Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*x - 1)
gp: K = bnfinit(y^14 - 3*y^13 + 8*y^12 - 17*y^11 + 29*y^10 - 36*y^9 + 38*y^8 - 23*y^7 - 3*y^6 + 35*y^5 - 46*y^4 + 42*y^3 - 21*y^2 + 8*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*x - 1)
\( x^{14} - 3 x^{13} + 8 x^{12} - 17 x^{11} + 29 x^{10} - 36 x^{9} + 38 x^{8} - 23 x^{7} - 3 x^{6} + \cdots - 1 \)
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: |
\(1278022980913757\)
\(\medspace = 13^{4}\cdot 109^{4}\cdot 317\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.00\) | 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: | $13^{1/2}109^{1/2}317^{1/2}\approx 670.2156369408282$ | ||
| Ramified primes: |
\(13\), \(109\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{317}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{3}{13}a^{11}-\frac{1}{13}a^{10}+\frac{4}{13}a^{8}+\frac{1}{13}a^{7}+\frac{1}{13}a^{6}-\frac{5}{13}a^{5}+\frac{5}{13}a^{4}+\frac{5}{13}a^{3}+\frac{5}{13}a^{2}+\frac{2}{13}a-\frac{1}{13}$, $\frac{1}{2561}a^{13}+\frac{87}{2561}a^{12}-\frac{633}{2561}a^{11}-\frac{448}{2561}a^{10}-\frac{1088}{2561}a^{9}-\frac{638}{2561}a^{8}+\frac{930}{2561}a^{7}+\frac{937}{2561}a^{6}-\frac{974}{2561}a^{5}+\frac{828}{2561}a^{4}-\frac{1174}{2561}a^{3}+\frac{565}{2561}a^{2}+\frac{791}{2561}a+\frac{475}{2561}$
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 $1$
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: |
$\frac{1299}{2561}a^{13}-\frac{4399}{2561}a^{12}+\frac{11240}{2561}a^{11}-\frac{24048}{2561}a^{10}+\frac{41336}{2561}a^{9}-\frac{51203}{2561}a^{8}+\frac{50892}{2561}a^{7}-\frac{29650}{2561}a^{6}-\frac{9745}{2561}a^{5}+\frac{55703}{2561}a^{4}-\frac{70969}{2561}a^{3}+\frac{54679}{2561}a^{2}-\frac{21713}{2561}a+\frac{4552}{2561}$, $\frac{228}{2561}a^{13}-\frac{1046}{2561}a^{12}+\frac{3032}{2561}a^{11}-\frac{6993}{2561}a^{10}+\frac{13158}{2561}a^{9}-\frac{18990}{2561}a^{8}+\frac{22132}{2561}a^{7}-\frac{17248}{2561}a^{6}+\frac{5266}{2561}a^{5}+\frac{12666}{2561}a^{4}-\frac{26347}{2561}a^{3}+\frac{29532}{2561}a^{2}-\frac{17637}{2561}a+\frac{6254}{2561}$, $\frac{105}{2561}a^{13}-\frac{518}{2561}a^{12}+\frac{1894}{2561}a^{11}-\frac{4094}{2561}a^{10}+\frac{8688}{2561}a^{9}-\frac{13406}{2561}a^{8}+\frac{1253}{197}a^{7}-\frac{13708}{2561}a^{6}+\frac{7459}{2561}a^{5}+\frac{611}{197}a^{4}-\frac{1375}{197}a^{3}+\frac{21304}{2561}a^{2}-\frac{10520}{2561}a+\frac{3186}{2561}$, $\frac{233}{2561}a^{13}-\frac{217}{2561}a^{12}+\frac{1049}{2561}a^{11}-\frac{1944}{2561}a^{10}+\frac{2596}{2561}a^{9}-\frac{2677}{2561}a^{8}+\frac{4127}{2561}a^{7}+\frac{636}{2561}a^{6}-\frac{1574}{2561}a^{5}+\frac{3410}{2561}a^{4}-\frac{2076}{2561}a^{3}+\frac{3595}{2561}a^{2}+\frac{2472}{2561}a+\frac{552}{2561}$, $\frac{1571}{2561}a^{13}-\frac{3981}{2561}a^{12}+\frac{10060}{2561}a^{11}-\frac{20218}{2561}a^{10}+\frac{32232}{2561}a^{9}-\frac{33452}{2561}a^{8}+\frac{32189}{2561}a^{7}-\frac{815}{197}a^{6}-\frac{17588}{2561}a^{5}+\frac{46883}{2561}a^{4}-\frac{37864}{2561}a^{3}+\frac{30665}{2561}a^{2}-\frac{9274}{2561}a+\frac{5899}{2561}$, $\frac{590}{2561}a^{13}-\frac{2254}{2561}a^{12}+\frac{473}{197}a^{11}-\frac{13539}{2561}a^{10}+\frac{23940}{2561}a^{9}-\frac{32458}{2561}a^{8}+\frac{34136}{2561}a^{7}-\frac{23198}{2561}a^{6}+\frac{580}{2561}a^{5}+\frac{28525}{2561}a^{4}-\frac{46303}{2561}a^{3}+\frac{39820}{2561}a^{2}-\frac{22067}{2561}a+\frac{6026}{2561}$, $\frac{11}{197}a^{13}-\frac{1349}{2561}a^{12}+\frac{3844}{2561}a^{11}-\frac{9298}{2561}a^{10}+\frac{1428}{197}a^{9}-\frac{28588}{2561}a^{8}+\frac{32126}{2561}a^{7}-\frac{28337}{2561}a^{6}+\frac{9059}{2561}a^{5}+\frac{18722}{2561}a^{4}-\frac{42196}{2561}a^{3}+\frac{42577}{2561}a^{2}-\frac{27151}{2561}a+\frac{4885}{2561}$
| 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: | \( 71.6776654006 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 71.6776654006 \cdot 1}{2\cdot\sqrt{1278022980913757}}\cr\approx \mathstrut & 0.246731003882 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*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 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*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 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*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 - 3*x^13 + 8*x^12 - 17*x^11 + 29*x^10 - 36*x^9 + 38*x^8 - 23*x^7 - 3*x^6 + 35*x^5 - 46*x^4 + 42*x^3 - 21*x^2 + 8*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
$C_2^7.\GL(3,2)$ (as 14T51):
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 21504 |
| The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
| Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
| 7.3.2007889.1 |
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 14 sibling: | data not computed |
| Degree 28 siblings: | data not computed |
| Degree 42 siblings: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
|
\(109\)
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(317\)
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{317}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |