Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*x - 1)
gp: K = bnfinit(y^14 - 5*y^13 + 7*y^12 - 4*y^11 + 15*y^10 - 22*y^9 - 11*y^8 + 23*y^7 - 9*y^6 + 16*y^5 - 6*y^3 - 2*y^2 - 3*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*x - 1)
\( x^{14} - 5 x^{13} + 7 x^{12} - 4 x^{11} + 15 x^{10} - 22 x^{9} - 11 x^{8} + 23 x^{7} - 9 x^{6} + \cdots - 1 \)
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: |
\(-16266899635859375\)
\(\medspace = -\,5^{7}\cdot 5011\cdot 41551849\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.39\) | 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: | $5^{1/2}5011^{1/2}41551849^{1/2}\approx 1020334.0515218533$ | ||
| Ramified primes: |
\(5\), \(5011\), \(41551849\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-1041081576695}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: |
$a^{13}-2a^{12}-4a^{11}+4a^{10}+10a^{9}+13a^{8}-31a^{7}-18a^{6}+21a^{5}+4a^{4}+6a^{3}+2a^{2}-4a-2$, $a$, $2a^{13}-10a^{12}+15a^{11}-11a^{10}+31a^{9}-47a^{8}-9a^{7}+46a^{6}-26a^{5}+26a^{4}-9a^{3}-5a^{2}-a-2$, $2a^{13}-8a^{12}+9a^{11}-9a^{10}+26a^{9}-23a^{8}-9a^{7}+27a^{6}-31a^{5}+10a^{4}+2a^{2}+4a+1$, $a^{13}-7a^{12}+15a^{11}-12a^{10}+21a^{9}-48a^{8}+12a^{7}+44a^{6}-34a^{5}+28a^{4}-15a^{3}-4a^{2}+a-2$, $2a^{13}-8a^{12}+9a^{11}-9a^{10}+26a^{9}-23a^{8}-9a^{7}+27a^{6}-31a^{5}+10a^{4}+2a^{2}+4a+2$, $a^{13}-3a^{12}-a^{11}+5a^{10}+6a^{9}+4a^{8}-34a^{7}+14a^{6}+18a^{5}-15a^{4}+10a^{3}-6a^{2}-a+1$, $a^{13}-5a^{12}+7a^{11}-a^{10}+5a^{9}-17a^{8}-15a^{7}+57a^{6}-19a^{5}-27a^{4}+15a^{3}-13a^{2}+6a+7$, $a^{13}-5a^{12}+7a^{11}-5a^{10}+19a^{9}-25a^{8}-10a^{7}+10a^{6}-a^{5}+33a^{4}-10a^{3}-3a^{2}-6a-6$, $a^{12}-3a^{11}+2a^{10}-6a^{9}+14a^{8}-2a^{7}+3a^{6}-6a^{5}-20a^{4}+9a^{3}-4a^{2}+5a+3$
| 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: | \( 637.300743349 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{3}\cdot 637.300743349 \cdot 1}{2\cdot\sqrt{16266899635859375}}\cr\approx \mathstrut & 0.158650558828 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*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 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*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 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*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 - 5*x^13 + 7*x^12 - 4*x^11 + 15*x^10 - 22*x^9 - 11*x^8 + 23*x^7 - 9*x^6 + 16*x^5 - 6*x^3 - 2*x^2 - 3*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
$S_7\wr C_2$ (as 14T61):
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 50803200 |
| The 135 conjugacy class representatives for $S_7\wr C_2$ |
| Character table for $S_7\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
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 28 siblings: | data not computed |
| Degree 42 sibling: | 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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.6.0.1}{6} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.14.7.1 | $x^{14} + 140 x^{13} + 8435 x^{12} + 284200 x^{11} + 5810525 x^{10} + 72852500 x^{9} + 534104381 x^{8} + 1994350486 x^{7} + 2670547075 x^{6} + 1822151870 x^{5} + 743294125 x^{4} + 386790250 x^{3} + 1508497384 x^{2} + 5074882448 x + 4401772109$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ |
|
\(5011\)
| $\Q_{5011}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{5011}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{5011}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(41551849\)
| $\Q_{41551849}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |