Properties

Label 26.0.191...128.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.920\times 10^{57}$
Root discriminant \(159.66\)
Ramified primes $2,79$
Class number not computed
Class group not computed
Galois group $C_{26}$ (as 26T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537)
 
gp: K = bnfinit(y^26 - 2*y^25 - 45*y^24 + 178*y^23 + 768*y^22 - 5448*y^21 + 801*y^20 + 65886*y^19 - 134835*y^18 - 349218*y^17 + 1833359*y^16 - 1872300*y^15 - 4196097*y^14 + 11424264*y^13 + 3723051*y^12 - 71922706*y^11 + 268110160*y^10 - 834583714*y^9 + 2196377085*y^8 - 4640948852*y^7 + 8189022689*y^6 - 12586842022*y^5 + 17351590713*y^4 - 20853231606*y^3 + 21481183702*y^2 - 16566954540*y + 8027125537, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537)
 

\( x^{26} - 2 x^{25} - 45 x^{24} + 178 x^{23} + 768 x^{22} - 5448 x^{21} + 801 x^{20} + 65886 x^{19} - 134835 x^{18} - 349218 x^{17} + 1833359 x^{16} - 1872300 x^{15} + \cdots + 8027125537 \) Copy content Toggle raw display

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

Invariants

Degree:  $26$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1919641021107487828653877081110544587155912070949008048128\) \(\medspace = -\,2^{39}\cdot 79^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(159.66\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}79^{12/13}\approx 159.66170770416778$
Ramified primes:   \(2\), \(79\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-2}) \)
$\card{ \Gal(K/\Q) }$:  $26$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(632=2^{3}\cdot 79\)
Dirichlet character group:    $\lbrace$$\chi_{632}(1,·)$, $\chi_{632}(259,·)$, $\chi_{632}(97,·)$, $\chi_{632}(179,·)$, $\chi_{632}(65,·)$, $\chi_{632}(275,·)$, $\chi_{632}(457,·)$, $\chi_{632}(459,·)$, $\chi_{632}(337,·)$, $\chi_{632}(403,·)$, $\chi_{632}(225,·)$, $\chi_{632}(89,·)$, $\chi_{632}(539,·)$, $\chi_{632}(289,·)$, $\chi_{632}(67,·)$, $\chi_{632}(283,·)$, $\chi_{632}(131,·)$, $\chi_{632}(561,·)$, $\chi_{632}(617,·)$, $\chi_{632}(299,·)$, $\chi_{632}(433,·)$, $\chi_{632}(563,·)$, $\chi_{632}(417,·)$, $\chi_{632}(441,·)$, $\chi_{632}(571,·)$, $\chi_{632}(475,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{23}a^{20}+\frac{1}{23}a^{19}+\frac{6}{23}a^{18}+\frac{1}{23}a^{17}+\frac{8}{23}a^{16}-\frac{5}{23}a^{15}+\frac{2}{23}a^{14}-\frac{10}{23}a^{13}+\frac{10}{23}a^{12}+\frac{10}{23}a^{11}-\frac{5}{23}a^{10}+\frac{5}{23}a^{8}-\frac{11}{23}a^{7}+\frac{2}{23}a^{6}+\frac{7}{23}a^{5}-\frac{11}{23}a^{4}+\frac{6}{23}a^{3}-\frac{9}{23}a^{2}+\frac{3}{23}a-\frac{6}{23}$, $\frac{1}{23}a^{21}+\frac{5}{23}a^{19}-\frac{5}{23}a^{18}+\frac{7}{23}a^{17}+\frac{10}{23}a^{16}+\frac{7}{23}a^{15}+\frac{11}{23}a^{14}-\frac{3}{23}a^{13}+\frac{8}{23}a^{11}+\frac{5}{23}a^{10}+\frac{5}{23}a^{9}+\frac{7}{23}a^{8}-\frac{10}{23}a^{7}+\frac{5}{23}a^{6}+\frac{5}{23}a^{5}-\frac{6}{23}a^{4}+\frac{8}{23}a^{3}-\frac{11}{23}a^{2}-\frac{9}{23}a+\frac{6}{23}$, $\frac{1}{23}a^{22}-\frac{10}{23}a^{19}+\frac{5}{23}a^{17}-\frac{10}{23}a^{16}-\frac{10}{23}a^{15}+\frac{10}{23}a^{14}+\frac{4}{23}a^{13}+\frac{4}{23}a^{12}+\frac{1}{23}a^{11}+\frac{7}{23}a^{10}+\frac{7}{23}a^{9}+\frac{11}{23}a^{8}-\frac{9}{23}a^{7}-\frac{5}{23}a^{6}+\frac{5}{23}a^{5}-\frac{6}{23}a^{4}+\frac