a - \frac{154}{355}$, $\frac{1}{89415625} a^{24} + \frac{3}{17883125} a^{23} - \frac{6449}{17883125} a^{22} + \frac{13596}{89415625} a^{21} - \frac{1699784}{89415625} a^{20} - \frac{1520668}{89415625} a^{19} - \frac{3627948}{89415625} a^{18} - \frac{3323793}{89415625} a^{17} + \frac{53267}{6878125} a^{16} + \frac{1234502}{89415625} a^{15} - \frac{4389131}{89415625} a^{14} + \frac{2832746}{89415625} a^{13} - \frac{30003249}{89415625} a^{12} - \frac{3550748}{89415625} a^{11} + \frac{22590774}{89415625} a^{10} - \frac{36366606}{89415625} a^{9} - \frac{273134}{1375625} a^{8} + \frac{2359}{40625} a^{7} - \frac{11699997}{89415625} a^{6} + \frac{42857729}{89415625} a^{5} - \frac{8071262}{17883125} a^{4} + \frac{25304402}{89415625} a^{3} + \frac{2045707}{6878125} a^{2} + \frac{2670188}{17883125} a - \frac{946562}{3576625}$, $\frac{1}{784803989526277132786021644580620722639830203197216328125} a^{25} - \frac{1911911312047144992295631852576040423383802275882}{784803989526277132786021644580620722639830203197216328125} a^{24} + \frac{1929019010572000732413235578865627817065408621927}{31392159581051085311440865783224828905593208127888653125} a^{23} + \frac{13207212660022549983008151462516478175435236024305601}{71345817229661557526001967689147338421802745745201484375} a^{22} + \frac{409600020839943461908499992025992783225049167746062229}{784803989526277132786021644580620722639830203197216328125} a^{21} + \frac{537998017938366206814935128476675633405607435163700216}{156960797905255426557204328916124144527966040639443265625} a^{20} - \frac{5508541992230012527326221822343359664904213935282858252}{784803989526277132786021644580620722639830203197216328125} a^{19} + \frac{47464683718048354185407806626218488539520945045238320188}{784803989526277132786021644580620722639830203197216328125} a^{18} + \frac{11629471588819314147221701428272628271801730408222367667}{784803989526277132786021644580620722639830203197216328125} a^{17} + \frac{14700821535716782153600492734720277559568431231924101953}{156960797905255426557204328916124144527966040639443265625} a^{16} + \frac{268425601927097551721194040979962996209836280194803}{12602231867142145849635032430038068609230513098309375} a^{15} + \frac{4069326205596102970421309979845004944331122744312357106}{60369537655867471752770895736970824818448477169016640625} a^{14} - \frac{65353893339408606893419174849902199049378779653047409161}{784803989526277132786021644580620722639830203197216328125} a^{13} - \frac{28304737764432365219540515200030681633826084552891487979}{156960797905255426557204328916124144527966040639443265625} a^{12} - \frac{35553757831050772194767060401241444056373470265326178829}{156960797905255426557204328916124144527966040639443265625} a^{11} - \frac{52696841442702763943069985113940380878982569493991704134}{784803989526277132786021644580620722639830203197216328125} a^{10} - \frac{331838192680291201375569140586598254507167098869904784378}{784803989526277132786021644580620722639830203197216328125} a^{9} + \frac{211733458476571929778996661366737543355306310432483818279}{784803989526277132786021644580620722639830203197216328125} a^{8} - \frac{1186144081297168640994429679467757530621440528508683742}{3339591444792668650153283593960088181446085971051984375} a^{7} - \frac{177526463346251054096029912069689836238519896409940937712}{784803989526277132786021644580620722639830203197216328125} a^{6} + \frac{18895037870411059097110472139404923648727560830308608777}{784803989526277132786021644580620722639830203197216328125} a^{5} + \frac{2052129287367135637235133347418720219476917403873336429}{5488139786897042886615535976088256801677134288092421875} a^{4} + \frac{168034406916441353007903694997153811211365207350403088847}{784803989526277132786021644580620722639830203197216328125} a^{3} - \frac{22467195984712454168751648014369880751620356188411921967}{71345817229661557526001967689147338421802745745201484375} a^{2} - \frac{50208697795280718145822003294281017408146219635552556196}{156960797905255426557204328916124144527966040639443265625} a + \frac{14925255832939004311646284809493023243617975482303495489}{31392159581051085311440865783224828905593208127888653125}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 195886918009.72314 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 195886918009.72314 \cdot 3}{2\sqrt{105838418476275527898387851073846090273368671}}\approx 0.679380864292949$ (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

\(\Q(\sqrt{-31}) \), 13.1.1847739844104888853729.1

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 ${\href{/LocalNumberField/2.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}$ $26$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $26$ $26$ R $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$
113.2.1.1$x^{2} - 113$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.113.2t1.a.a$1$ $ 113 $ \(\Q(\sqrt{113}) \) $C_2$ (as 2T1) $1$ $1$
1.3503.2t1.a.a$1$ $ 31 \cdot 113 $ \(\Q(\sqrt{-3503}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3503.26t3.a.a$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3503.13t2.a.d$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.13t2.a.a$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.26t3.a.d$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3503.13t2.a.e$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.26t3.a.b$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3503.13t2.a.c$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.13t2.a.f$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.13t2.a.b$2$ $ 31 \cdot 113 $ 13.1.1847739844104888853729.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.3503.26t3.a.f$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3503.26t3.a.e$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.3503.26t3.a.c$2$ $ 31 \cdot 113 $ 26.0.105838418476275527898387851073846090273368671.1 $D_{26}$ (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 *.