Properties

Label 26.2.173...056.1
Degree $26$
Signature $[2, 12]$
Discriminant $1.733\times 10^{51}$
Root discriminant \(93.48\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{26}$ (as 26T96)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x - 4)
 
gp: K = bnfinit(y^26 - 2*y - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 2*x - 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 2*x - 4)
 

\( x^{26} - 2x - 4 \) 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:   \(1732793616956939114861181647068785799342044590637056\) \(\medspace = 2^{24}\cdot 71593\cdot 343928521\cdot 41\!\cdots\!97\) 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:  \(93.48\)
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:  $2^{24/25}71593^{1/2}343928521^{1/2}4194577183575531075991971526697^{1/2}\approx 1.9769799626305397e+22$
Ramified primes:   \(2\), \(71593\), \(343928521\), \(41945\!\cdots\!26697\) 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:  $\Q(\sqrt{10328\!\cdots\!33241}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{2}a^{25}$ 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

Trivial group, which has order $1$ (assuming GRH)

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:   $a+1$, $a^{25}+2a^{12}-1$, $a^{25}+a^{24}-a^{22}+a^{20}-2a^{18}-a^{17}+a^{16}-3a^{14}-2a^{13}+a^{12}+a^{11}-2a^{10}-2a^{9}+a^{8}+3a^{7}-2a^{5}+2a^{4}+6a^{3}+2a^{2}-2a-1$, $a^{25}-a^{23}+2a^{21}-a^{20}-2a^{19}+2a^{18}+a^{17}-3a^{16}-a^{15}+4a^{14}-2a^{13}-3a^{12}+3a^{11}+2a^{10}-3a^{9}-a^{8}+4a^{7}-a^{6}-3a^{5}+2a^{3}-a^{2}-4a-1$, $4a^{25}-3a^{24}+2a^{23}-a^{22}+a^{20}-2a^{19}+4a^{18}-4a^{17}+5a^{16}-6a^{15}+6a^{14}-7a^{13}+6a^{12}-4a^{11}+3a^{10}-2a^{9}+2a^{8}-2a^{6}+6a^{5}-7a^{4}+7a^{3}-7a^{2}+7a-17$, $a^{25}+2a^{24}-2a^{23}-2a^{22}+3a^{21}+2a^{20}-4a^{19}+4a^{17}-a^{16}-6a^{15}+3a^{14}+6a^{13}-4a^{12}-6a^{11}+5a^{10}+5a^{9}-7a^{8}-2a^{7}+9a^{6}+a^{5}-10a^{4}+9a^{2}-4a-9$, $a^{25}+a^{24}-a^{22}+a^{20}-a^{19}-a^{18}+2a^{17}+a^{16}-2a^{15}-2a^{14}+3a^{13}+a^{12}-4a^{11}+2a^{10}+4a^{9}-5a^{8}-3a^{7}+5a^{6}+3a^{5}-5a^{4}+5a^{2}-5a-7$, $a^{25}-5a^{24}-a^{23}-6a^{22}-11a^{21}-6a^{20}-13a^{19}-14a^{18}-8a^{17}-14a^{16}-12a^{15}-3a^{14}-9a^{13}+9a^{11}+3a^{10}+16a^{9}+22a^{8}+17a^{7}+30a^{6}+31a^{5}+19a^{4}+33a^{3}+21a^{2}+9a+13$, $2a^{25}-2a^{24}+4a^{23}-4a^{22}+2a^{21}-2a^{20}-a^{19}+4a^{18}-3a^{17}+4a^{16}-3a^{15}-a^{14}+2a^{12}-2a^{11}+7a^{10}-8a^{9}+3a^{8}-4a^{7}-a^{6}+7a^{5}-3a^{4}+5a^{3}-5a^{2}-2a-5$, $2a^{24}+2a^{23}-a^{21}+a^{20}-a^{19}+2a^{17}+3a^{16}+a^{15}+2a^{14}+a^{13}-a^{12}+a^{9}+3a^{8}+5a^{7}+2a^{6}+4a^{5}+2a^{4}-3a^{3}-3a^{2}+4a+3$, $3a^{25}+3a^{24}+5a^{23}+6a^{22}+6a^{21}+7a^{20}+7a^{19}+6a^{18}+5a^{17}+4a^{16}+a^{15}-2a^{14}-2a^{13}-6a^{12}-11a^{11}-12a^{10}-14a^{9}-17a^{8}-16a^{7}-14a^{6}-16a^{5}-14a^{4}-7a^{3}-2a^{2}+3a+5$, $5a^{25}-9a^{24}+17a^{23}-13a^{22}+14a^{21}-13a^{20}-8a^{19}+14a^{18}-12a^{17}+23a^{16}-17a^{15}+8a^{14}-11a^{13}-10a^{12}+19a^{11}-9a^{10}+28a^{9}-32a^{8}+8a^{7}-8a^{6}-15a^{5}+36a^{4}-14a^{3}+27a^{2}-40a-15$, $1156a^{25}+97a^{24}-1033a^{23}-1104a^{22}+696a^{21}+1408a^{20}+601a^{19}-1509a^{18}-1453a^{17}+344a^{16}+2119a^{15}+990a^{14}-1655a^{13}-2287a^{12}+96a^{11}+2931a^{10}+1675a^{9}-1659a^{8}-3541a^{7}-165a^{6}+3344a^{5}+3116a^{4}-2008a^{3}-4673a^{2}-1434a+2107$ Copy content Toggle raw display (assuming GRH)
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:  \( 6568002949534731.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 6568002949534731.0 \cdot 1}{2\cdot\sqrt{1732793616956939114861181647068785799342044590637056}}\cr\approx \mathstrut & 1.19467103798012 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 2*x - 4)
 
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 - 4, 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 - 4);
 
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 - 4);
 
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

$S_{26}$ (as 26T96):

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 non-solvable group of order 403291461126605635584000000
The 2436 conjugacy class representatives for $S_{26}$ are not computed
Character table for $S_{26}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ $21{,}\,{\href{/padicField/7.5.0.1}{5} }$ $23{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $21{,}\,{\href{/padicField/13.5.0.1}{5} }$ $22{,}\,{\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.6.0.1}{6} }$ $25{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $22{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
Deg $25$$25$$1$$24$
\(71593\) Copy content Toggle raw display $\Q_{71593}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{71593}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{71593}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(343928521\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $22$$1$$22$$0$22T1$[\ ]^{22}$
\(419\!\cdots\!697\) Copy content Toggle raw display $\Q_{41\!\cdots\!97}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $19$$1$$19$$0$$C_{19}$$[\ ]^{19}$