{5}{23}a^{3}-\frac{10}{23}a^{2}-\frac{9}{23}a+\frac{7}{23}$, $\frac{1}{23}a^{23}+\frac{10}{23}a^{19}-\frac{4}{23}a^{18}+\frac{1}{23}a^{16}+\frac{6}{23}a^{15}+\frac{1}{23}a^{14}-\frac{4}{23}a^{13}+\frac{9}{23}a^{12}-\frac{8}{23}a^{11}+\frac{3}{23}a^{10}+\frac{11}{23}a^{9}-\frac{5}{23}a^{8}+\frac{2}{23}a^{6}-\frac{5}{23}a^{5}+\frac{10}{23}a^{4}+\frac{4}{23}a^{3}-\frac{7}{23}a^{2}-\frac{9}{23}a+\frac{9}{23}$, $\frac{1}{1253201}a^{24}-\frac{22033}{1253201}a^{23}+\frac{26467}{1253201}a^{22}+\frac{16710}{1253201}a^{21}-\frac{22941}{1253201}a^{20}+\frac{272495}{1253201}a^{19}-\frac{637}{2369}a^{18}+\frac{469626}{1253201}a^{17}+\frac{303}{1253201}a^{16}-\frac{80824}{1253201}a^{15}+\frac{334265}{1253201}a^{14}-\frac{260376}{1253201}a^{13}+\frac{515943}{1253201}a^{12}+\frac{14986}{54487}a^{11}-\frac{383738}{1253201}a^{10}-\frac{461126}{1253201}a^{9}+\frac{130323}{1253201}a^{8}-\frac{46281}{1253201}a^{7}+\frac{430331}{1253201}a^{6}-\frac{104341}{1253201}a^{5}+\frac{89829}{1253201}a^{4}+\frac{518892}{1253201}a^{3}-\frac{10219}{54487}a^{2}-\frac{339893}{1253201}a-\frac{150560}{1253201}$, $\frac{1}{23\!\cdots\!53}a^{25}-\frac{75\!\cdots\!21}{23\!\cdots\!53}a^{24}-\frac{30\!\cdots\!43}{23\!\cdots\!53}a^{23}+\frac{65\!\cdots\!09}{23\!\cdots\!51}a^{22}-\frac{67\!\cdots\!62}{23\!\cdots\!53}a^{21}-\frac{42\!\cdots\!53}{23\!\cdots\!53}a^{20}-\frac{80\!\cdots\!55}{23\!\cdots\!53}a^{19}+\frac{61\!\cdots\!97}{23\!\cdots\!53}a^{18}+\frac{60\!\cdots\!73}{23\!\cdots\!53}a^{17}+\frac{17\!\cdots\!83}{23\!\cdots\!53}a^{16}+\frac{75\!\cdots\!91}{23\!\cdots\!53}a^{15}+\frac{13\!\cdots\!15}{10\!\cdots\!11}a^{14}-\frac{82\!\cdots\!94}{23\!\cdots\!53}a^{13}-\frac{83\!\cdots\!58}{23\!\cdots\!53}a^{12}+\frac{56\!\cdots\!00}{23\!\cdots\!53}a^{11}-\frac{39\!\cdots\!18}{23\!\cdots\!53}a^{10}+\frac{43\!\cdots\!83}{10\!\cdots\!11}a^{9}+\frac{10\!\cdots\!44}{23\!\cdots\!53}a^{8}+\frac{10\!\cdots\!13}{23\!\cdots\!53}a^{7}-\frac{88\!\cdots\!44}{23\!\cdots\!53}a^{6}-\frac{53\!\cdots\!73}{23\!\cdots\!53}a^{5}-\frac{16\!\cdots\!71}{23\!\cdots\!53}a^{4}+\frac{28\!\cdots\!58}{23\!\cdots\!53}a^{3}+\frac{98\!\cdots\!41}{23\!\cdots\!53}a^{2}-\frac{24\!\cdots\!66}{23\!\cdots\!53}a+\frac{99\!\cdots\!24}{23\!\cdots\!53}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x^25 - 45*x^24 + 178*x^23 + 768*x^22 - 5448*x^21 + 801*x^20 + 65886*x^19 - 134835*x^18 - 349218*x^17 + 1833359*x^16 - 1872300*x^15 - 4196097*x^14 + 11424264*x^13 + 3723051*x^12 - 71922706*x^11 + 268110160*x^10 - 834583714*x^9 + 2196377085*x^8 - 4640948852*x^7 + 8189022689*x^6 - 12586842022*x^5 + 17351590713*x^4 - 20853231606*x^3 + 21481183702*x^2 - 16566954540*x + 8027125537);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-2}) \), 13.13.59091511031674153381441.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/11.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/17.13.0.1}{13} }^{2}$ ${\href{/padicField/19.13.0.1}{13} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{13}$ $26$ $26$ $26$ ${\href{/padicField/41.13.0.1}{13} }^{2}$ ${\href{/padicField/43.13.0.1}{13} }^{2}$ $26$ $26$ ${\href{/padicField/59.13.0.1}{13} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $26$$2$$13$$39$
\(79\) Copy content Toggle raw display Deg $26$$13$$2$$24$