Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 2*x^12 + x^11 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*x^2 + x - 1)
gp: K = bnfinit(y^14 - y^13 + 2*y^12 + y^11 - 6*y^10 + 7*y^9 - 14*y^8 - 36*y^7 - 24*y^6 - 12*y^5 + 2*y^4 + 5*y^3 + 4*y^2 + y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + 2*x^12 + x^11 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*x^2 + x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + 2*x^12 + x^11 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*x^2 + x - 1)
\( x^{14} - x^{13} + 2 x^{12} + x^{11} - 6 x^{10} + 7 x^{9} - 14 x^{8} - 36 x^{7} - 24 x^{6} - 12 x^{5} + \cdots - 1 \)
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: | $[6, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(29283534866328125\)
\(\medspace = 5^{7}\cdot 71^{2}\cdot 8623^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.00\) | 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: | $5^{1/2}71^{1/2}8623^{1/2}\approx 1749.6185298515788$ | ||
| Ramified primes: |
\(5\), \(71\), \(8623\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{49206103}a^{13}+\frac{18057423}{49206103}a^{12}+\frac{3547670}{49206103}a^{11}+\frac{11463660}{49206103}a^{10}-\frac{1376806}{49206103}a^{9}+\frac{10657425}{49206103}a^{8}-\frac{17333050}{49206103}a^{7}-\frac{3894733}{49206103}a^{6}+\frac{10893097}{49206103}a^{5}+\frac{22077925}{49206103}a^{4}-\frac{1664257}{49206103}a^{3}-\frac{9741640}{49206103}a^{2}-\frac{8870433}{49206103}a+\frac{12924099}{49206103}$
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: | $9$ | 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: |
$\frac{7106568}{49206103}a^{13}-\frac{13773041}{49206103}a^{12}+\frac{27102450}{49206103}a^{11}-\frac{17815422}{49206103}a^{10}-\frac{27116876}{49206103}a^{9}+\frac{75378624}{49206103}a^{8}-\frac{171947058}{49206103}a^{7}-\frac{96411565}{49206103}a^{6}-\frac{81273800}{49206103}a^{5}-\frac{9477782}{49206103}a^{4}+\frac{23377104}{49206103}a^{3}+\frac{13814476}{49206103}a^{2}+\frac{95310386}{49206103}a-\frac{7615139}{49206103}$, $\frac{26922944}{49206103}a^{13}-\frac{15762353}{49206103}a^{12}+\frac{49443798}{49206103}a^{11}+\frac{36368140}{49206103}a^{10}-\frac{122973728}{49206103}a^{9}+\frac{104828690}{49206103}a^{8}-\frac{323417070}{49206103}a^{7}-\frac{1047556248}{49206103}a^{6}-\frac{1219041264}{49206103}a^{5}-\frac{687447616}{49206103}a^{4}-\frac{161887941}{49206103}a^{3}+\frac{18236552}{49206103}a^{2}+\frac{107156332}{49206103}a+\frac{36231934}{49206103}$, $\frac{7615139}{49206103}a^{13}-\frac{14721707}{49206103}a^{12}+\frac{29003319}{49206103}a^{11}-\frac{19487311}{49206103}a^{10}-\frac{27875412}{49206103}a^{9}+\frac{80422849}{49206103}a^{8}-\frac{181990570}{49206103}a^{7}-\frac{102197946}{49206103}a^{6}-\frac{86351771}{49206103}a^{5}-\frac{10107868}{49206103}a^{4}+\frac{24708060}{49206103}a^{3}+\frac{14698591}{49206103}a^{2}+\frac{16646080}{49206103}a-\frac{38489144}{49206103}$, $\frac{14477262}{49206103}a^{13}+\frac{7415735}{49206103}a^{12}+\frac{5065788}{49206103}a^{11}+\frac{65100520}{49206103}a^{10}-\frac{71394138}{49206103}a^{9}-\frac{27359832}{49206103}a^{8}-\frac{21998678}{49206103}a^{7}-\frac{879167612}{49206103}a^{6}-\frac{1077373848}{49206103}a^{5}-\frac{670676780}{49206103}a^{4}-\frac{411527002}{49206103}a^{3}-\frac{5651818}{49206103}a^{2}+\frac{79610250}{49206103}a+\frac{49564880}{49206103}$, $\frac{10319400}{49206103}a^{13}-\frac{18941195}{49206103}a^{12}+\frac{42311073}{49206103}a^{11}-\frac{21431905}{49206103}a^{10}-\frac{41656180}{49206103}a^{9}+\frac{131034850}{49206103}a^{8}-\frac{277490365}{49206103}a^{7}-\frac{156438624}{49206103}a^{6}-\frac{132999640}{49206103}a^{5}-\frac{354117729}{49206103}a^{4}-\frac{170410637}{49206103}a^{3}-\frac{106623618}{49206103}a^{2}+\frac{25843567}{49206103}a-\frac{12442145}{49206103}$, $\frac{3781351}{49206103}a^{13}+\frac{15217287}{49206103}a^{12}-\frac{25152617}{49206103}a^{11}+\frac{54972913}{49206103}a^{10}-\frac{33429197}{49206103}a^{9}-\frac{87645310}{49206103}a^{8}+\frac{134039347}{49206103}a^{7}-\frac{457957413}{49206103}a^{6}-\frac{592884901}{49206103}a^{5}-\frac{68344009}{49206103}a^{4}+\frac{271496390}{49206103}a^{3}+\frac{112672220}{49206103}a^{2}-\frac{93287385}{49206103}a-\frac{11905894}{49206103}$, $\frac{10319400}{49206103}a^{13}-\frac{18941195}{49206103}a^{12}+\frac{42311073}{49206103}a^{11}-\frac{21431905}{49206103}a^{10}-\frac{41656180}{49206103}a^{9}+\frac{131034850}{49206103}a^{8}-\frac{277490365}{49206103}a^{7}-\frac{156438624}{49206103}a^{6}-\frac{132999640}{49206103}a^{5}-\frac{354117729}{49206103}a^{4}-\frac{170410637}{49206103}a^{3}-\frac{106623618}{49206103}a^{2}+\frac{75049670}{49206103}a-\frac{12442145}{49206103}$, $\frac{13559813}{49206103}a^{13}-\frac{46500152}{49206103}a^{12}+\frac{84073202}{49206103}a^{11}-\frac{93465909}{49206103}a^{10}-\frac{42770254}{49206103}a^{9}+\frac{270282885}{49206103}a^{8}-\frac{556213386}{49206103}a^{7}+\frac{222681220}{49206103}a^{6}+\frac{395791195}{49206103}a^{5}+\frac{45058669}{49206103}a^{4}+\frac{332108846}{49206103}a^{3}+\frac{49324446}{49206103}a^{2}+\frac{2914394}{49206103}a-\frac{6704764}{49206103}$, $\frac{566778}{49206103}a^{13}-\frac{24094288}{49206103}a^{12}+\frac{32320371}{49206103}a^{11}-\frac{69583155}{49206103}a^{10}+\frac{16236409}{49206103}a^{9}+\frac{98852885}{49206103}a^{8}-\frac{189773362}{49206103}a^{7}+\frac{428861336}{49206103}a^{6}+\frac{570049086}{49206103}a^{5}+\frac{613037677}{49206103}a^{4}+\frac{410389388}{49206103}a^{3}-\frac{28830496}{49206103}a^{2}+\frac{18093048}{49206103}a-\frac{119158382}{49206103}$
| 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: | \( 647.937455607 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{4}\cdot 647.937455607 \cdot 1}{2\cdot\sqrt{29283534866328125}}\cr\approx \mathstrut & 0.188838606894 \end{aligned}\]
# 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 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*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 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*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 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*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 - 6*x^10 + 7*x^9 - 14*x^8 - 36*x^7 - 24*x^6 - 12*x^5 + 2*x^4 + 5*x^3 + 4*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 $S_7\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 7.3.612233.1 |
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 | ${\href{/padicField/2.14.0.1}{14} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(71\)
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 71.5.0.1 | $x^{5} + 18 x + 64$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(8623\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |