Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + x^2 + 3*x + 1)
gp: K = bnfinit(y^14 - 4*y^13 + 8*y^12 - 10*y^11 + 5*y^10 + 3*y^9 - 10*y^8 + 7*y^7 - y^6 - 2*y^5 + 5*y^4 + y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + x^2 + 3*x + 1)
\( x^{14} - 4 x^{13} + 8 x^{12} - 10 x^{11} + 5 x^{10} + 3 x^{9} - 10 x^{8} + 7 x^{7} - x^{6} - 2 x^{5} + \cdots + 1 \)
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: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(552331698375977\)
\(\medspace = 13^{4}\cdot 109^{4}\cdot 137\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.30\) | 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: | $13^{1/2}109^{1/2}137^{1/2}\approx 440.6007262817437$ | ||
| Ramified primes: |
\(13\), \(109\), \(137\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{137}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19}a^{13}+\frac{5}{19}a^{12}-\frac{4}{19}a^{11}-\frac{8}{19}a^{10}+\frac{9}{19}a^{9}+\frac{8}{19}a^{8}+\frac{5}{19}a^{7}-\frac{5}{19}a^{6}-\frac{8}{19}a^{5}+\frac{2}{19}a^{4}+\frac{4}{19}a^{3}-\frac{2}{19}a^{2}+\frac{2}{19}a+\frac{2}{19}$
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: |
$\frac{18}{19}a^{13}-\frac{100}{19}a^{12}+\frac{289}{19}a^{11}-\frac{562}{19}a^{10}+\frac{751}{19}a^{9}-\frac{692}{19}a^{8}+\frac{337}{19}a^{7}+\frac{81}{19}a^{6}-\frac{315}{19}a^{5}+\frac{321}{19}a^{4}-\frac{137}{19}a^{3}+\frac{21}{19}a^{2}+\frac{74}{19}a-\frac{40}{19}$, $a$, $\frac{36}{19}a^{13}-\frac{143}{19}a^{12}+\frac{274}{19}a^{11}-\frac{307}{19}a^{10}+\frac{58}{19}a^{9}+\frac{307}{19}a^{8}-\frac{580}{19}a^{7}+\frac{428}{19}a^{6}-\frac{98}{19}a^{5}-\frac{99}{19}a^{4}+\frac{258}{19}a^{3}-\frac{72}{19}a^{2}+\frac{53}{19}a+\frac{91}{19}$, $\frac{119}{19}a^{13}-\frac{545}{19}a^{12}+\frac{1272}{19}a^{11}-\frac{1940}{19}a^{10}+\frac{1736}{19}a^{9}-\frac{663}{19}a^{8}-\frac{811}{19}a^{7}+\frac{1324}{19}a^{6}-\frac{914}{19}a^{5}+\frac{314}{19}a^{4}+\frac{400}{19}a^{3}-\frac{219}{19}a^{2}+\frac{276}{19}a+\frac{200}{19}$, $\frac{69}{19}a^{13}-\frac{320}{19}a^{12}+\frac{750}{19}a^{11}-\frac{1141}{19}a^{10}+\frac{1020}{19}a^{9}-\frac{379}{19}a^{8}-\frac{491}{19}a^{7}+\frac{795}{19}a^{6}-\frac{552}{19}a^{5}+\frac{195}{19}a^{4}+\frac{219}{19}a^{3}-\frac{157}{19}a^{2}+\frac{157}{19}a+\frac{100}{19}$, $\frac{41}{19}a^{13}-\frac{175}{19}a^{12}+\frac{368}{19}a^{11}-\frac{480}{19}a^{10}+\frac{274}{19}a^{9}+\frac{138}{19}a^{8}-\frac{536}{19}a^{7}+\frac{498}{19}a^{6}-\frac{195}{19}a^{5}-\frac{51}{19}a^{4}+\frac{278}{19}a^{3}-\frac{139}{19}a^{2}+\frac{82}{19}a+\frac{120}{19}$, $\frac{64}{19}a^{13}-\frac{288}{19}a^{12}+\frac{656}{19}a^{11}-\frac{968}{19}a^{10}+\frac{804}{19}a^{9}-\frac{210}{19}a^{8}-\frac{535}{19}a^{7}+\frac{706}{19}a^{6}-\frac{398}{19}a^{5}+\frac{52}{19}a^{4}+\frac{313}{19}a^{3}-\frac{147}{19}a^{2}+\frac{128}{19}a+\frac{128}{19}$
| 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: | \( 43.5555878552 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 43.5555878552 \cdot 1}{2\cdot\sqrt{552331698375977}}\cr\approx \mathstrut & 0.228062090750 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + 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 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + 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 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + 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 - 4*x^13 + 8*x^12 - 10*x^11 + 5*x^10 + 3*x^9 - 10*x^8 + 7*x^7 - x^6 - 2*x^5 + 5*x^4 + 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
$C_2^7.\GL(3,2)$ (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
| 7.3.2007889.1 |
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 | ${\href{/padicField/2.7.0.1}{7} }^{2}$ | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.3.0.1}{3} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.14.0.1}{14} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.14.0.1}{14} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.8.4.1 | $x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(109\)
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 109.2.0.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 109.2.1.2 | $x^{2} + 218$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 109.4.0.1 | $x^{4} + 11 x^{2} + 98 x + 6$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 109.4.2.1 | $x^{4} + 14604 x^{3} + 54096386 x^{2} + 5674982964 x + 401153281$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(137\)
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 137.8.0.1 | $x^{8} + 4 x^{4} + 105 x^{3} + 21 x^{2} + 34 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |