Properties

Label 26.2.117...632.1
Degree $26$
Signature $[2, 12]$
Discriminant $1.172\times 10^{43}$
Root discriminant \(45.34\)
Ramified primes $2,3,263$
Class number not computed
Class group not computed
Galois group $D_{26}$ (as 26T3)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323)
 
gp: K = bnfinit(y^26 - 18*y^24 + 99*y^22 - 81*y^20 - 1377*y^18 + 9963*y^16 - 16038*y^14 - 56862*y^12 + 774198*y^10 - 747954*y^8 + 2657205*y^6 + 17714700*y^4 + 17006112*y^2 - 1594323, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323)
 

\( x^{26} - 18 x^{24} + 99 x^{22} - 81 x^{20} - 1377 x^{18} + 9963 x^{16} - 16038 x^{14} - 56862 x^{12} + 774198 x^{10} - 747954 x^{8} + 2657205 x^{6} + 17714700 x^{4} + \cdots - 1594323 \) 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:   \(11717236087522974259320261669002899863109632\) \(\medspace = 2^{26}\cdot 3^{13}\cdot 263^{12}\) 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:  \(45.34\)
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:  $2\cdot 3^{1/2}263^{1/2}\approx 56.178287620752556$
Ramified primes:   \(2\), \(3\), \(263\) 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:  \(\Q(\sqrt{3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{10935}a^{14}+\frac{2}{3645}a^{12}+\frac{1}{405}a^{8}+\frac{2}{135}a^{6}-\frac{2}{15}a^{2}+\frac{1}{5}$, $\frac{1}{10935}a^{15}+\frac{2}{3645}a^{13}+\frac{1}{405}a^{9}+\frac{2}{135}a^{7}-\frac{2}{15}a^{3}+\frac{1}{5}a$, $\frac{1}{32805}a^{16}+\frac{1}{3645}a^{12}+\frac{1}{1215}a^{10}+\frac{1}{135}a^{6}-\frac{2}{45}a^{4}-\frac{2}{5}$, $\frac{1}{32805}a^{17}+\frac{1}{3645}a^{13}+\frac{1}{1215}a^{11}+\frac{1}{135}a^{7}-\frac{2}{45}a^{5}-\frac{2}{5}a$, $\frac{1}{98415}a^{18}-\frac{1}{3645}a^{12}+\frac{1}{135}a^{6}-\frac{1}{5}$, $\frac{1}{98415}a^{19}-\frac{1}{3645}a^{13}+\frac{1}{135}a^{7}-\frac{1}{5}a$, $\frac{1}{295245}a^{20}+\frac{2}{3645}a^{12}+\frac{2}{405}a^{8}+\frac{2}{135}a^{6}+\frac{2}{15}a^{2}+\frac{1}{5}$, $\frac{1}{295245}a^{21}+\frac{2}{3645}a^{13}+\frac{2}{405}a^{9}+\frac{2}{135}a^{7}+\frac{2}{15}a^{3}+\frac{1}{5}a$, $\frac{1}{4428675}a^{22}-\frac{2}{1476225}a^{20}-\frac{1}{492075}a^{18}+\frac{1}{164025}a^{16}+\frac{1}{54675}a^{14}-\frac{4}{18225}a^{12}-\frac{7}{6075}a^{10}+\frac{2}{2025}a^{8}+\frac{1}{75}a^{6}-\frac{2}{25}a^{2}-\frac{9}{25}$, $\frac{1}{4428675}a^{23}-\frac{2}{1476225}a^{21}-\frac{1}{492075}a^{19}+\frac{1}{164025}a^{17}+\frac{1}{54675}a^{15}-\frac{4}{18225}a^{13}-\frac{7}{6075}a^{11}+\frac{2}{2025}a^{9}+\frac{1}{75}a^{7}-\frac{2}{25}a^{3}-\frac{9}{25}a$, $\frac{1}{535033607590125}a^{24}-\frac{15094789}{178344535863375}a^{22}-\frac{4680527}{59448178621125}a^{20}-\frac{67253512}{19816059540375}a^{18}-\frac{74073196}{6605353180125}a^{16}-\frac{49449866}{2201784393375}a^{14}-\frac{316860844}{733928131125}a^{12}-\frac{321924899}{244642710375}a^{10}-\frac{5408989}{2329930575}a^{8}-\frac{3650606}{9060841125}a^{6}+\frac{89946638}{3020280375}a^{4}+\frac{44935846}{1006760125}a^{2}+\frac{214299153}{1006760125}$, $\frac{1}{535033607590125}a^{25}-\frac{15094789}{178344535863375}a^{23}-\frac{4680527}{59448178621125}a^{21}-\frac{67253512}{19816059540375}a^{19}-\frac{74073196}{6605353180125}a^{17}-\frac{49449866}{2201784393375}a^{15}-\frac{316860844}{733928131125}a^{13}-\frac{321924899}{244642710375}a^{11}-\frac{5408989}{2329930575}a^{9}-\frac{3650606}{9060841125}a^{7}+\frac{89946638}{3020280375}a^{5}+\frac{44935846}{1006760125}a^{3}+\frac{214299153}{1006760125}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $5$

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 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323)
 
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 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323, 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 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323);
 
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 - 18*x^24 + 99*x^22 - 81*x^20 - 1377*x^18 + 9963*x^16 - 16038*x^14 - 56862*x^12 + 774198*x^10 - 747954*x^8 + 2657205*x^6 + 17714700*x^4 + 17006112*x^2 - 1594323);
 
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

\(\Q(\sqrt{3}) \), 13.1.330928743953809.1

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: 26.0.3081633091018542230201228818947762663997833216.1
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 R R ${\href{/padicField/5.2.0.1}{2} }^{13}$ ${\href{/padicField/7.2.0.1}{2} }^{13}$ ${\href{/padicField/11.13.0.1}{13} }^{2}$ ${\href{/padicField/13.13.0.1}{13} }^{2}$ $26$ ${\href{/padicField/19.2.0.1}{2} }^{13}$ ${\href{/padicField/23.13.0.1}{13} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{13}$ $26$ ${\href{/padicField/37.13.0.1}{13} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{13}$ $26$ ${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{13}$ ${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(2\) Copy content Toggle raw display Deg $26$$2$$13$$26$
\(3\) Copy content Toggle raw display Deg $26$$2$$13$$13$
\(263\) Copy content Toggle raw display $\Q_{263}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{263}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3156.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 263 $ \(\Q(\sqrt{-789}) \) $C_2$ (as 2T1) $1$ $-1$
1.263.2t1.a.a$1$ $ 263 $ \(\Q(\sqrt{-263}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 2.37872.26t3.b.d$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.f$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.e$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.37872.26t3.b.a$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.37872.26t3.b.e$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.b$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.c$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.263.13t2.a.d$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.37872.26t3.b.c$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.37872.26t3.b.f$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (as 26T3) $1$ $0$
* 2.263.13t2.a.a$2$ $ 263 $ 13.1.330928743953809.1 $D_{13}$ (as 13T2) $1$ $0$
* 2.37872.26t3.b.b$2$ $ 2^{4} \cdot 3^{2} \cdot 263 $ 26.2.11717236087522974259320261669002899863109632.1 $D_{26}$ (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 *.