Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681)
gp: K = bnfinit(y^14 - y^13 + 16*y^12 - 17*y^11 + 66*y^10 - 84*y^9 + 38*y^8 - 315*y^7 + 65*y^6 - 503*y^5 + 1556*y^4 + 665*y^3 + 2377*y^2 - 3516*y + 1681, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681)
\( x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} + \cdots + 1681 \)
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: |
\(-22439994995240462987343\)
\(\medspace = -\,3^{7}\cdot 29^{13}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.49\) | 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: | $3^{1/2}29^{13/14}\approx 39.49131773649383$ | ||
| Ramified primes: |
\(3\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-87}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(87=3\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{87}(1,·)$, $\chi_{87}(35,·)$, $\chi_{87}(5,·)$, $\chi_{87}(38,·)$, $\chi_{87}(71,·)$, $\chi_{87}(7,·)$, $\chi_{87}(80,·)$, $\chi_{87}(49,·)$, $\chi_{87}(82,·)$, $\chi_{87}(52,·)$, $\chi_{87}(86,·)$, $\chi_{87}(25,·)$, $\chi_{87}(62,·)$, $\chi_{87}(16,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{64}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17}a^{9}-\frac{6}{17}a^{8}-\frac{7}{17}a^{7}-\frac{7}{17}a^{6}+\frac{5}{17}a^{5}+\frac{6}{17}a^{4}-\frac{6}{17}a^{3}-\frac{6}{17}a^{2}-\frac{2}{17}a+\frac{4}{17}$, $\frac{1}{17}a^{10}+\frac{8}{17}a^{8}+\frac{2}{17}a^{7}-\frac{3}{17}a^{6}+\frac{2}{17}a^{5}-\frac{4}{17}a^{4}-\frac{8}{17}a^{3}-\frac{4}{17}a^{2}-\frac{8}{17}a+\frac{7}{17}$, $\frac{1}{17}a^{11}-\frac{1}{17}a^{8}+\frac{2}{17}a^{7}+\frac{7}{17}a^{6}+\frac{7}{17}a^{5}-\frac{5}{17}a^{4}-\frac{7}{17}a^{3}+\frac{6}{17}a^{2}+\frac{6}{17}a+\frac{2}{17}$, $\frac{1}{697}a^{12}-\frac{12}{697}a^{11}+\frac{9}{697}a^{10}-\frac{6}{697}a^{9}-\frac{258}{697}a^{8}+\frac{308}{697}a^{7}-\frac{1}{697}a^{6}+\frac{57}{697}a^{5}-\frac{13}{697}a^{4}+\frac{286}{697}a^{3}+\frac{217}{697}a^{2}+\frac{344}{697}a-\frac{7}{17}$, $\frac{1}{31\!\cdots\!79}a^{13}+\frac{856902660014}{31\!\cdots\!79}a^{12}+\frac{910220185274448}{31\!\cdots\!79}a^{11}-\frac{18054269152595}{770365705304719}a^{10}-\frac{681555880715611}{31\!\cdots\!79}a^{9}+\frac{989312706677007}{31\!\cdots\!79}a^{8}-\frac{58\!\cdots\!49}{31\!\cdots\!79}a^{7}-\frac{116597290667229}{31\!\cdots\!79}a^{6}+\frac{74\!\cdots\!79}{31\!\cdots\!79}a^{5}-\frac{87\!\cdots\!30}{31\!\cdots\!79}a^{4}-\frac{10\!\cdots\!26}{31\!\cdots\!79}a^{3}+\frac{13\!\cdots\!58}{31\!\cdots\!79}a^{2}-\frac{15\!\cdots\!25}{31\!\cdots\!79}a+\frac{333489547292032}{770365705304719}$
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
$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$
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: |
\( -1 \)
(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: |
$\frac{1112114970210}{31\!\cdots\!79}a^{13}-\frac{6935735935728}{31\!\cdots\!79}a^{12}+\frac{41918097299628}{31\!\cdots\!79}a^{11}-\frac{174697919717574}{31\!\cdots\!79}a^{10}+\frac{482681897203975}{31\!\cdots\!79}a^{9}-\frac{14\!\cdots\!89}{31\!\cdots\!79}a^{8}+\frac{22\!\cdots\!04}{31\!\cdots\!79}a^{7}-\frac{41\!\cdots\!68}{31\!\cdots\!79}a^{6}+\frac{61\!\cdots\!38}{31\!\cdots\!79}a^{5}-\frac{44\!\cdots\!95}{31\!\cdots\!79}a^{4}+\frac{90\!\cdots\!84}{31\!\cdots\!79}a^{3}-\frac{47\!\cdots\!32}{31\!\cdots\!79}a^{2}-\frac{477737409127038}{31\!\cdots\!79}a-\frac{195783900862757}{770365705304719}$, $\frac{7708799736526}{31\!\cdots\!79}a^{13}-\frac{10928791841837}{31\!\cdots\!79}a^{12}+\frac{124973199420763}{31\!\cdots\!79}a^{11}-\frac{142638384395782}{31\!\cdots\!79}a^{10}+\frac{446352051059720}{31\!\cdots\!79}a^{9}-\frac{146469250997243}{31\!\cdots\!79}a^{8}-\frac{830068629530720}{31\!\cdots\!79}a^{7}+\frac{681871725647319}{31\!\cdots\!79}a^{6}-\frac{22\!\cdots\!11}{31\!\cdots\!79}a^{5}-\frac{35\!\cdots\!48}{31\!\cdots\!79}a^{4}+\frac{10\!\cdots\!96}{31\!\cdots\!79}a^{3}-\frac{18\!\cdots\!42}{31\!\cdots\!79}a^{2}-\frac{43\!\cdots\!43}{31\!\cdots\!79}a+\frac{514901033877651}{770365705304719}$, $\frac{1204596686635}{31\!\cdots\!79}a^{13}+\frac{5541076490938}{31\!\cdots\!79}a^{12}+\frac{30172434956940}{31\!\cdots\!79}a^{11}+\frac{107294955569948}{31\!\cdots\!79}a^{10}+\frac{272008992922456}{31\!\cdots\!79}a^{9}+\frac{570441168179015}{31\!\cdots\!79}a^{8}+\frac{781778897992559}{31\!\cdots\!79}a^{7}-\frac{390882092739519}{31\!\cdots\!79}a^{6}-\frac{28\!\cdots\!67}{31\!\cdots\!79}a^{5}-\frac{81\!\cdots\!14}{31\!\cdots\!79}a^{4}-\frac{10\!\cdots\!76}{31\!\cdots\!79}a^{3}-\frac{14\!\cdots\!10}{31\!\cdots\!79}a^{2}+\frac{12\!\cdots\!64}{31\!\cdots\!79}a+\frac{328652293343553}{770365705304719}$, $\frac{4089134603620}{31\!\cdots\!79}a^{13}+\frac{4189926238325}{31\!\cdots\!79}a^{12}+\frac{55000898421676}{31\!\cdots\!79}a^{11}+\frac{75484476973130}{31\!\cdots\!79}a^{10}+\frac{109285221750210}{31\!\cdots\!79}a^{9}+\frac{418044553432312}{31\!\cdots\!79}a^{8}-\frac{499186568292005}{31\!\cdots\!79}a^{7}-\frac{38316052814779}{31\!\cdots\!79}a^{6}-\frac{23\!\cdots\!23}{31\!\cdots\!79}a^{5}-\frac{12\!\cdots\!75}{31\!\cdots\!79}a^{4}-\frac{27\!\cdots\!94}{31\!\cdots\!79}a^{3}-\frac{969056671578775}{31\!\cdots\!79}a^{2}+\frac{35\!\cdots\!94}{31\!\cdots\!79}a-\frac{240510494617842}{770365705304719}$, $\frac{3707590263250}{31\!\cdots\!79}a^{13}-\frac{12001593407322}{31\!\cdots\!79}a^{12}+\frac{68838527820103}{31\!\cdots\!79}a^{11}-\frac{194607482452942}{31\!\cdots\!79}a^{10}+\frac{437236274859701}{31\!\cdots\!79}a^{9}-\frac{882336826300499}{31\!\cdots\!79}a^{8}+\frac{12\!\cdots\!39}{31\!\cdots\!79}a^{7}-\frac{22\!\cdots\!01}{31\!\cdots\!79}a^{6}+\frac{31\!\cdots\!70}{31\!\cdots\!79}a^{5}-\frac{42\!\cdots\!04}{31\!\cdots\!79}a^{4}+\frac{82\!\cdots\!32}{31\!\cdots\!79}a^{3}-\frac{46\!\cdots\!79}{31\!\cdots\!79}a^{2}+\frac{57177471132876}{31\!\cdots\!79}a-\frac{487100612944443}{770365705304719}$, $\frac{9587506393099}{31\!\cdots\!79}a^{13}-\frac{13896636982205}{31\!\cdots\!79}a^{12}+\frac{129189124937983}{31\!\cdots\!79}a^{11}-\frac{203625627854443}{31\!