Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4)
gp: K = bnfinit(y^14 - y^13 - 10*y^12 + 5*y^11 + 77*y^10 - 56*y^9 - 309*y^8 + 293*y^7 + 550*y^6 - 549*y^5 - 389*y^4 + 352*y^3 + 60*y^2 - 36*y - 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4)
\( x^{14} - x^{13} - 10 x^{12} + 5 x^{11} + 77 x^{10} - 56 x^{9} - 309 x^{8} + 293 x^{7} + 550 x^{6} + \cdots - 4 \)
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: | $[10, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(13025427850756358144\)
\(\medspace = 2^{20}\cdot 29\cdot 809^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.19\) | 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: | $2^{13/6}29^{1/2}809^{1/2}\approx 687.7093310587969$ | ||
| Ramified primes: |
\(2\), \(29\), \(809\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{4}a^{7}+\frac{1}{8}a^{6}+\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2167736}a^{13}+\frac{16433}{2167736}a^{12}-\frac{94187}{2167736}a^{11}-\frac{105649}{2167736}a^{10}-\frac{37505}{541934}a^{9}-\frac{85485}{1083868}a^{8}-\frac{335433}{2167736}a^{7}-\frac{494915}{2167736}a^{6}-\frac{358055}{2167736}a^{5}-\frac{1040915}{2167736}a^{4}-\frac{531845}{1083868}a^{3}-\frac{14501}{541934}a^{2}-\frac{64713}{270967}a+\frac{104057}{541934}$
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: | $11$ | 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: |
$\frac{144149}{1083868}a^{13}+\frac{7003}{1083868}a^{12}-\frac{311049}{270967}a^{11}-\frac{290667}{541934}a^{10}+\frac{2193075}{270967}a^{9}+\frac{94497}{270967}a^{8}-\frac{31870275}{1083868}a^{7}+\frac{9528105}{1083868}a^{6}+\frac{21261925}{541934}a^{5}-\frac{5749037}{270967}a^{4}-\frac{1606116}{270967}a^{3}+\frac{7185687}{541934}a^{2}-\frac{2989227}{270967}a+\frac{876142}{270967}$, $\frac{10376}{270967}a^{13}+\frac{11293}{1083868}a^{12}-\frac{438273}{1083868}a^{11}-\frac{339069}{1083868}a^{10}+\frac{1550625}{541934}a^{9}+\frac{885359}{541934}a^{8}-\frac{6537561}{541934}a^{7}-\frac{4634197}{1083868}a^{6}+\frac{25941293}{1083868}a^{5}+\frac{10868099}{1083868}a^{4}-\frac{9123037}{541934}a^{3}-\frac{7379703}{541934}a^{2}-\frac{317863}{270967}a+\frac{922583}{270967}$, $\frac{5197}{541934}a^{13}-\frac{80183}{2167736}a^{12}-\frac{123437}{541934}a^{11}+\frac{498057}{2167736}a^{10}+\frac{1001987}{541934}a^{9}-\frac{1415429}{1083868}a^{8}-\frac{10796893}{1083868}a^{7}+\frac{13601353}{2167736}a^{6}+\frac{14279805}{541934}a^{5}-\frac{26484067}{2167736}a^{4}-\frac{26824059}{1083868}a^{3}+\frac{7595075}{1083868}a^{2}+\frac{1007523}{541934}a-\frac{131806}{270967}$, $\frac{11439}{541934}a^{13}+\frac{516857}{2167736}a^{12}-\frac{40301}{541934}a^{11}-\frac{4630803}{2167736}a^{10}-\frac{157868}{270967}a^{9}+\frac{3850526}{270967}a^{8}+\frac{2729903}{1083868}a^{7}-\frac{110902119}{2167736}a^{6}-\frac{1146338}{270967}a^{5}+\frac{164781753}{2167736}a^{4}+\frac{6136757}{1083868}a^{3}-\frac{9303148}{270967}a^{2}+\frac{509663}{541934}a+\frac{70935}{541934}$, $\frac{57081}{1083868}a^{13}+\frac{97643}{541934}a^{12}-\frac{302867}{1083868}a^{11}-\frac{902893}{541934}a^{10}+\frac{1162349}{1083868}a^{9}+\frac{10576615}{1083868}a^{8}-\frac{1406721}{1083868}a^{7}-\frac{17089219}{541934}a^{6}-\frac{10477259}{1083868}a^{5}+\frac{11471050}{270967}a^{4}+\frac{28080269}{1083868}a^{3}-\frac{14612769}{1083868}a^{2}-\frac{7016611}{541934}a-\frac{1192881}{541934}$, $\frac{408715}{2167736}a^{13}+\frac{124125}{541934}a^{12}-\frac{3693487}{2167736}a^{11}-\frac{2902045}{1083868}a^{10}+\frac{12229679}{1083868}a^{9}+\frac{15741225}{1083868}a^{8}-\frac{99376933}{2167736}a^{7}-\frac{20698981}{541934}a^{6}+\frac{185123661}{2167736}a^{5}+\frac{21911935}{541934}a^{4}-\frac{63456449}{1083868}a^{3}-\frac{4521463}{541934}a^{2}+\frac{2353778}{270967}a+\frac{286602}{270967}$, $\frac{125413}{1083868}a^{13}+\frac{103897}{541934}a^{12}-\frac{1916369}{2167736}a^{11}-\frac{4633781}{2167736}a^{10}+\frac{1456423}{270967}a^{9}+\frac{3191129}{270967}a^{8}-\frac{9906335}{541934}a^{7}-\frac{20182045}{541934}a^{6}+\frac{48302143}{2167736}a^{5}+\frac{123371243}{2167736}a^{4}-\frac{1749935}{541934}a^{3}-\frac{31778709}{1083868}a^{2}+\frac{337493}{541934}a+\frac{600788}{270967}$, $\frac{98781}{1083868}a^{13}-\frac{23781}{270967}a^{12}-\frac{1823039}{2167736}a^{11}+\frac{631239}{2167736}a^{10}+\frac{6833785}{1083868}a^{9}-\frac{1158450}{270967}a^{8}-\frac{6169619}{270967}a^{7}+\frac{5403963}{270967}a^{6}+\frac{68053893}{2167736}a^{5}-\frac{58521825}{2167736}a^{4}-\frac{12765683}{1083868}a^{3}+\frac{12910507}{1083868}a^{2}-\frac{64712}{270967}a-\frac{278628}{270967}$, $\frac{7533}{1083868}a^{13}+\frac{228837}{1083868}a^{12}+\frac{32837}{2167736}a^{11}-\frac{4112181}{2167736}a^{10}-\frac{626069}{541934}a^{9}+\frac{14079425}{1083868}a^{8}+\frac{1206973}{270967}a^{7}-\frac{52256373}{1083868}a^{6}-\frac{16032647}{2167736}a^{5}+\frac{162369935}{2167736}a^{4}+\frac{6090951}{541934}a^{3}-\frac{20395189}{541934}a^{2}-\frac{3034221}{541934}a+\frac{720601}{541934}$, $\frac{92329}{1083868}a^{13}-\frac{74519}{2167736}a^{12}-\frac{1450019}{2167736}a^{11}-\frac{196455}{1083868}a^{10}+\frac{5242899}{1083868}a^{9}-\frac{559723}{541934}a^{8}-\frac{16007245}{1083868}a^{7}+\frac{9593733}{2167736}a^{6}+\frac{26679345}{2167736}a^{5}+\frac{11315139}{1083868}a^{4}-\frac{1161133}{541934}a^{3}-\frac{25801393}{1083868}a^{2}+\frac{4891465}{541934}a+\frac{958503}{541934}$, $\frac{32150}{270967}a^{13}+\frac{12167}{1083868}a^{12}-\frac{2343369}{2167736}a^{11}-\frac{1164941}{2167736}a^{10}+\frac{4203705}{541934}a^{9}+\frac{573091}{541934}a^{8}-\frac{32608275}{1083868}a^{7}+\frac{5434135}{1083868}a^{6}+\frac{109924123}{2167736}a^{5}-\frac{31516129}{2167736}a^{4}-\frac{17557451}{541934}a^{3}+\frac{9890955}{1083868}a^{2}+\frac{2367453}{541934}a+\frac{295872}{270967}$
| 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: | \( 134389.922333 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{2}\cdot 134389.922333 \cdot 1}{2\cdot\sqrt{13025427850756358144}}\cr\approx \mathstrut & 0.752662713452 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4)
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 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4, 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 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4);
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 - 10*x^12 + 5*x^11 + 77*x^10 - 56*x^9 - 309*x^8 + 293*x^7 + 550*x^6 - 549*x^5 - 389*x^4 + 352*x^3 + 60*x^2 - 36*x - 4);
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
| 7.7.670188544.2 |
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 | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.14.0.1}{14} }$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.14.0.1}{14} }$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.8.8 | $x^{4} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
| 2.4.8.8 | $x^{4} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(29\)
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(809\)
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{809}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |