Properties

Label 26.0.526...875.1
Degree $26$
Signature $[0, 13]$
Discriminant $-5.267\times 10^{44}$
Root discriminant $52.49$
Ramified primes $5, 19, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $D_{26}$ (as 26T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261)
 
gp: K = bnfinit(x^26 - 8*x^25 + 14*x^24 + 18*x^23 - 92*x^22 + 514*x^21 - 1212*x^20 - 596*x^19 + 954*x^18 - 2053*x^17 + 29744*x^16 - 4491*x^15 + 71256*x^14 - 216720*x^13 - 344608*x^12 + 50717*x^11 + 1942970*x^10 + 2307612*x^9 + 264538*x^8 - 5680405*x^7 - 6094469*x^6 - 2337568*x^5 + 3586683*x^4 + 5626157*x^3 + 5313221*x^2 + 843733*x + 496261, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![496261, 843733, 5313221, 5626157, 3586683, -2337568, -6094469, -5680405, 264538, 2307612, 1942970, 50717, -344608, -216720, 71256, -4491, 29744, -2053, 954, -596, -1212, 514, -92, 18, 14, -8, 1]);
 

\(x^{26} - 8 x^{25} + 14 x^{24} + 18 x^{23} - 92 x^{22} + 514 x^{21} - 1212 x^{20} - 596 x^{19} + 954 x^{18} - 2053 x^{17} + 29744 x^{16} - 4491 x^{15} + 71256 x^{14} - 216720 x^{13} - 344608 x^{12} + 50717 x^{11} + 1942970 x^{10} + 2307612 x^{9} + 264538 x^{8} - 5680405 x^{7} - 6094469 x^{6} - 2337568 x^{5} + 3586683 x^{4} + 5626157 x^{3} + 5313221 x^{2} + 843733 x + 496261\)  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:  \(-526721251926767614625217802292605164794921875\)\(\medspace = -\,5^{13}\cdot 19^{13}\cdot 29^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.49$
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:  $5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{5}{19} a^{17} + \frac{4}{19} a^{16} + \frac{9}{19} a^{15} - \frac{7}{19} a^{14} + \frac{6}{19} a^{13} + \frac{2}{19} a^{12} - \frac{4}{19} a^{11} + \frac{9}{19} a^{10} - \frac{7}{19} a^{9} - \frac{3}{19} a^{8} - \frac{8}{19} a^{7} - \frac{9}{19} a^{6} - \frac{9}{19} a^{5} + \frac{6}{19} a^{4} + \frac{8}{19} a^{3} - \frac{3}{19} a^{2}$, $\frac{1}{19} a^{19} - \frac{2}{19} a^{17} + \frac{8}{19} a^{16} + \frac{5}{19} a^{15} + \frac{3}{19} a^{14} - \frac{9}{19} a^{13} + \frac{5}{19} a^{12} - \frac{9}{19} a^{11} + \frac{5}{19} a^{10} - \frac{6}{19} a^{9} + \frac{7}{19} a^{8} - \frac{7}{19} a^{7} - \frac{2}{19} a^{6} - \frac{6}{19} a^{5} - \frac{3}{19} a^{4} - \frac{5}{19} a^{3} - \frac{4}{19} a^{2}$, $\frac{1}{19} a^{20} - \frac{1}{19} a^{17} - \frac{6}{19} a^{16} + \frac{2}{19} a^{15} - \frac{4}{19} a^{14} - \frac{2}{19} a^{13} - \frac{5}{19} a^{12} - \frac{3}{19} a^{11} - \frac{7}{19} a^{10} - \frac{7}{19} a^{9} + \frac{6}{19} a^{8} + \frac{1}{19} a^{7} - \frac{5}{19} a^{6} - \frac{2}{19} a^{5} + \frac{7}{19} a^{4} - \frac{7}{19} a^{3} - \frac{6}{19} a^{2}$, $\frac{1}{19} a^{21} - \frac{1}{19} a^{17} + \frac{6}{19} a^{16} + \frac{5}{19} a^{15} - \frac{9}{19} a^{14} + \frac{1}{19} a^{13} - \frac{1}{19} a^{12} + \frac{8}{19} a^{11} + \frac{2}{19} a^{10} - \frac{1}{19} a^{9} - \frac{2}{19} a^{8} + \frac{6}{19} a^{7} + \frac{8}{19} a^{6} - \frac{2}{19} a^{5} - \frac{1}{19} a^{4} + \frac{2}{19} a^{3} - \frac{3}{19} a^{2}$, $\frac{1}{551} a^{22} - \frac{5}{551} a^{21} - \frac{6}{551} a^{20} + \frac{11}{551} a^{19} - \frac{13}{551} a^{18} + \frac{49}{551} a^{17} + \frac{127}{551} a^{16} - \frac{118}{551} a^{15} - \frac{193}{551} a^{14} - \frac{260}{551} a^{13} + \frac{131}{551} a^{12} + \frac{157}{551} a^{11} - \frac{155}{551} a^{10} - \frac{241}{551} a^{9} - \frac{173}{551} a^{8} + \frac{1}{19} a^{7} - \frac{268}{551} a^{6} + \frac{272}{551} a^{5} - \frac{102}{551} a^{4} + \frac{258}{551} a^{3} - \frac{90}{551} a^{2} + \frac{14}{29} a - \frac{12}{29}$, $\frac{1}{7163} a^{23} + \frac{2}{7163} a^{22} + \frac{46}{7163} a^{21} + \frac{27}{7163} a^{20} - \frac{168}{7163} a^{19} + \frac{45}{7163} a^{18} + \frac{673}{7163} a^{17} + \frac{1090}{7163} a^{16} - \frac{2498}{7163} a^{15} - \frac{74}{7163} a^{14} - \frac{2414}{7163} a^{13} - \frac{840}{7163} a^{12} + \frac{3206}{7163} a^{11} + \frac{1922}{7163} a^{10} - \frac{2121}{7163} a^{9} - \frac{1240}{7163} a^{8} + \frac{892}{7163} a^{7} + \frac{1238}{7163} a^{6} - \frac{1736}{7163} a^{5} - \frac{2225}{7163} a^{4} - \frac{2170}{7163} a^{3} + \frac{3000}{7163} a^{2} - \frac{88}{377} a - \frac{113}{377}$, $\frac{1}{436943} a^{24} + \frac{20}{436943} a^{23} + \frac{30}{436943} a^{22} + \frac{8278}{436943} a^{21} + \frac{1384}{436943} a^{20} - \frac{535}{436943} a^{19} + \frac{159}{7163} a^{18} - \frac{94527}{436943} a^{17} + \frac{160564}{436943} a^{16} - \frac{205159}{436943} a^{15} + \frac{203084}{436943} a^{14} - \frac{157444}{436943} a^{13} - \frac{171411}{436943} a^{12} - \frac{36752}{436943} a^{11} + \frac{3589}{436943} a^{10} - \frac{60062}{436943} a^{9} - \frac{209980}{436943} a^{8} - \frac{29077}{436943} a^{7} + \frac{128734}{436943} a^{6} - \frac{106429}{436943} a^{5} - \frac{124380}{436943} a^{4} - \frac{109796}{436943} a^{3} - \frac{82482}{436943} a^{2} - \frac{3179}{22997} a - \frac{2918}{22997}$, $\frac{1}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{25} - \frac{18031227577960129778038653601363686644780231824011863970762043809267189617180}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{24} + \frac{107441780873728438833099835882444962537077533851015818708467534497170482834321}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{23} + \frac{1782064713943047344876924285483317010381574833083455526308441649906463374710120}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{22} + \frac{52244256808308559797905107969038008697833148520403252451486516774379821448612691}{3229732149903872094880549156877889658749222266972011276472850897466492864385831069} a^{21} - \frac{1250925199282087658833225522125091651004085226032647853722638472877866484401249850}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{20} + \frac{2168729663244700594390773871211514168875265847711349613471656262237555138363879385}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{19} + \frac{804100378104238811980202228247685581608208676866369201709979173748581321990184965}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{18} + \frac{19208026585164890410118294334804472209031275603088392899225449126639374851154359283}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{17} + \frac{6534883333427522045763500727130354033490219073746054408736234254813306427650395956}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{16} - \frac{17035695100123004302195309373402044923953235061731853858870921970376441419936175036}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{15} + \frac{41127388745788689462467177556126091779961164108043514152700657546816575123258023812}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{14} - \frac{10124962771068482008864831007443271855725387218127548463972254660736414392177915925}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{13} + \frac{37667950544479709711627489769362074347101037428869659215455138136759688201313455202}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{12} - \frac{7541010440934560253682510292357692373310349496978184545721389767130378616394108677}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{11} + \frac{27984249922146520826736960857569915341663413921977680943978429224549799751980627418}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{10} - \frac{35286522456209219653599010239187521928240485912170588179679035542047448880598938659}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{9} - \frac{34067549690787041131819331477717164547799611407794517965317877046067621236346597668}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{8} + \frac{41494150751724774181373916130129012418684509096837573645901106915958424206945920103}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{7} + \frac{19955451924276650950094934008693714000838832119247629444222791467827178524492496334}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{6} - \frac{3801940455329254987592429169592771149070206626227501372434062492310469135997595979}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{5} - \frac{40323079202191947985961772444800938888587560551884651770926448541328252662460054825}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{4} - \frac{16237532783510931370034600524296155601955373775454022543755888277332607198736842786}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{3} - \frac{39184866422859579265961644015499646319975014617148302366347853438270205379885369317}{93662232347212290751535925549458800103727445742188327017712676026528293067189101001} a^{2} - \frac{128408034680802842363088885638581720960466237518345252818051930131149379543361467}{379199321243774456483951115584853441715495731749750311812601927232908069097931583} a - \frac{27720855185441900037081909870432971604102614114023208732385821815983692333421065}{379199321243774456483951115584853441715495731749750311812601927232908069097931583}$  Toggle raw display

