Properties

Label 26.0.120...736.1
Degree $26$
Signature $[0, 13]$
Discriminant $-1.202\times 10^{42}$
Root discriminant \(41.54\)
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 - 4*x + 4)
 
gp: K = bnfinit(y^26 - 4*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 4*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 4*x + 4)
 

\( x^{26} - 4x + 4 \) 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:   \(-1201507202980564696903888106396747076468736\) \(\medspace = -\,2^{26}\cdot 11\cdot 5351\cdot 322710411593\cdot 942553212724915163\) 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:  \(41.54\)
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:  not computed
Ramified primes:   \(2\), \(11\), \(5351\), \(322710411593\), \(942553212724915163\) 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:  $\Q(\sqrt{-17903\!\cdots\!13399}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$, $\frac{1}{18}a^{25}-\frac{2}{9}a^{24}-\frac{1}{9}a^{23}-\frac{1}{18}a^{22}+\frac{2}{9}a^{21}+\frac{1}{9}a^{20}+\frac{1}{18}a^{19}-\frac{2}{9}a^{18}-\frac{1}{9}a^{17}-\frac{1}{18}a^{16}+\frac{2}{9}a^{15}+\frac{1}{9}a^{14}+\frac{1}{18}a^{13}-\frac{2}{9}a^{12}-\frac{1}{9}a^{11}+\frac{4}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{2}{9}a^{6}-\frac{1}{9}a^{5}+\frac{4}{9}a^{4}+\frac{2}{9}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$ 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:   $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{13}-a$, $\frac{2}{9}a^{25}+\frac{1}{9}a^{24}+\frac{1}{18}a^{23}+\frac{5}{18}a^{22}+\frac{7}{18}a^{21}+\frac{4}{9}a^{20}+\frac{2}{9}a^{19}+\frac{1}{9}a^{18}+\frac{1}{18}a^{17}+\frac{5}{18}a^{16}+\frac{7}{18}a^{15}+\frac{4}{9}a^{14}+\frac{2}{9}a^{13}+\frac{1}{9}a^{12}-\frac{4}{9}a^{11}-\frac{2}{9}a^{10}-\frac{1}{9}a^{9}+\frac{4}{9}a^{8}+\frac{2}{9}a^{7}+\frac{1}{9}a^{6}-\frac{4}{9}a^{5}-\frac{2}{9}a^{4}-\frac{1}{9}a^{3}+\frac{4}{9}a^{2}+\frac{2}{9}a-\frac{7}{9}$, $\frac{1}{3}a^{25}+\frac{1}{6}a^{24}+\frac{1}{3}a^{23}+\frac{1}{6}a^{22}+\frac{1}{3}a^{21}+\frac{1}{6}a^{20}+\frac{1}{3}a^{19}+\frac{1}{6}a^{18}+\frac{1}{3}a^{17}+\frac{1}{6}a^{16}+\frac{1}{3}a^{15}+\frac{1}{6}a^{14}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a-\frac{5}{3}$, $\frac{1}{6}a^{25}+\frac{1}{3}a^{24}+\frac{1}{6}a^{23}+\frac{1}{3}a^{22}+\frac{1}{6}a^{21}-\frac{1}{6}a^{20}+\frac{1}{6}a^{19}-\frac{1}{6}a^{18}+\frac{1}{6}a^{17}-\frac{1}{6}a^{16}+\frac{1}{6}a^{15}-\frac{1}{6}a^{14}+\frac{1}{6}a^{13}-\frac{2}{3}a^{12}+\frac{2}{3}a^{11}-\frac{2}{3}a^{10}+\frac{2}{3}a^{9}-\frac{2}{3}a^{8}+\frac{2}{3}a^{7}-\frac{2}{3}a^{6}+\frac{2}{3}a^{5}-\frac{2}{3}a^{4}+\frac{2}{3}a^{3}-\frac{2}{3}a^{2}+\frac{2}{3}a-\frac{4}{3}$, $\frac{28}{9}a^{25}+\frac{55}{18}a^{24}+\frac{59}{18}a^{23}+\frac{26}{9}a^{22}+\frac{53}{18}a^{21}+\frac{49}{18}a^{20}+\frac{47}{18}a^{19}+\frac{23}{9}a^{18}+\frac{41}{18}a^{17}+\frac{43}{18}a^{16}+\frac{35}{18}a^{15}+\frac{20}{9}a^{14}+\frac{29}{18}a^{13}+\frac{14}{9}a^{12}+\frac{16}{9}a^{11}+\frac{8}{9}a^{10}+\frac{13}{9}a^{9}+\frac{11}{9}a^{8}+\frac{1}{9}a^{7}+\frac{14}{9}a^{6}-\frac{2}{9}a^{5}+\frac{8}{9}a^{4}+\frac{4}{9}a^{3}-\frac{7}{9}a^{2}+\frac{10}{9}a-\frac{116}{9}$, $\frac{1}{18}a^{25}+\frac{5}{18}a^{24}+\frac{7}{18}a^{23}-\frac{1}{18}a^{22}-\frac{5}{18}a^{21}+\frac{1}{9}a^{20}+\frac{1}{18}a^{19}-\frac{2}{9}a^{18}-\frac{1}{9}a^{17}-\frac{1}{18}a^{16}-\frac{5}{18}a^{15}+\frac{1}{9}a^{14}+\frac{1}{18}a^{13}-\frac{2}{9}a^{12}-\frac{1}{9}a^{11}+\frac{4}{9}a^{10}+\frac{2}{9}a^{9}+\frac{1}{9}a^{8}-\frac{4}{9}a^{7}-\frac{2}{9}a^{6}+\frac{8}{9}a^{5}+\frac{4}{9}a^{4}-\frac{7}{9}a^{3}+\frac{1}{9}a^{2}-\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{21}+\frac{1}{2}a^{18}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+a^{7}-a^{6}-a^{2}$, $\frac{4}{9}a^{25}+\frac{2}{9}a^{24}+\frac{11}{18}a^{23}+\frac{1}{18}a^{22}-\frac{2}{9}a^{21}-\frac{1}{9}a^{20}-\frac{5}{9}a^{19}+\frac{2}{9}a^{18}+\frac{1}{9}a^{17}+\frac{1}{18}a^{16}-\frac{2}{9}a^{15}-\frac{1}{9}a^{14}-\frac{5}{9}a^{13}+\frac{2}{9}a^{12}+\frac{1}{9}a^{11}-\frac{4}{9}a^{10}+\frac{7}{9}a^{9}-\frac{10}{9}a^{8}+\frac{4}{9}a^{7}+\frac{2}{9}a^{6}+\frac{1}{9}a^{5}+\frac{5}{9}a^{4}-\frac{2}{9}a^{3}-\frac{1}{9}a^{2}+\frac{4}{9}a-\frac{5}{9}$, $\frac{1}{6}a^{25}-\frac{1}{6}a^{24}-\frac{1}{3}a^{23}-\frac{1}{6}a^{22}+\frac{1}{6}a^{21}+\frac{1}{3}a^{20}+\frac{1}{6}a^{19}+\frac{1}{3}a^{18}+\frac{1}{6}a^{17}+\frac{5}{6}a^{16}+\frac{1}{6}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{4}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{4}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{4}{3}a^{3}+\frac{4}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{10}{9}a^{25}+\frac{5}{9}a^{24}+\frac{5}{18}a^{23}+\frac{7}{18}a^{22}+\frac{4}{9}a^{21}+\frac{2}{9}a^{20}+\frac{1}{9}a^{19}+\frac{5}{9}a^{18}-\frac{2}{9}a^{17}-\frac{1}{9}a^{16}+\frac{4}{9}a^{15}+\frac{13}{18}a^{14}-\frac{8}{9}a^{13}-\frac{4}{9}a^{12}+\frac{7}{9}a^{11}-\frac{1}{9}a^{10}-\frac{5}{9}a^{9}-\frac{7}{9}a^{8}+\frac{10}{9}a^{7}-\frac{13}{9}a^{6}-\frac{2}{9}a^{5}+\frac{8}{9}a^{4}-\frac{5}{9}a^{3}-\frac{7}{9}a^{2}-\frac{8}{9}a-\frac{17}{9}$, $\frac{1}{9}a^{25}+\frac{5}{9}a^{24}+\frac{5}{18}a^{23}-\frac{1}{9}a^{22}-\frac{1}{18}a^{21}+\frac{2}{9}a^{20}+\frac{11}{18}a^{19}+\frac{1}{18}a^{18}-\frac{13}{18}a^{17}-\frac{1}{9}a^{16}-\frac{5}{9}a^{15}-\frac{7}{9}a^{14}-\frac{7}{18}a^{13}-\frac{4}{9}a^{12}-\frac{2}{9}a^{11}-\frac{10}{9}a^{10}-\frac{14}{9}a^{9}+\frac{2}{9}a^{8}+\frac{10}{9}a^{7}-\frac{4}{9}a^{6}-\frac{2}{9}a^{5}-\frac{1}{9}a^{4}+\frac{4}{9}a^{3}+\frac{11}{9}a^{2}-\frac{8}{9}a+\frac{10}{9}$ 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:  \( 80495462904.86972 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 80495462904.86972 \cdot 1}{2\cdot\sqrt{1201507202980564696903888106396747076468736}}\cr\approx \mathstrut & 0.873406673313651 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 4*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 - 4*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 - 4*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 - 4*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 $20{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ $22{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R $25{,}\,{\href{/padicField/13.1.0.1}{1} }$ $15{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ $17{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ $18{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ $24{,}\,{\href{/padicField/41.2.0.1}{2} }$ $17{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ $17{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 Deg $26$$26$$1$$26$
\(11\) Copy content Toggle raw display 11.2.1.1$x^{2} + 22$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} + 8 x^{2} + 10 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.10.0.1$x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.0.1$x^{10} + 7 x^{5} + 8 x^{4} + 10 x^{3} + 6 x^{2} + 6 x + 2$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(5351\) Copy content Toggle raw display $\Q_{5351}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(322710411593\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(942553212724915163\) Copy content Toggle raw display $\Q_{942553212724915163}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{942553212724915163}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$