Properties

Label 26.2.273...125.1
Degree $26$
Signature $[2, 12]$
Discriminant $2.734\times 10^{46}$
Root discriminant \(61.10\)
Ramified primes $5,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times F_{13}$ (as 26T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 5)
 
gp: K = bnfinit(y^26 - 5, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 5);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 5)
 

\( x^{26} - 5 \) 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:   \(27338662801166956620931510927975177764892578125\) \(\medspace = 5^{25}\cdot 13^{26}\) 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:  \(61.10\)
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:  $5^{25/26}13^{167/156}\approx 73.21114221183463$
Ramified primes:   \(5\), \(13\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{11}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{12}$ 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:   $\frac{1}{2}a^{13}-\frac{1}{2}$, $\frac{1}{2}a^{13}-a^{12}+a^{11}-a^{10}+a^{9}-a^{8}+a^{7}-a^{6}+a^{5}-a^{4}+a^{3}-a^{2}+a-\frac{1}{2}$, $\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-a^{14}+\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{7}+a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{2}a^{3}+1$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+a^{20}-a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}+a^{13}-\frac{3}{2}a^{12}+\frac{1}{2}a^{11}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}-2a^{7}+2a^{6}-\frac{3}{2}a^{5}+\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{5}{2}a^{2}+\frac{5}{2}a-3$, $\frac{1}{2}a^{23}-\frac{1}{2}a^{22}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-a^{15}+a^{14}-\frac{1}{2}a^{10}+\frac{1}{2}a^{9}+a^{7}-a^{6}-\frac{1}{2}a^{5}+\frac{1}{2}a^{4}+\frac{1}{2}a^{3}-1$, $\frac{1}{2}a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{3}{2}a^{18}-a^{17}-a^{16}-\frac{3}{2}a^{15}-a^{14}-2a^{13}-\frac{3}{2}a^{12}-\frac{1}{2}a^{11}-a^{10}+\frac{1}{2}a^{9}+\frac{1}{2}a^{7}+\frac{5}{2}a^{6}+\frac{3}{2}a^{5}+3a^{4}+3a^{3}+\frac{5}{2}a^{2}+5a+3$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{23}+\frac{1}{2}a^{21}-a^{19}+a^{18}+\frac{1}{2}a^{17}-\frac{3}{2}a^{16}+\frac{1}{2}a^{15}+\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-a^{11}+\frac{5}{2}a^{10}-2a^{9}-\frac{1}{2}a^{8}+a^{7}+a^{5}-\frac{5}{2}a^{4}+\frac{3}{2}a^{3}+\frac{3}{2}a^{2}-3a+\frac{1}{2}$, $\frac{7}{2}a^{25}+3a^{24}+\frac{1}{2}a^{23}-\frac{5}{2}a^{22}-4a^{21}-4a^{20}-2a^{19}+\frac{3}{2}a^{18}+5a^{17}+6a^{16}+4a^{15}-\frac{1}{2}a^{14}-\frac{11}{2}a^{13}-\frac{15}{2}a^{12}-6a^{11}-\frac{3}{2}a^{10}+\frac{9}{2}a^{9}+10a^{8}+10a^{7}+4a^{6}-\frac{9}{2}a^{5}-11a^{4}-13a^{3}-8a^{2}+\frac{3}{2}a+\frac{23}{2}$, $\frac{1}{2}a^{25}-\frac{1}{2}a^{24}-a^{23}-a^{22}-a^{21}+\frac{1}{2}a^{20}+a^{19}+a^{18}+\frac{3}{2}a^{17}-a^{15}-\frac{3}{2}a^{14}-2a^{13}-\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+2a^{10}+3a^{9}+a^{8}+\frac{1}{2}a^{7}-2a^{6}-4a^{5}-\frac{3}{2}a^{4}-a^{3}+2a^{2}+\frac{9}{2}a+3$, $a^{25}+\frac{3}{2}a^{24}-2a^{23}+\frac{1}{2}a^{22}+\frac{3}{2}a^{21}-\frac{3}{2}a^{20}-\frac{1}{2}a^{19}+\frac{3}{2}a^{18}-\frac{1}{2}a^{17}-a^{16}+a^{15}+a^{14}-a^{13}-a^{12}+\frac{5}{2}a^{11}-a^{10}-\frac{7}{2}a^{9}+\frac{9}{2}a^{8}+\frac{1}{2}a^{7}-\frac{11}{2}a^{6}+\frac{11}{2}a^{5}+\frac{5}{2}a^{4}-9a^{3}+5a^{2}+5a-11$, $2a^{25}-2a^{24}+\frac{3}{2}a^{23}-a^{22}-\frac{1}{2}a^{21}+2a^{20}-\frac{5}{2}a^{19}+3a^{18}-3a^{17}+\frac{5}{2}a^{16}-\frac{3}{2}a^{14}+3a^{13}-4a^{12}+5a^{11}-\frac{7}{2}a^{10}+a^{9}+\frac{1}{2}a^{8}-3a^{7}+\frac{11}{2}a^{6}-8a^{5}+5a^{4}-\frac{7}{2}a^{3}+\frac{5}{2}a-9$, $\frac{3}{2}a^{25}-a^{24}+\frac{3}{2}a^{22}-2a^{21}+2a^{19}-\frac{5}{2}a^{18}+\frac{1}{2}a^{17}+2a^{16}-3a^{15}+2a^{14}+\frac{3}{2}a^{13}-\frac{7}{2}a^{12}+3a^{11}+a^{10}-\frac{7}{2}a^{9}+3a^{8}-a^{7}-4a^{6}+\frac{11}{2}a^{5}-\frac{5}{2}a^{4}-5a^{3}+8a^{2}-3a-\frac{7}{2}$, $\frac{3}{2}a^{25}+\frac{9}{2}a^{24}+3a^{23}+5a^{22}+\frac{11}{2}a^{21}+4a^{20}+\frac{13}{2}a^{19}+4a^{18}+4a^{17}+4a^{16}-\frac{5}{2}a^{13}-\frac{15}{2}a^{12}-\frac{9}{2}a^{11}-12a^{10}-9a^{9}-\frac{21}{2}a^{8}-14a^{7}-\frac{17}{2}a^{6}-13a^{5}-9a^{4}-5a^{3}-5a^{2}+3a+\frac{17}{2}$ 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:  \( 9612138497230.408 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 9612138497230.408 \cdot 1}{2\cdot\sqrt{27338662801166956620931510927975177764892578125}}\cr\approx \mathstrut & 0.440169353206470 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^26 - 5)
 
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 - 5, 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 - 5);
 
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 - 5);
 
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

$C_2\times F_{13}$ (as 26T10):

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 312
The 26 conjugacy class representatives for $C_2\times F_{13}$
Character table for $C_2\times F_{13}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 13.1.73944117820374267578125.1

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: data not computed
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 ${\href{/padicField/2.12.0.1}{12} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.6.0.1}{6} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }$ R ${\href{/padicField/7.12.0.1}{12} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.12.0.1}{12} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.12.0.1}{12} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.3.0.1}{3} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.12.0.1}{12} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.12.0.1}{12} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{6}{,}\,{\href{/padicField/47.2.0.1}{2} }$ $26$ ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(5\) Copy content Toggle raw display Deg $26$$26$$1$$25$
\(13\) Copy content Toggle raw display Deg $26$$13$$2$$26$