Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16)
gp: K = bnfinit(y^14 - 4*y^13 + 52*y^10 - 26*y^8 - 208*y^7 + 260*y^4 + 169*y^2 - 4*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16)
\( x^{14} - 4x^{13} + 52x^{10} - 26x^{8} - 208x^{7} + 260x^{4} + 169x^{2} - 4x + 16 \)
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: |
\(536561726709678316933\)
\(\medspace = 11^{6}\cdot 13^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.25\) | 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: | $11^{1/2}13^{13/14}\approx 35.8981702227843$ | ||
| Ramified primes: |
\(11\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{4}a^{7}-\frac{1}{4}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{9}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{1}{8}a^{6}+\frac{1}{8}a^{5}+\frac{1}{16}a^{4}-\frac{5}{16}a^{3}+\frac{7}{16}a^{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{8}-\frac{1}{16}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{7}{16}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{9}-\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{7}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{15232}a^{13}+\frac{313}{15232}a^{12}+\frac{213}{15232}a^{11}-\frac{71}{15232}a^{10}+\frac{393}{15232}a^{9}+\frac{821}{15232}a^{8}+\frac{335}{15232}a^{7}-\frac{91}{2176}a^{6}-\frac{15}{2176}a^{5}+\frac{5}{2176}a^{4}-\frac{69}{15232}a^{3}-\frac{929}{15232}a^{2}-\frac{39}{3808}a+\frac{3}{952}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
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 $2$)
| 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: |
$\frac{3}{896}a^{13}+\frac{43}{896}a^{12}-\frac{201}{896}a^{11}-\frac{45}{896}a^{10}+\frac{115}{896}a^{9}+\frac{2575}{896}a^{8}+\frac{893}{896}a^{7}-\frac{129}{128}a^{6}-\frac{1189}{128}a^{5}-\frac{345}{128}a^{4}-\frac{1047}{896}a^{3}+\frac{6397}{896}a^{2}-\frac{61}{224}a+\frac{65}{56}$, $\frac{275}{15232}a^{13}-\frac{557}{15232}a^{12}-\frac{2353}{15232}a^{11}+\frac{2371}{15232}a^{10}+\frac{9067}{15232}a^{9}+\frac{22999}{15232}a^{8}-\frac{12595}{15232}a^{7}+\frac{135}{2176}a^{6}-\frac{8205}{2176}a^{5}+\frac{1239}{2176}a^{4}-\frac{85615}{15232}a^{3}+\frac{36789}{15232}a^{2}-\frac{8821}{3808}a+\frac{825}{952}$, $\frac{1}{476}a^{13}-\frac{11}{119}a^{12}+\frac{545}{952}a^{11}-\frac{975}{952}a^{10}+\frac{191}{952}a^{9}-\frac{2523}{952}a^{8}+\frac{1006}{119}a^{7}-\frac{57}{68}a^{6}+\frac{565}{136}a^{5}-\frac{2387}{136}a^{4}+\frac{6407}{952}a^{3}-\frac{9117}{952}a^{2}+\frac{439}{476}a-\frac{107}{119}$, $\frac{225}{15232}a^{13}-\frac{975}{15232}a^{12}+\frac{1277}{15232}a^{11}-\frac{3599}{15232}a^{10}+\frac{10361}{15232}a^{9}+\frac{37}{15232}a^{8}+\frac{31583}{15232}a^{7}-\frac{4563}{2176}a^{6}-\frac{247}{2176}a^{5}-\frac{10707}{2176}a^{4}+\frac{30171}{15232}a^{3}-\frac{52897}{15232}a^{2}+\frac{4553}{3808}a-\frac{277}{952}$, $\frac{545}{15232}a^{13}-\frac{2679}{15232}a^{12}+\frac{893}{15232}a^{11}+\frac{6049}{15232}a^{10}+\frac{19025}{15232}a^{9}-\frac{19035}{15232}a^{8}-\frac{47809}{15232}a^{7}-\frac{7707}{2176}a^{6}+\frac{2025}{2176}a^{5}+\frac{9117}{2176}a^{4}+\frac{83299}{15232}a^{3}+\frac{110591}{15232}a^{2}+\frac{6353}{3808}a+\frac{2587}{952}$, $\frac{171}{2176}a^{13}-\frac{877}{2176}a^{12}+\frac{791}{2176}a^{11}+\frac{99}{2176}a^{10}+\frac{7635}{2176}a^{9}-\frac{7305}{2176}a^{8}-\frac{5819}{2176}a^{7}-\frac{20799}{2176}a^{6}+\frac{14957}{2176}a^{5}+\frac{9521}{2176}a^{4}+\frac{18393}{2176}a^{3}+\frac{3525}{2176}a^{2}+\frac{1083}{544}a-\frac{31}{136}$, $\frac{319}{15232}a^{13}-\frac{1065}{15232}a^{12}-\frac{597}{15232}a^{11}+\frac{199}{15232}a^{10}+\frac{13031}{15232}a^{9}+\frac{8667}{15232}a^{8}+\frac{241}{15232}a^{7}-\frac{3733}{2176}a^{6}+\frac{111}{2176}a^{5}+\frac{2139}{2176}a^{4}+\frac{29397}{15232}a^{3}-\frac{1231}{15232}a^{2}+\frac{1839}{3808}a+\frac{5}{952}$
| 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: | \( 277701.766085 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 277701.766085 \cdot 1}{2\cdot\sqrt{536561726709678316933}}\cr\approx \mathstrut & 1.47529204984 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16)
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 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16, 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 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16);
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 - 4*x^13 + 52*x^10 - 26*x^8 - 208*x^7 + 260*x^4 + 169*x^2 - 4*x + 16);
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
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 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), 7.1.6424482779.1 |
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
| Galois closure: | deg 28 |
| Degree 14 sibling: | deg 14 |
| 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 | ${\href{/padicField/2.2.0.1}{2} }^{7}$ | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.2.0.1}{2} }^{7}$ | R | R | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.14.13.1 | $x^{14} + 13$ | $14$ | $1$ | $13$ | $D_{14}$ | $[\ ]_{14}^{2}$ |