Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + x + 1)
gp: K = bnfinit(y^14 - y^13 + 2*y^12 - y^11 + 2*y^10 - 2*y^9 + 2*y^8 - 2*y^7 - 3*y^5 - y^3 + 2*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + x + 1)
\( x^{14} - x^{13} + 2x^{12} - x^{11} + 2x^{10} - 2x^{9} + 2x^{8} - 2x^{7} - 3x^{5} - x^{3} + 2x^{2} + x + 1 \)
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: | $[0, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-74532401109963\)
\(\medspace = -\,3^{7}\cdot 184607^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(9.79\) | 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: | $3^{1/2}184607^{1/2}\approx 744.1915076107762$ | ||
| Ramified primes: |
\(3\), \(184607\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{47}a^{13}-\frac{16}{47}a^{12}+\frac{7}{47}a^{11}-\frac{12}{47}a^{10}-\frac{6}{47}a^{9}-\frac{6}{47}a^{8}-\frac{2}{47}a^{7}-\frac{19}{47}a^{6}+\frac{3}{47}a^{5}-\frac{1}{47}a^{4}+\frac{15}{47}a^{3}+\frac{9}{47}a^{2}+\frac{8}{47}a+\frac{22}{47}$
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: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{38}{47} a^{13} + \frac{44}{47} a^{12} - \frac{78}{47} a^{11} + \frac{33}{47} a^{10} - \frac{54}{47} a^{9} + \frac{40}{47} a^{8} - \frac{65}{47} a^{7} + \frac{64}{47} a^{6} - \frac{20}{47} a^{5} + \frac{85}{47} a^{4} - \frac{6}{47} a^{3} + \frac{34}{47} a^{2} - \frac{69}{47} a + \frac{10}{47} \)
(order $6$)
| 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: |
$\frac{6}{47}a^{13}-\frac{2}{47}a^{12}-\frac{5}{47}a^{11}+\frac{22}{47}a^{10}-\frac{36}{47}a^{9}+\frac{11}{47}a^{8}-\frac{12}{47}a^{7}-\frac{20}{47}a^{6}-\frac{29}{47}a^{5}-\frac{6}{47}a^{4}-\frac{4}{47}a^{3}+\frac{7}{47}a^{2}+\frac{48}{47}a+\frac{38}{47}$, $\frac{7}{47}a^{13}-\frac{18}{47}a^{12}+\frac{2}{47}a^{11}+\frac{10}{47}a^{10}-\frac{42}{47}a^{9}+\frac{5}{47}a^{8}-\frac{14}{47}a^{7}+\frac{8}{47}a^{6}-\frac{26}{47}a^{5}-\frac{7}{47}a^{4}+\frac{58}{47}a^{3}+\frac{16}{47}a^{2}+\frac{9}{47}a+\frac{13}{47}$, $\frac{55}{47}a^{13}-\frac{81}{47}a^{12}+\frac{103}{47}a^{11}-\frac{49}{47}a^{10}+\frac{46}{47}a^{9}-\frac{95}{47}a^{8}+\frac{78}{47}a^{7}-\frac{58}{47}a^{6}-\frac{70}{47}a^{5}-\frac{55}{47}a^{4}+\frac{26}{47}a^{3}+\frac{25}{47}a^{2}+\frac{64}{47}a+\frac{35}{47}$, $\frac{27}{47}a^{13}-\frac{56}{47}a^{12}+\frac{95}{47}a^{11}-\frac{89}{47}a^{10}+\frac{73}{47}a^{9}-\frac{68}{47}a^{8}+\frac{40}{47}a^{7}-\frac{43}{47}a^{6}-\frac{13}{47}a^{5}-\frac{27}{47}a^{4}+\frac{29}{47}a^{3}+\frac{8}{47}a^{2}+\frac{28}{47}a+\frac{30}{47}$, $\frac{7}{47}a^{13}-\frac{18}{47}a^{12}+\frac{2}{47}a^{11}+\frac{10}{47}a^{10}-\frac{42}{47}a^{9}+\frac{5}{47}a^{8}-\frac{14}{47}a^{7}+\frac{8}{47}a^{6}-\frac{73}{47}a^{5}+\frac{40}{47}a^{4}+\frac{11}{47}a^{3}+\frac{16}{47}a^{2}+\frac{9}{47}a+\frac{60}{47}$, $\frac{51}{47}a^{13}-\frac{64}{47}a^{12}+\frac{75}{47}a^{11}-\frac{1}{47}a^{10}+\frac{23}{47}a^{9}-\frac{71}{47}a^{8}+\frac{86}{47}a^{7}-\frac{29}{47}a^{6}-\frac{82}{47}a^{5}-\frac{51}{47}a^{4}+\frac{60}{47}a^{3}-\frac{11}{47}a^{2}+\frac{32}{47}a+\frac{41}{47}$
| 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: | \( 26.6828967522 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 26.6828967522 \cdot 1}{6\cdot\sqrt{74532401109963}}\cr\approx \mathstrut & 0.199144468401 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + 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 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + 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 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + 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 + 2*x^12 - x^11 + 2*x^10 - 2*x^9 + 2*x^8 - 2*x^7 - 3*x^5 - x^3 + 2*x^2 + 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
| \(\Q(\sqrt{-3}) \), 7.1.184607.1 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(184607\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |