Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1)
gp: K = bnfinit(y^14 - y^13 + 13*y^12 + 52*y^10 + 78*y^8 + 78*y^6 + 52*y^4 + 13*y^2 + 64*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1)
\( x^{14} - x^{13} + 13x^{12} + 52x^{10} + 78x^{8} + 78x^{6} + 52x^{4} + 13x^{2} + 64x + 1 \)
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: |
\(35632953415471973197\)
\(\medspace = 7^{6}\cdot 13^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.92\) | 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: | $7^{1/2}13^{13/14}\approx 28.636833207042958$ | ||
| Ramified primes: |
\(7\), \(13\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}+\frac{1}{9}a^{9}+\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{1}{3}a^{5}+\frac{4}{9}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}+\frac{2}{9}a+\frac{4}{9}$, $\frac{1}{45}a^{11}+\frac{2}{45}a^{10}+\frac{2}{45}a^{9}-\frac{4}{45}a^{8}-\frac{1}{45}a^{7}-\frac{2}{45}a^{6}-\frac{2}{45}a^{5}-\frac{7}{15}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{16}{45}$, $\frac{1}{135}a^{12}-\frac{7}{135}a^{10}+\frac{2}{135}a^{9}-\frac{13}{135}a^{8}+\frac{2}{27}a^{7}-\frac{1}{45}a^{6}+\frac{13}{135}a^{5}-\frac{2}{135}a^{4}-\frac{11}{27}a^{3}+\frac{28}{135}a^{2}+\frac{4}{45}a-\frac{67}{135}$, $\frac{1}{10219635}a^{13}+\frac{1489}{3406545}a^{12}-\frac{102196}{10219635}a^{11}+\frac{18481}{2043927}a^{10}-\frac{386167}{10219635}a^{9}+\frac{132869}{2043927}a^{8}-\frac{31358}{3406545}a^{7}-\frac{6136}{2043927}a^{6}-\frac{1270568}{10219635}a^{5}-\frac{106808}{2043927}a^{4}+\frac{4319809}{10219635}a^{3}-\frac{48584}{681309}a^{2}-\frac{2016481}{10219635}a-\frac{1137791}{3406545}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{250802}{10219635}a^{13}-\frac{572173}{10219635}a^{12}+\frac{3517036}{10219635}a^{11}-\frac{706076}{2043927}a^{10}+\frac{11531513}{10219635}a^{9}-\frac{3299552}{3406545}a^{8}+\frac{16210439}{10219635}a^{7}-\frac{979556}{2043927}a^{6}+\frac{12053692}{10219635}a^{5}-\frac{366952}{378505}a^{4}+\frac{3708592}{3406545}a^{3}+\frac{665590}{2043927}a^{2}-\frac{2448760}{2043927}a+\frac{12522631}{10219635}$, $\frac{386}{9855}a^{13}-\frac{121}{3285}a^{12}+\frac{995}{1971}a^{11}+\frac{457}{9855}a^{10}+\frac{4013}{1971}a^{9}+\frac{2186}{9855}a^{8}+\frac{2266}{657}a^{7}+\frac{4763}{9855}a^{6}+\frac{8251}{1971}a^{5}-\frac{3281}{9855}a^{4}+\frac{5428}{1971}a^{3}-\frac{562}{1095}a^{2}+\frac{8791}{9855}a+\frac{347}{1095}$, $\frac{7654}{1135515}a^{13}+\frac{14420}{681309}a^{12}+\frac{106448}{3406545}a^{11}+\frac{1267198}{3406545}a^{10}+\frac{221383}{3406545}a^{9}+\frac{3739327}{3406545}a^{8}-\frac{41962}{378505}a^{7}+\frac{908194}{1135515}a^{6}+\frac{1212709}{1135515}a^{5}-\frac{1486912}{3406545}a^{4}+\frac{3461998}{3406545}a^{3}+\frac{146033}{3406545}a^{2}-\frac{29435}{681309}a+\frac{3171677}{3406545}$, $\frac{1225}{75701}a^{13}-\frac{187336}{10219635}a^{12}+\frac{568126}{3406545}a^{11}+\frac{321613}{10219635}a^{10}+\frac{3462334}{10219635}a^{9}-\frac{1002254}{10219635}a^{8}+\frac{1868687}{10219635}a^{7}+\frac{76048}{1135515}a^{6}+\frac{14892056}{10219635}a^{5}-\frac{6425131}{10219635}a^{4}+\frac{5873578}{10219635}a^{3}+\frac{10403273}{10219635}a^{2}+\frac{1060378}{3406545}a-\frac{928429}{2043927}$, $\frac{791}{681309}a^{13}+\frac{237358}{10219635}a^{12}-\frac{1949}{378505}a^{11}+\frac{2732978}{10219635}a^{10}+\frac{364819}{2043927}a^{9}+\frac{7505498}{10219635}a^{8}+\frac{4035388}{10219635}a^{7}+\frac{533008}{1135515}a^{6}+\frac{1626986}{2043927}a^{5}+\frac{3347047}{10219635}a^{4}+\frac{5865197}{10219635}a^{3}-\frac{7133672}{10219635}a^{2}+\frac{286403}{681309}a-\frac{1452469}{10219635}$, $\frac{157756}{10219635}a^{13}-\frac{307361}{10219635}a^{12}+\frac{319903}{2043927}a^{11}-\frac{1585307}{10219635}a^{10}+\frac{1429714}{10219635}a^{9}-\frac{721085}{681309}a^{8}-\frac{1970446}{2043927}a^{7}-\frac{13106918}{10219635}a^{6}+\frac{2442206}{10219635}a^{5}+\frac{278048}{681309}a^{4}-\frac{3760}{681309}a^{3}-\frac{8389382}{10219635}a^{2}-\frac{3259787}{2043927}a-\frac{2430871}{10219635}$, $\frac{151081}{3406545}a^{13}-\frac{29398}{681309}a^{12}+\frac{213212}{378505}a^{11}+\frac{12787}{3406545}a^{10}+\frac{7380067}{3406545}a^{9}-\frac{60896}{227103}a^{8}+\frac{9228577}{3406545}a^{7}-\frac{1562104}{1135515}a^{6}+\frac{527167}{378505}a^{5}-\frac{1748308}{681309}a^{4}-\frac{4207027}{3406545}a^{3}-\frac{7402828}{3406545}a^{2}-\frac{7750352}{3406545}a+\frac{695741}{681309}$
| 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: | \( 45027.4148013 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 45027.4148013 \cdot 1}{2\cdot\sqrt{35632953415471973197}}\cr\approx \mathstrut & 0.928240063449 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1)
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 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1, 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 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1);
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 - x^13 + 13*x^12 + 52*x^10 + 78*x^8 + 78*x^6 + 52*x^4 + 13*x^2 + 64*x + 1);
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
| \(\Q(\sqrt{13}) \), 7.1.1655595487.1 |
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: | 14.0.249430673908303812379.1 |
| 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.2.0.1}{2} }^{6}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | ${\href{/padicField/5.2.0.1}{2} }^{7}$ | R | ${\href{/padicField/11.14.0.1}{14} }$ | R | ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.7.0.1}{7} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(13\)
| 13.14.13.1 | $x^{14} + 13$ | $14$ | $1$ | $13$ | $D_{14}$ | $[\ ]_{14}^{2}$ |