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Properties

Label 14.2.203...592.1
Degree $14$
Signature $[2, 6]$
Discriminant $2.033\times 10^{22}$
Root discriminant \(39.21\)
Ramified primes $2,13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\PGL(2,13)$ (as 14T39)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445)
 
gp: K = bnfinit(y^14 - 2*y^13 + 13*y^12 - 52*y^11 + 104*y^10 - 364*y^9 + 806*y^8 - 1300*y^7 + 2990*y^6 - 4212*y^5 + 4680*y^4 - 7020*y^3 + 5265*y^2 - 1802*y + 445, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445)
 

\( x^{14} - 2 x^{13} + 13 x^{12} - 52 x^{11} + 104 x^{10} - 364 x^{9} + 806 x^{8} - 1300 x^{7} + 2990 x^{6} + \cdots + 445 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(20325604337285010030592\) \(\medspace = 2^{26}\cdot 13^{13}\) 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:  \(39.21\)
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:  $2^{13/6}13^{167/156}\approx 69.93946361168216$
Ramified primes:   \(2\), \(13\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{40}a^{8}-\frac{3}{40}a^{6}+\frac{1}{20}a^{5}+\frac{1}{5}a^{4}-\frac{1}{20}a^{3}-\frac{19}{40}a^{2}+\frac{1}{10}a-\frac{3}{8}$, $\frac{1}{40}a^{9}+\frac{1}{20}a^{7}-\frac{3}{40}a^{6}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{9}{40}a^{3}-\frac{2}{5}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{80}a^{10}-\frac{1}{80}a^{9}-\frac{1}{16}a^{7}-\frac{3}{80}a^{6}-\frac{1}{20}a^{5}-\frac{13}{80}a^{4}+\frac{17}{80}a^{3}-\frac{9}{20}a^{2}-\frac{23}{80}a+\frac{7}{16}$, $\frac{1}{160}a^{11}+\frac{1}{160}a^{9}-\frac{1}{160}a^{8}+\frac{3}{80}a^{7}-\frac{3}{32}a^{6}+\frac{7}{160}a^{5}+\frac{1}{5}a^{4}-\frac{5}{32}a^{3}-\frac{47}{160}a^{2}-\frac{11}{80}a-\frac{13}{32}$, $\frac{1}{160}a^{12}-\frac{1}{160}a^{10}+\frac{1}{160}a^{9}-\frac{1}{80}a^{8}-\frac{1}{32}a^{7}-\frac{3}{160}a^{6}+\frac{3}{20}a^{5}+\frac{17}{160}a^{4}+\frac{3}{32}a^{3}+\frac{21}{80}a^{2}+\frac{29}{160}a+\frac{1}{16}$, $\frac{1}{160}a^{13}-\frac{1}{160}a^{10}+\frac{1}{160}a^{9}-\frac{1}{80}a^{8}-\frac{7}{160}a^{7}-\frac{17}{160}a^{6}-\frac{1}{4}a^{5}-\frac{3}{32}a^{4}+\frac{3}{32}a^{3}-\frac{11}{80}a^{2}-\frac{5}{16}a-\frac{11}{32}$ 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:  $7$
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:   $\frac{1}{160}a^{13}-\frac{1}{80}a^{12}+\frac{7}{160}a^{11}-\frac{31}{160}a^{10}+\frac{11}{80}a^{9}-\frac{21}{160}a^{8}+\frac{53}{160}a^{7}+\frac{119}{40}a^{6}-\frac{1031}{160}a^{5}+\frac{259}{32}a^{4}-\frac{1599}{80}a^{3}+\frac{3517}{160}a^{2}-\frac{107}{16}a+\frac{27}{16}$, $\frac{1}{80}a^{13}-\frac{7}{160}a^{12}+\frac{33}{160}a^{11}-\frac{131}{160}a^{10}+\frac{41}{20}a^{9}-\frac{911}{160}a^{8}+\frac{2047}{160}a^{7}-\frac{917}{40}a^{6}+\frac{6783}{160}a^{5}-\frac{9477}{160}a^{4}+\frac{1371}{20}a^{3}-\frac{2517}{32}a^{2}+\frac{7807}{160}a-\frac{321}{32}$, $\frac{1}{80}a^{13}-\frac{1}{80}a^{12}+\frac{11}{80}a^{11}-\frac{41}{80}a^{10}+\frac{29}{40}a^{9}-\frac{57}{16}a^{8}+\frac{551}{80}a^{7}-\frac{361}{40}a^{6}+\frac{2277}{80}a^{5}-\frac{2631}{80}a^{4}+\frac{1211}{40}a^{3}-\frac{1103}{16}a^{2}+\frac{181}{5}a+\frac{87}{16}$, $\frac{1}{160}a^{13}-\frac{3}{160}a^{12}+\frac{3}{40}a^{11}-\frac{7}{20}a^{10}+\frac{27}{40}a^{9}-\frac{37}{20}a^{8}+\frac{73}{16}a^{7}-\frac{91}{16}a^{6}+\frac{447}{40}a^{5}-\frac{157}{10}a^{4}+\frac{73}{8}a^{3}-\frac{117}{10}a^{2}+\frac{541}{160}a-\frac{35}{32}$, $\frac{1}{40}a^{13}+\frac{13}{40}a^{11}-\frac{51}{80}a^{10}+\frac{21}{16}a^{9}-\frac{51}{8}a^{8}+\frac{563}{80}a^{7}-\frac{1441}{80}a^{6}+\frac{1481}{40}a^{5}-\frac{2257}{80}a^{4}+\frac{923}{16}a^{3}-\frac{423}{8}a^{2}+\frac{299}{16}a-\frac{83}{16}$, $\frac{1}{40}a^{13}+\frac{1}{160}a^{12}+\frac{13}{40}a^{11}-\frac{99}{160}a^{10}+\frac{183}{160}a^{9}-\frac{539}{80}a^{8}+\frac{1053}{160}a^{7}-\frac{2937}{160}a^{6}+\frac{1667}{40}a^{5}-\frac{4389}{160}a^{4}+\frac{10841}{160}a^{3}-\frac{5277}{80}a^{2}+\frac{3743}{160}a-7$, $\frac{1}{40}a^{13}-\frac{1}{16}a^{12}+\frac{13}{40}a^{11}-\frac{117}{80}a^{10}+\frac{233}{80}a^{9}-\frac{193}{20}a^{8}+\frac{1857}{80}a^{7}-\frac{2813}{80}a^{6}+\frac{3259}{40}a^{5}-\frac{9679}{80}a^{4}+\frac{9739}{80}a^{3}-\frac{3837}{20}a^{2}+\frac{2349}{16}a-\frac{257}{8}$ 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:  \( 1523520.8954 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 1523520.8954 \cdot 1}{2\cdot\sqrt{20325604337285010030592}}\cr\approx \mathstrut & 1.3150306863 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445)
 
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^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445, 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^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445);
 
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^14 - 2*x^13 + 13*x^12 - 52*x^11 + 104*x^10 - 364*x^9 + 806*x^8 - 1300*x^7 + 2990*x^6 - 4212*x^5 + 4680*x^4 - 7020*x^3 + 5265*x^2 - 1802*x + 445);
 
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

$\PGL(2,13)$ (as 14T39):

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 2184
The 15 conjugacy class representatives for $\PGL(2,13)$
Character table for $\PGL(2,13)$

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]
 

Sibling fields

Degree 28 sibling: data not computed
Degree 42 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 R ${\href{/padicField/3.7.0.1}{7} }^{2}$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.3.0.1}{3} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.7.0.1}{7} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{7}$ ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.7.0.1}{7} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.6.10.3$x^{6} + 2 x^{5} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.8.16.8$x^{8} + 4 x^{7} + 10 x^{6} + 24 x^{5} + 51 x^{4} + 48 x^{3} - 18 x^{2} + 63$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$[\ ]$
13.13.13.5$x^{13} + 130 x + 13$$13$$1$$13$$F_{13}$$[13/12]_{12}$

Additional information

This field is unusual in that it has non-solvable Galois group and is ramified at only two small primes.