\cdots\!79}a^{10}+\frac{351771849167120}{31\!\cdots\!79}a^{9}-\frac{710595197734625}{31\!\cdots\!79}a^{8}+\frac{43998762578293}{31\!\cdots\!79}a^{7}-\frac{21\!\cdots\!87}{31\!\cdots\!79}a^{6}+\frac{24\!\cdots\!48}{31\!\cdots\!79}a^{5}+\frac{968814086392364}{31\!\cdots\!79}a^{4}+\frac{12\!\cdots\!99}{31\!\cdots\!79}a^{3}-\frac{61\!\cdots\!84}{31\!\cdots\!79}a^{2}-\frac{32\!\cdots\!48}{31\!\cdots\!79}a-\frac{635639900758060}{770365705304719}$
| 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: | \( 6020.98510015 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 6020.98510015 \cdot 48}{2\cdot\sqrt{22439994995240462987343}}\cr\approx \mathstrut & 0.372929290422 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681)
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 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681, 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 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681);
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 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681);
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 cyclic group of order 14 |
| The 14 conjugacy class representatives for $C_{14}$ |
| Character table for $C_{14}$ |
Intermediate fields
| \(\Q(\sqrt{-87}) \), 7.7.594823321.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | R | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.7.0.1}{7} }^{2}$ | ${\href{/padicField/11.7.0.1}{7} }^{2}$ | ${\href{/padicField/13.7.0.1}{7} }^{2}$ | ${\href{/padicField/17.1.0.1}{1} }^{14}$ | ${\href{/padicField/19.14.0.1}{14} }$ | ${\href{/padicField/23.14.0.1}{14} }$ | R | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.1.0.1}{1} }^{14}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.14.7.1 | $x^{14} - 486 x^{4} - 2187$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(29\)
| 29.14.13.11 | $x^{14} + 348$ | $14$ | $1$ | $13$ | $C_{14}$ | $[\ ]_{14}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.87.2t1.a.a | $1$ | $ 3 \cdot 29 $ | \(\Q(\sqrt{-87}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.29.7t1.a.a | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.a | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.29.7t1.a.b | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.b | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.29.7t1.a.c | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.c | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.29.7t1.a.d | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.d | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.29.7t1.a.e | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.e | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
| * | 1.29.7t1.a.f | $1$ | $ 29 $ | 7.7.594823321.1 | $C_7$ (as 7T1) | $0$ | $1$ |
| * | 1.87.14t1.a.f | $1$ | $ 3 \cdot 29 $ | 14.0.22439994995240462987343.1 | $C_{14}$ (as 14T1) | $0$ | $-1$ |
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 *.