Properties

Label 26.0.173...056.1
Degree $26$
Signature $[0, 13]$
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:  $[0, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1732793613976706876091650397068785799342044590637056\) \(\medspace = -\,2^{24}\cdot 3\cdot 7\cdot 331\cdot 347\cdot 1627\cdot 49019\cdot 1249111\cdot 102017147\cdot 4213315416817943\) 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:  $2^{24/25}3^{1/2}7^{1/2}331^{1/2}347^{1/2}1627^{1/2}49019^{1/2}1249111^{1/2}102017147^{1/2}4213315416817943^{1/2}\approx 1.976979960930435e+22$
Ramified primes:   \(2\), \(3\), \(7\), \(331\), \(347\), \(1627\), \(49019\), \(1249111\), \(102017147\), \(4213315416817943\) 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:  $\Q(\sqrt{-10328\!\cdots\!01991}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $12$
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:   $a^{25}-a^{24}+a^{22}-a^{21}+a^{19}-a^{18}+a^{16}-a^{15}+a^{13}-a^{12}+a^{10}-a^{9}+a^{7}-a^{6}+a^{4}-a^{3}+1$, $4a^{25}-2a^{24}+4a^{23}+2a^{22}-3a^{21}+4a^{20}-2a^{19}-4a^{18}+3a^{17}-6a^{16}-3a^{15}+3a^{14}-8a^{13}+2a^{12}+3a^{11}-7a^{10}+10a^{9}-3a^{7}+15a^{6}-7a^{5}+3a^{4}+12a^{3}-16a^{2}+10a-7$, $3a^{25}-2a^{23}-3a^{22}-5a^{21}-5a^{20}+3a^{18}+2a^{17}+4a^{16}+7a^{15}+3a^{14}-2a^{13}-3a^{12}-5a^{11}-8a^{10}-5a^{9}+2a^{8}+2a^{7}+3a^{6}+11a^{5}+10a^{4}-2a^{3}-4a^{2}-2a-15$, $2a^{25}+7a^{24}-4a^{23}-4a^{22}+a^{21}-2a^{20}+7a^{19}+2a^{18}-8a^{17}+3a^{16}-2a^{15}-2a^{14}+8a^{13}-6a^{12}+a^{11}+6a^{10}-13a^{9}+3a^{8}+5a^{7}-4a^{6}+14a^{5}-12a^{4}-17a^{3}+16a^{2}-a+5$, $2a^{25}-2a^{24}-3a^{23}+3a^{21}+4a^{20}-6a^{19}-4a^{17}+12a^{16}-7a^{15}+4a^{14}-13a^{13}+11a^{12}+5a^{10}-10a^{9}-4a^{8}+9a^{7}+2a^{6}+4a^{5}-16a^{4}+3a^{3}+4a^{2}+13a-13$, $4a^{25}+5a^{24}+5a^{23}+2a^{22}+7a^{21}+6a^{20}+9a^{18}+4a^{17}+a^{16}+8a^{15}+2a^{14}+2a^{13}+3a^{12}+4a^{11}-a^{10}-a^{9}+6a^{8}-9a^{7}+a^{6}+3a^{5}-16a^{4}+3a^{3}-4a^{2}-14a-11$, $a^{25}-a^{24}-a^{23}+a^{21}-3a^{19}+2a^{18}-2a^{15}-2a^{14}+5a^{13}-5a^{12}-2a^{10}+a^{9}+2a^{8}-9a^{7}+4a^{6}-3a^{3}-3a^{2}+7a-3$, $5a^{24}-18a^{23}+20a^{22}-7a^{21}-2a^{20}+21a^{19}-16a^{18}+13a^{17}-10a^{16}-2a^{15}+a^{14}-20a^{13}+34a^{12}-28a^{11}+16a^{10}+27a^{9}-47a^{8}+45a^{7}-28a^{6}-14a^{5}+18a^{4}-37a^{3}+39a^{2}-42a+45$, $2a^{25}-10a^{24}+a^{23}+5a^{22}+9a^{21}-a^{20}+6a^{19}-10a^{18}-14a^{17}+11a^{16}-18a^{15}+27a^{14}+a^{13}+a^{12}+5a^{11}-20a^{10}-7a^{9}-2a^{8}+12a^{7}-4a^{6}+43a^{5}-36a^{4}+13a^{3}-14a^{2}-43a+45$, $3a^{25}+8a^{24}+11a^{23}+8a^{22}+5a^{21}+3a^{20}-a^{19}-7a^{18}-6a^{17}-2a^{16}-3a^{15}-2a^{14}+3a^{13}+4a^{12}-2a^{11}-6a^{10}-6a^{9}-9a^{8}-9a^{7}-a^{6}+3a^{5}+10a^{4}+19a^{3}+24a^{2}+13a-1$, $11a^{25}-7a^{24}-6a^{23}+13a^{22}-9a^{21}+3a^{20}-9a^{19}-a^{18}+11a^{17}-23a^{16}+3a^{15}-8a^{14}+a^{13}+a^{12}-36a^{11}+24a^{10}-11a^{9}-19a^{8}+3a^{7}-27a^{6}+32a^{5}-31a^{4}-17a^{3}+21a^{2}-26a+9$, $125a^{25}+83a^{24}+85a^{23}+51a^{22}-20a^{21}-24a^{20}-89a^{19}-137a^{18}-112a^{17}-164a^{16}-136a^{15}-70a^{14}-92a^{13}+16a^{12}+74a^{11}+70a^{10}+197a^{9}+179a^{8}+151a^{7}+231a^{6}+86a^{5}+62a^{4}+59a^{3}-148a^{2}-116a-417$ 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:  \( 6042375351216186.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 6042375351216186.0 \cdot 1}{2\cdot\sqrt{1732793613976706876091650397068785799342044590637056}}\cr\approx \mathstrut & 1.72640455695576 \end{aligned}\] (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 R ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ R $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.6.0.1}{6} }$ $25{,}\,{\href{/padicField/23.1.0.1}{1} }$ $23{,}\,{\href{/padicField/29.3.0.1}{3} }$ $22{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ $26$ ${\href{/padicField/43.13.0.1}{13} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ $20{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.9.0.1}{9} }^{2}{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ $25{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.20.0.1$x^{20} + 2 x^{13} + x^{11} + x^{10} + x^{9} + x^{8} + 2 x^{5} + 2 x^{4} + 2 x^{3} + x + 2$$1$$20$$0$20T1$[\ ]^{20}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} + 6 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} + 21$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $21$$1$$21$$0$$C_{21}$$[\ ]^{21}$
\(331\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(347\) Copy content Toggle raw display $\Q_{347}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(1627\) Copy content Toggle raw display $\Q_{1627}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1627}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1627}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(49019\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(1249111\) Copy content Toggle raw display $\Q_{1249111}$$x$$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}$
\(102017147\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(4213315416817943\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $13$$1$$13$$0$$C_{13}$$[\ ]^{13}$