Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*x + 1)
gp: K = bnfinit(y^14 - 4*y^13 - 7*y^12 + 30*y^11 + 6*y^10 + 6*y^9 - 35*y^8 - 70*y^7 - 35*y^6 + 6*y^5 + 6*y^4 + 30*y^3 - 7*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*x + 1)
\( x^{14} - 4 x^{13} - 7 x^{12} + 30 x^{11} + 6 x^{10} + 6 x^{9} - 35 x^{8} - 70 x^{7} - 35 x^{6} + \cdots + 1 \)
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: | $[8, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-19273578224371707904\)
\(\medspace = -\,2^{12}\cdot 19\cdot 3967^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.85\) | 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: | $2^{3/2}19^{1/2}3967^{1/2}\approx 776.5204440322225$ | ||
| Ramified primes: |
\(2\), \(19\), \(3967\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{16}a^{13}+\frac{3}{16}a^{12}-\frac{1}{8}a^{11}+\frac{3}{8}a^{9}-\frac{3}{16}a^{7}+\frac{5}{16}a^{6}+\frac{3}{8}a^{4}-\frac{1}{8}a^{2}-\frac{5}{16}a-\frac{7}{16}$
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: | $10$ | 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: |
$a$, $\frac{1641}{16}a^{13}-\frac{6085}{16}a^{12}-\frac{6609}{8}a^{11}+2825a^{10}+\frac{11323}{8}a^{9}+1120a^{8}-\frac{51355}{16}a^{7}-\frac{130995}{16}a^{6}-6096a^{5}-\frac{10413}{8}a^{4}+184a^{3}+\frac{25751}{8}a^{2}+\frac{3987}{16}a-\frac{5519}{16}$, $\frac{685}{16}a^{13}-\frac{2521}{16}a^{12}-\frac{2805}{8}a^{11}+1174a^{10}+\frac{5111}{8}a^{9}+446a^{8}-\frac{22103}{16}a^{7}-\frac{54847}{16}a^{6}-2558a^{5}-\frac{4153}{8}a^{4}+117a^{3}+\frac{10443}{8}a^{2}+\frac{1551}{16}a-\frac{2219}{16}$, $\frac{2449}{8}a^{13}-\frac{9045}{8}a^{12}-\frac{9957}{4}a^{11}+8418a^{10}+\frac{17679}{4}a^{9}+3212a^{8}-\frac{78091}{8}a^{7}-\frac{195563}{8}a^{6}-18194a^{5}-\frac{14861}{4}a^{4}+738a^{3}+\frac{37739}{4}a^{2}+\frac{5715}{8}a-\frac{8079}{8}$, $\frac{4289}{8}a^{13}-\frac{15869}{8}a^{12}-\frac{17401}{4}a^{11}+14790a^{10}+\frac{30639}{4}a^{9}+5418a^{8}-\frac{136499}{8}a^{7}-\frac{340411}{8}a^{6}-31537a^{5}-\frac{25073}{4}a^{4}+1230a^{3}+\frac{65395}{4}a^{2}+\frac{9971}{8}a-\frac{14023}{8}$, $\frac{1519}{8}a^{13}-\frac{5595}{8}a^{12}-\frac{6211}{4}a^{11}+5211a^{10}+\frac{11261}{4}a^{9}+1975a^{8}-\frac{48965}{8}a^{7}-\frac{121589}{8}a^{6}-11322a^{5}-\frac{9095}{4}a^{4}+533a^{3}+\frac{23273}{4}a^{2}+\frac{3477}{8}a-\frac{4985}{8}$, $\frac{1873}{8}a^{13}-\frac{6925}{8}a^{12}-\frac{7609}{4}a^{11}+6454a^{10}+\frac{13463}{4}a^{9}+2373a^{8}-\frac{59715}{8}a^{7}-\frac{148867}{8}a^{6}-13801a^{5}-\frac{10973}{4}a^{4}+560a^{3}+\frac{28591}{4}a^{2}+\frac{4395}{8}a-\frac{6143}{8}$, $643a^{13}-2382a^{12}-5205a^{11}+17759a^{10}+9081a^{9}+6485a^{8}-20372a^{7}-50944a^{6}-37728a^{5}-7522a^{4}+1364a^{3}+19620a^{2}+1509a-2105$, $\frac{347}{16}a^{13}-\frac{1279}{16}a^{12}-\frac{1395}{8}a^{11}+585a^{10}+\frac{2369}{8}a^{9}+316a^{8}-\frac{10401}{16}a^{7}-\frac{28953}{16}a^{6}-1401a^{5}-\frac{3079}{8}a^{4}+20a^{3}+\frac{6061}{8}a^{2}+\frac{1129}{16}a-\frac{1341}{16}$, $\frac{6779}{16}a^{13}-\frac{25087}{16}a^{12}-\frac{27459}{8}a^{11}+11674a^{10}+\frac{48065}{8}a^{9}+4420a^{8}-\frac{214689}{16}a^{7}-\frac{539641}{16}a^{6}-25067a^{5}-\frac{41071}{8}a^{4}+910a^{3}+\frac{104437}{8}a^{2}+\frac{16041}{16}a-\frac{22365}{16}$
| 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: | \( 34886.0050183 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{3}\cdot 34886.0050183 \cdot 1}{2\cdot\sqrt{19273578224371707904}}\cr\approx \mathstrut & 0.252301387134 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*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 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*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 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*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 - 4*x^13 - 7*x^12 + 30*x^11 + 6*x^10 + 6*x^9 - 35*x^8 - 70*x^7 - 35*x^6 + 6*x^5 + 6*x^4 + 30*x^3 - 7*x^2 - 4*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
$C_2^7.\GL(3,2)$ (as 14T51):
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 21504 |
| The 48 conjugacy class representatives for $C_2^7.\GL(3,2)$ |
| Character table for $C_2^7.\GL(3,2)$ |
Intermediate fields
| 7.7.1007173696.1 |
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 siblings: | 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 | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.7.0.1}{7} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.14.0.1}{14} }$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.12.12.11 | $x^{12} + 28 x^{10} + 40 x^{9} + 356 x^{8} + 896 x^{7} + 2720 x^{6} + 6656 x^{5} + 12464 x^{4} + 19456 x^{3} + 26304 x^{2} + 19840 x + 5824$ | $2$ | $6$ | $12$ | $A_4 \times C_2$ | $[2, 2]^{6}$ | |
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(3967\)
| 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 $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |