Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*x + 1)
gp: K = bnfinit(y^14 - 3*y^13 + 2*y^12 + 3*y^11 - 7*y^10 + 11*y^9 - 25*y^8 + 33*y^7 - 4*y^6 - 56*y^5 + 93*y^4 - 78*y^3 + 38*y^2 - 10*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 3*x^13 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 3*x^13 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*x + 1)
\( x^{14} - 3 x^{13} + 2 x^{12} + 3 x^{11} - 7 x^{10} + 11 x^{9} - 25 x^{8} + 33 x^{7} - 4 x^{6} - 56 x^{5} + \cdots + 1 \)
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: |
\(321952970703125\)
\(\medspace = 5^{9}\cdot 37^{2}\cdot 347^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.87\) | 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: | $5^{5/6}37^{1/2}347^{1/2}\approx 433.25202677491876$ | ||
| Ramified primes: |
\(5\), \(37\), \(347\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{853}a^{13}+\frac{60}{853}a^{12}+\frac{370}{853}a^{11}+\frac{282}{853}a^{10}-\frac{154}{853}a^{9}-\frac{308}{853}a^{8}+\frac{190}{853}a^{7}+\frac{61}{853}a^{6}-\frac{426}{853}a^{5}+\frac{402}{853}a^{4}-\frac{171}{853}a^{3}+\frac{238}{853}a^{2}-\frac{322}{853}a+\frac{176}{853}$
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: |
$\frac{2870}{853}a^{13}-\frac{6077}{853}a^{12}-\frac{85}{853}a^{11}+\frac{9226}{853}a^{10}-\frac{11215}{853}a^{9}+\frac{20220}{853}a^{8}-\frac{53506}{853}a^{7}+\frac{45414}{853}a^{6}+\frac{35555}{853}a^{5}-\frac{130878}{853}a^{4}+\frac{140450}{853}a^{3}-\frac{84640}{853}a^{2}+\frac{27808}{853}a-\frac{4121}{853}$, $\frac{140}{853}a^{13}+\frac{1576}{853}a^{12}-\frac{3645}{853}a^{11}-\frac{611}{853}a^{10}+\frac{5736}{853}a^{9}-\frac{4735}{853}a^{8}+\frac{8687}{853}a^{7}-\frac{29845}{853}a^{6}+\frac{23954}{853}a^{5}+\frac{25572}{853}a^{4}-\frac{74267}{853}a^{3}+\frac{66587}{853}a^{2}-\frac{28873}{853}a+\frac{5021}{853}$, $\frac{140}{853}a^{13}+\frac{1576}{853}a^{12}-\frac{3645}{853}a^{11}-\frac{611}{853}a^{10}+\frac{5736}{853}a^{9}-\frac{4735}{853}a^{8}+\frac{8687}{853}a^{7}-\frac{29845}{853}a^{6}+\frac{23954}{853}a^{5}+\frac{25572}{853}a^{4}-\frac{74267}{853}a^{3}+\frac{66587}{853}a^{2}-\frac{28873}{853}a+\frac{4168}{853}$, $\frac{2030}{853}a^{13}-\frac{5297}{853}a^{12}+\frac{1313}{853}a^{11}+\frac{7774}{853}a^{10}-\frac{10658}{853}a^{9}+\frac{16216}{853}a^{8}-\frac{42506}{853}a^{7}+\frac{46207}{853}a^{6}+\frac{20634}{853}a^{5}-\frac{112004}{853}a^{4}+\frac{133962}{853}a^{3}-\frac{78987}{853}a^{2}+\frac{22769}{853}a-\frac{980}{853}$, $a-1$, $\frac{717}{853}a^{13}-\frac{3895}{853}a^{12}+\frac{4272}{853}a^{11}+\frac{4298}{853}a^{10}-\frac{10617}{853}a^{9}+\frac{11180}{853}a^{8}-\frac{26693}{853}a^{7}+\frac{50561}{853}a^{6}-\frac{13716}{853}a^{5}-\frac{77703}{853}a^{4}+\frac{131587}{853}a^{3}-\frac{96343}{853}a^{2}+\frac{35262}{853}a-\frac{5170}{853}$, $\frac{1833}{853}a^{13}-\frac{3469}{853}a^{12}-\frac{1631}{853}a^{11}+\frac{6812}{853}a^{10}-\frac{5057}{853}a^{9}+\frac{9505}{853}a^{8}-\frac{29609}{853}a^{7}+\frac{17983}{853}a^{6}+\frac{38875}{853}a^{5}-\frac{82014}{853}a^{4}+\frac{58465}{853}a^{3}-\frac{9012}{853}a^{2}-\frac{9333}{853}a+\frac{4439}{853}$
| 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: | \( 30.836203822 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 30.836203822 \cdot 1}{2\cdot\sqrt{321952970703125}}\cr\approx \mathstrut & 0.21148223674 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*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 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*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 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*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 + 2*x^12 + 3*x^11 - 7*x^10 + 11*x^9 - 25*x^8 + 33*x^7 - 4*x^6 - 56*x^5 + 93*x^4 - 78*x^3 + 38*x^2 - 10*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
$C_{7236}$ (as 14T49):
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 10080 |
| The 30 conjugacy class representatives for $S_7\times C_2$ |
| Character table for $S_7\times C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 7.1.320975.1 |
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 sibling: | 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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.14.0.1}{14} }$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.6.5.1 | $x^{6} + 5$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(37\)
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 37.2.0.1 | $x^{2} + 33 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 37.4.2.1 | $x^{4} + 1916 x^{3} + 948367 x^{2} + 29317674 x + 2943243$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 37.6.0.1 | $x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(347\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |