Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*x - 1)
gp: K = bnfinit(y^14 - 2*y^13 + 6*y^11 - 3*y^10 - 8*y^9 + 13*y^8 + 4*y^7 - 16*y^6 + 7*y^5 + 13*y^4 - 11*y^3 - 4*y^2 + 6*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*x - 1)
\( x^{14} - 2 x^{13} + 6 x^{11} - 3 x^{10} - 8 x^{9} + 13 x^{8} + 4 x^{7} - 16 x^{6} + 7 x^{5} + 13 x^{4} + \cdots - 1 \)
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: |
\(493856488203125\)
\(\medspace = 5^{7}\cdot 43^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.21\) | 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: | $5^{1/2}43^{1/2}\approx 14.66287829861518$ | ||
| Ramified primes: |
\(5\), \(43\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}$, $\frac{1}{1235}a^{13}-\frac{84}{1235}a^{12}-\frac{28}{1235}a^{11}-\frac{83}{247}a^{10}+\frac{188}{1235}a^{9}+\frac{137}{1235}a^{8}-\frac{353}{1235}a^{7}+\frac{51}{1235}a^{6}+\frac{1}{1235}a^{5}-\frac{322}{1235}a^{4}-\frac{506}{1235}a^{3}-\frac{3}{247}a^{2}-\frac{9}{1235}a-\frac{491}{1235}$
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: |
$\frac{496}{1235}a^{13}-\frac{909}{1235}a^{12}-\frac{303}{1235}a^{11}+\frac{575}{247}a^{10}-\frac{612}{1235}a^{9}-\frac{4913}{1235}a^{8}+\frac{5222}{1235}a^{7}+\frac{3066}{1235}a^{6}-\frac{8149}{1235}a^{5}+\frac{838}{1235}a^{4}+\frac{7139}{1235}a^{3}-\frac{994}{247}a^{2}-\frac{3229}{1235}a+\frac{2229}{1235}$, $\frac{417}{1235}a^{13}-\frac{448}{1235}a^{12}-\frac{561}{1235}a^{11}+\frac{463}{247}a^{10}+\frac{591}{1235}a^{9}-\frac{3386}{1235}a^{8}+\frac{3469}{1235}a^{7}+\frac{3977}{1235}a^{6}-\frac{4523}{1235}a^{5}+\frac{341}{1235}a^{4}+\frac{6358}{1235}a^{3}-\frac{510}{247}a^{2}-\frac{2518}{1235}a+\frac{1498}{1235}$, $\frac{263}{1235}a^{13}-\frac{109}{1235}a^{12}-\frac{448}{1235}a^{11}+\frac{1017}{1235}a^{10}+\frac{1526}{1235}a^{9}-\frac{1513}{1235}a^{8}+\frac{33}{1235}a^{7}+\frac{4521}{1235}a^{6}-\frac{231}{1235}a^{5}-\frac{2682}{1235}a^{4}+\frac{752}{247}a^{3}+\frac{693}{247}a^{2}-\frac{2367}{1235}a-\frac{188}{247}$, $\frac{131}{1235}a^{13}-\frac{383}{1235}a^{12}+\frac{284}{1235}a^{11}+\frac{469}{1235}a^{10}-\frac{813}{1235}a^{9}-\frac{331}{1235}a^{8}+\frac{1181}{1235}a^{7}-\frac{1223}{1235}a^{6}-\frac{857}{1235}a^{5}+\frac{796}{1235}a^{4}-\frac{265}{247}a^{3}-\frac{393}{247}a^{2}+\frac{56}{1235}a-\frac{119}{247}$, $\frac{47}{247}a^{13}-\frac{474}{1235}a^{12}-\frac{158}{1235}a^{11}+\frac{1769}{1235}a^{10}-\frac{1021}{1235}a^{9}-\frac{2138}{1235}a^{8}+\frac{2754}{1235}a^{7}+\frac{1611}{1235}a^{6}-\frac{4458}{1235}a^{5}+\frac{653}{1235}a^{4}+\frac{2861}{1235}a^{3}-\frac{458}{247}a^{2}-\frac{423}{247}a+\frac{1446}{1235}$, $\frac{54}{65}a^{13}-\frac{18}{13}a^{12}-\frac{6}{13}a^{11}+\frac{314}{65}a^{10}-\frac{79}{65}a^{9}-\frac{83}{13}a^{8}+\frac{542}{65}a^{7}+\frac{62}{13}a^{6}-\frac{674}{65}a^{5}+\frac{35}{13}a^{4}+\frac{587}{65}a^{3}-\frac{71}{13}a^{2}-\frac{226}{65}a+\frac{97}{65}$, $\frac{5}{247}a^{13}-\frac{124}{1235}a^{12}+\frac{782}{1235}a^{11}-\frac{1236}{1235}a^{10}+\frac{254}{1235}a^{9}+\frac{2437}{1235}a^{8}-\frac{921}{1235}a^{7}-\frac{2924}{1235}a^{6}+\frac{5212}{1235}a^{5}+\frac{348}{1235}a^{4}-\frac{3264}{1235}a^{3}+\frac{419}{247}a^{2}+\frac{449}{247}a-\frac{1654}{1235}$
| 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: | \( 39.9535247267 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 39.9535247267 \cdot 1}{2\cdot\sqrt{493856488203125}}\cr\approx \mathstrut & 0.221240139170 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*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 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*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 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*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 - 2*x^13 + 6*x^11 - 3*x^10 - 8*x^9 + 13*x^8 + 4*x^7 - 16*x^6 + 7*x^5 + 13*x^4 - 11*x^3 - 4*x^2 + 6*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
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 solvable group of order 28 |
| The 10 conjugacy class representatives for $D_{14}$ |
| Character table for $D_{14}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 7.1.9938375.1 |
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
| Galois closure: | deg 28 |
| Degree 14 sibling: | 14.0.4247165798546875.1 |
| 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 | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{7}$ | ${\href{/padicField/17.2.0.1}{2} }^{7}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{7}$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/47.2.0.1}{2} }^{7}$ | ${\href{/padicField/53.2.0.1}{2} }^{7}$ | ${\href{/padicField/59.7.0.1}{7} }^{2}$ |
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 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(43\)
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.43.2t1.a.a | $1$ | $ 43 $ | \(\Q(\sqrt{-43}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.215.2t1.a.a | $1$ | $ 5 \cdot 43 $ | \(\Q(\sqrt{-215}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 2.215.7t2.a.c | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.215.7t2.a.a | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.215.14t3.a.b | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.215.7t2.a.b | $2$ | $ 5 \cdot 43 $ | 7.1.9938375.1 | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.215.14t3.a.c | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
| * | 2.215.14t3.a.a | $2$ | $ 5 \cdot 43 $ | 14.2.493856488203125.1 | $D_{14}$ (as 14T3) | $1$ | $0$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.