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Properties

Label 26.2.713...893.1
Degree $26$
Signature $[2, 12]$
Discriminant $7.139\times 10^{41}$
Root discriminant \(40.72\)
Ramified primes $13,191$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513)
 
gp: K = bnfinit(y^26 - y^25 - 37*y^24 + 36*y^23 + 656*y^22 - 602*y^21 - 7532*y^20 + 5987*y^19 + 63242*y^18 - 39168*y^17 - 407978*y^16 + 177750*y^15 + 2044391*y^14 - 593805*y^13 - 7863783*y^12 + 1620832*y^11 + 22802242*y^10 - 3804296*y^9 - 48757198*y^8 + 7281003*y^7 + 74301119*y^6 - 12442481*y^5 - 74591007*y^4 + 16397553*y^3 + 41639217*y^2 - 9412671*y - 11716513, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513)
 

\( x^{26} - x^{25} - 37 x^{24} + 36 x^{23} + 656 x^{22} - 602 x^{21} - 7532 x^{20} + 5987 x^{19} + \cdots - 11716513 \) 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:  $[2, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(713943735055873357083710772983186869616893\) \(\medspace = 13^{13}\cdot 191^{12}\) 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:  \(40.72\)
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:  $13^{1/2}191^{1/2}\approx 49.82971001320397$
Ramified primes:   \(13\), \(191\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{7}a^{21}+\frac{2}{7}a^{20}-\frac{2}{7}a^{19}-\frac{1}{7}a^{18}+\frac{3}{7}a^{17}-\frac{1}{7}a^{15}+\frac{3}{7}a^{14}-\frac{3}{7}a^{13}-\frac{3}{7}a^{12}-\frac{2}{7}a^{11}-\frac{3}{7}a^{10}-\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{2}{7}$, $\frac{1}{7}a^{22}+\frac{1}{7}a^{20}+\frac{3}{7}a^{19}-\frac{2}{7}a^{18}+\frac{1}{7}a^{17}-\frac{1}{7}a^{16}-\frac{2}{7}a^{15}-\frac{2}{7}a^{14}+\frac{3}{7}a^{13}-\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{23}+\frac{1}{7}a^{20}+\frac{2}{7}a^{18}+\frac{3}{7}a^{17}-\frac{2}{7}a^{16}-\frac{1}{7}a^{15}-\frac{3}{7}a^{12}+\frac{1}{7}a^{11}+\frac{3}{7}a^{10}-\frac{2}{7}a^{9}+\frac{1}{7}a^{8}-\frac{3}{7}a^{7}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{2}{7}a^{2}-\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{407030407}a^{24}+\frac{26500680}{407030407}a^{23}-\frac{26746959}{407030407}a^{22}-\frac{16747158}{407030407}a^{21}-\frac{809895}{3602039}a^{20}-\frac{168841075}{407030407}a^{19}+\frac{121204474}{407030407}a^{18}+\frac{2485405}{8306743}a^{17}-\frac{72052441}{407030407}a^{16}-\frac{152047013}{407030407}a^{15}+\frac{61556484}{407030407}a^{14}+\frac{4123975}{58147201}a^{13}+\frac{16250571}{407030407}a^{12}+\frac{179770040}{407030407}a^{11}-\frac{414085}{8306743}a^{10}-\frac{23961327}{407030407}a^{9}-\frac{94350002}{407030407}a^{8}-\frac{2417505}{407030407}a^{7}+\frac{41840655}{407030407}a^{6}+\frac{21041534}{407030407}a^{5}+\frac{5485625}{21422653}a^{4}-\frac{85395049}{407030407}a^{3}-\frac{73695659}{407030407}a^{2}+\frac{26168305}{407030407}a-\frac{17597504}{407030407}$, $\frac{1}{44\!\cdots\!83}a^{25}+\frac{28\!\cdots\!51}{44\!\cdots\!83}a^{24}+\frac{25\!\cdots\!00}{63\!\cdots\!69}a^{23}+\frac{28\!\cdots\!50}{44\!\cdots\!83}a^{22}-\frac{15\!\cdots\!12}{33\!\cdots\!51}a^{21}+\frac{24\!\cdots\!90}{63\!\cdots\!69}a^{20}-\frac{27\!\cdots\!74}{91\!\cdots\!67}a^{19}+\frac{19\!\cdots\!32}{40\!\cdots\!53}a^{18}-\frac{12\!\cdots\!35}{44\!\cdots\!83}a^{17}-\frac{11\!\cdots\!99}{44\!\cdots\!83}a^{16}+\frac{11\!\cdots\!56}{44\!\cdots\!83}a^{15}+\frac{21\!\cdots\!68}{44\!\cdots\!83}a^{14}+\frac{18\!\cdots\!46}{44\!\cdots\!83}a^{13}+\frac{11\!\cdots\!79}{63\!\cdots\!69}a^{12}+\frac{12\!\cdots\!22}{44\!\cdots\!83}a^{11}-\frac{13\!\cdots\!67}{44\!\cdots\!83}a^{10}+\frac{20\!\cdots\!75}{44\!\cdots\!83}a^{9}-\frac{14\!\cdots\!10}{44\!\cdots\!83}a^{8}-\frac{19\!\cdots\!12}{44\!\cdots\!83}a^{7}-\frac{44\!\cdots\!73}{44\!\cdots\!83}a^{6}+\frac{17\!\cdots\!96}{44\!\cdots\!83}a^{5}-\frac{22\!\cdots\!82}{44\!\cdots\!83}a^{4}-\frac{20\!\cdots\!08}{44\!\cdots\!83}a^{3}+\frac{51\!\cdots\!86}{44\!\cdots\!83}a^{2}+\frac{67\!\cdots\!06}{39\!\cdots\!91}a+\frac{12\!\cdots\!79}{44\!\cdots\!83}$ 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:  $13$
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 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513)
 