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 154406339448.51657 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{13}\cdot 154406339448.51657 \cdot 8}{2\sqrt{526721251926767614625217802292605164794921875}}\approx 0.640137082872717$ (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

\(\Q(\sqrt{-2755}) \), 13.1.27983987175790801.1

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

Sibling fields

Degree 26 sibling: 26.2.955936936346220716198217427028321533203125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ $26$ R $26$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R $26$ R ${\href{/LocalNumberField/31.13.0.1}{13} }^{2}$ $26$ ${\href{/LocalNumberField/41.13.0.1}{13} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}$

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
5Data not computed
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.2755.2t1.a.a$1$ $ 5 \cdot 19 \cdot 29 $ \(\Q(\sqrt{-2755}) \) $C_2$ (as 2T1) $1$ $-1$
1.551.2t1.a.a$1$ $ 19 \cdot 29 $ \(\Q(\sqrt{-551}) \) $C_2$ (as 2T1) $1$ $-1$
1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.13775.26t3.a.d$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.f$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.e$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.a.e$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.13775.26t3.a.f$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.b$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.c$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.551.13t2.a.d$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.a.a$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.13775.26t3.a.c$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.551.13t2.a.a$2$ $ 19 \cdot 29 $ 13.1.27983987175790801.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.13775.26t3.a.b$2$ $ 5^{2} \cdot 19 \cdot 29 $ 26.0.526721251926767614625217802292605164794921875.1 $D_{26}$ (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 *.