Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2)
gp: K = bnfinit(y^14 - 2*y^13 - 8*y^12 + 4*y^11 + 6*y^10 + 23*y^9 + 83*y^8 + 42*y^7 + 105*y^6 - 46*y^5 + 162*y^4 + 113*y^3 + 42*y^2 + 3*y + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2)
\( x^{14} - 2 x^{13} - 8 x^{12} + 4 x^{11} + 6 x^{10} + 23 x^{9} + 83 x^{8} + 42 x^{7} + 105 x^{6} + \cdots + 2 \)
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: |
\(23410829236277007605\)
\(\medspace = 5\cdot 181^{4}\cdot 257^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(24.18\) | 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: | $5^{1/2}181^{1/2}257^{1/2}\approx 482.2706708892839$ | ||
| Ramified primes: |
\(5\), \(181\), \(257\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{103873886553326}a^{13}-\frac{12877817265428}{51936943276663}a^{12}-\frac{5889591824858}{51936943276663}a^{11}+\frac{6380991809305}{103873886553326}a^{10}+\frac{13592262353767}{103873886553326}a^{9}+\frac{12434123185463}{103873886553326}a^{8}+\frac{9995439081481}{51936943276663}a^{7}-\frac{23114674406163}{51936943276663}a^{6}-\frac{9856486853083}{103873886553326}a^{5}-\frac{703757909733}{103873886553326}a^{4}+\frac{20297878707225}{51936943276663}a^{3}+\frac{16953401367243}{51936943276663}a^{2}+\frac{2858469601728}{51936943276663}a+\frac{19835646932265}{51936943276663}$
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: |
$\frac{3387969991483}{103873886553326}a^{13}-\frac{1281624428168}{51936943276663}a^{12}-\frac{18572390383496}{51936943276663}a^{11}-\frac{7669617558949}{51936943276663}a^{10}+\frac{47057105739431}{103873886553326}a^{9}+\frac{83631655393783}{103873886553326}a^{8}+\frac{373500919634525}{103873886553326}a^{7}+\frac{466332917849397}{103873886553326}a^{6}+\frac{221352483812231}{51936943276663}a^{5}+\frac{359646028051435}{103873886553326}a^{4}+\frac{135302036889886}{51936943276663}a^{3}+\frac{12\!\cdots\!65}{103873886553326}a^{2}+\frac{213688668287711}{103873886553326}a+\frac{44285303062620}{51936943276663}$, $\frac{5488612984285}{103873886553326}a^{13}-\frac{6194743012730}{51936943276663}a^{12}-\frac{41066520166627}{103873886553326}a^{11}+\frac{16836627201310}{51936943276663}a^{10}+\frac{26072204605223}{103873886553326}a^{9}+\frac{56618714490124}{51936943276663}a^{8}+\frac{214809963699384}{51936943276663}a^{7}+\frac{63617430763687}{51936943276663}a^{6}+\frac{254304478269413}{51936943276663}a^{5}-\frac{398715491764721}{103873886553326}a^{4}+\frac{978219155124203}{103873886553326}a^{3}+\frac{214365444058505}{51936943276663}a^{2}+\frac{10090027981531}{103873886553326}a+\frac{26830232550054}{51936943276663}$, $\frac{215109923311}{51936943276663}a^{13}+\frac{1965622687373}{51936943276663}a^{12}-\frac{16013745691663}{103873886553326}a^{11}-\frac{15458056407582}{51936943276663}a^{10}+\frac{23141003043668}{51936943276663}a^{9}+\frac{29060490572509}{103873886553326}a^{8}+\frac{126903509558109}{103873886553326}a^{7}+\frac{341287885516307}{103873886553326}a^{6}-\frac{6960094382279}{51936943276663}a^{5}+\frac{168104475532112}{51936943276663}a^{4}-\frac{463610390894159}{103873886553326}a^{3}+\frac{991726004908215}{103873886553326}a^{2}+\frac{101491641502661}{51936943276663}a-\frac{1522960884987}{51936943276663}$, $\frac{4681269551437}{103873886553326}a^{13}-\frac{5435593671034}{51936943276663}a^{12}-\frac{17178272743124}{51936943276663}a^{11}+\frac{31499641362823}{103873886553326}a^{10}+\frac{18049787236121}{103873886553326}a^{9}+\frac{97380430243113}{103873886553326}a^{8}+\frac{182746417387438}{51936943276663}a^{7}+\frac{30628048631068}{51936943276663}a^{6}+\frac{458687471464903}{103873886553326}a^{5}-\frac{338225463046239}{103873886553326}a^{4}+\frac{397670466913873}{51936943276663}a^{3}+\frac{173340727481023}{51936943276663}a^{2}+\frac{4095753800804}{51936943276663}a+\frac{39913261360125}{51936943276663}$, $\frac{13063478106178}{51936943276663}a^{13}-\frac{29543764292621}{51936943276663}a^{12}-\frac{98624006556395}{51936943276663}a^{11}+\frac{163016845505885}{103873886553326}a^{10}+\frac{72506289032038}{51936943276663}a^{9}+\frac{275773126737125}{51936943276663}a^{8}+\frac{19\!\cdots\!23}{103873886553326}a^{7}+\frac{485488513072759}{103873886553326}a^{6}+\frac{23\!\cdots\!39}{103873886553326}a^{5}-\frac{992111820874255}{51936943276663}a^{4}+\frac{21\!\cdots\!87}{51936943276663}a^{3}+\frac{19\!\cdots\!63}{103873886553326}a^{2}+\frac{43421695958269}{103873886553326}a-\frac{189926790522350}{51936943276663}$, $\frac{7831702158913}{103873886553326}a^{13}-\frac{17270619606313}{103873886553326}a^{12}-\frac{60822234279311}{103873886553326}a^{11}+\frac{46336150987531}{103873886553326}a^{10}+\frac{26972604014770}{51936943276663}a^{9}+\frac{166287902985029}{103873886553326}a^{8}+\frac{297925171619874}{51936943276663}a^{7}+\frac{181902020470467}{103873886553326}a^{6}+\frac{616510057012015}{103873886553326}a^{5}-\frac{331366914371093}{51936943276663}a^{4}+\frac{650214205391491}{51936943276663}a^{3}+\frac{580392871496949}{103873886553326}a^{2}+\frac{6025039364659}{51936943276663}a+\frac{7038406054115}{51936943276663}$, $\frac{922927998797}{103873886553326}a^{13}-\frac{714139656921}{103873886553326}a^{12}-\frac{4562517875066}{51936943276663}a^{11}-\frac{2422620925445}{51936943276663}a^{10}+\frac{1819650757512}{51936943276663}a^{9}+\frac{7422570713652}{51936943276663}a^{8}+\frac{53410236767403}{51936943276663}a^{7}+\frac{164862202637701}{103873886553326}a^{6}+\frac{114100163347013}{51936943276663}a^{5}+\frac{124129020058047}{51936943276663}a^{4}+\frac{306481970402933}{103873886553326}a^{3}+\frac{349979650567319}{103873886553326}a^{2}+\frac{140040113351883}{103873886553326}a+\frac{15752292983746}{51936943276663}$
| 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: | \( 39247.4267217 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 39247.4267217 \cdot 1}{2\cdot\sqrt{23410829236277007605}}\cr\approx \mathstrut & 0.998187195426 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2)
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 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2, 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 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2);
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 - 8*x^12 + 4*x^11 + 6*x^10 + 23*x^9 + 83*x^8 + 42*x^7 + 105*x^6 - 46*x^5 + 162*x^4 + 113*x^3 + 42*x^2 + 3*x + 2);
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 $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
| 7.7.2163831289.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{3}{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | R | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(181\)
| 181.2.0.1 | $x^{2} + 177 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 181.4.2.2 | $x^{4} - 32037 x^{2} + 65522$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 181.4.2.2 | $x^{4} - 32037 x^{2} + 65522$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 181.4.0.1 | $x^{4} + 6 x^{2} + 105 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(257\)
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{257}$ | $x$ | $1$ | $1$ | $0$ | 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 $4$ | $2$ | $2$ | $2$ |