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 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513, 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 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513);
 
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 - x^25 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513);
 
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

$D_{26}$ (as 26T3):

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 solvable group of order 52
The 16 conjugacy class representatives for $D_{26}$
Character table for $D_{26}$

Intermediate fields

\(\Q(\sqrt{13}) \), 13.1.48551226272641.1

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]
 

Sibling fields

Degree 26 sibling: deg 26
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $26$ ${\href{/padicField/3.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/7.2.0.1}{2} }^{13}$ ${\href{/padicField/11.2.0.1}{2} }^{13}$ R ${\href{/padicField/17.13.0.1}{13} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{13}$ ${\href{/padicField/23.13.0.1}{13} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{12}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{13}$ ${\href{/padicField/37.2.0.1}{2} }^{13}$ ${\href{/padicField/41.2.0.1}{2} }^{13}$ ${\href{/padicField/43.13.0.1}{13} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{13}$ ${\href{/padicField/53.2.0.1}{2} }^{12}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $26$

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
\(13\) Copy content Toggle raw display Deg $26$$2$$13$$13$
\(191\) Copy content Toggle raw display $\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
$\Q_{191}$$x + 172$$1$$1$$0$Trivial$[\ ]$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$
191.2.1.2$x^{2} + 191$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
1.191.2t1.a.a$1$ $ 191 $ \(\Q(\sqrt{-191}) \) $C_2$ (as 2T1) $1$ $-1$
1.2483.2t1.a.a$1$ $ 13 \cdot 191 $ \(\Q(\sqrt{-2483}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.32279.26t3.a.d$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.f$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.e$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.e$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.32279.26t3.a.f$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.b$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.c$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.191.13t2.a.d$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.a$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.32279.26t3.a.c$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.191.13t2.a.a$2$ $ 191 $ 13.1.48551226272641.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.32279.26t3.a.b$2$ $ 13^{2} \cdot 191 $ 26.2.713943735055873357083710772983186869616893.1 $D_{26}$ (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 *.