Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2)
gp: K = bnfinit(y^14 - y^13 + y^12 + 3*y^11 - 3*y^10 - y^9 + 5*y^8 - 5*y^7 + 2*y^5 + 2*y^3 + 2*y^2 + 2, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2)
\( x^{14} - x^{13} + x^{12} + 3x^{11} - 3x^{10} - x^{9} + 5x^{8} - 5x^{7} + 2x^{5} + 2x^{3} + 2x^{2} + 2 \)
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: |
\(-951316069511168\)
\(\medspace = -\,2^{12}\cdot 23\cdot 317^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.75\) | 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: | $2^{6/7}23^{1/2}317^{1/2}\approx 154.67469192665862$ | ||
| Ramified primes: |
\(2\), \(23\), \(317\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{692}a^{13}-\frac{17}{692}a^{12}+\frac{25}{173}a^{11}-\frac{10}{173}a^{10}-\frac{55}{692}a^{9}+\frac{187}{692}a^{8}+\frac{75}{173}a^{7}+\frac{53}{173}a^{6}+\frac{17}{173}a^{5}+\frac{149}{346}a^{4}+\frac{19}{173}a^{3}-\frac{44}{173}a^{2}+\frac{25}{346}a-\frac{27}{173}$
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: | $6$ | 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: |
$\frac{107}{692}a^{13}-\frac{89}{692}a^{12}-\frac{13}{346}a^{11}+\frac{109}{346}a^{10}-\frac{3}{692}a^{9}-\frac{751}{692}a^{8}+\frac{67}{173}a^{7}+\frac{135}{173}a^{6}-\frac{84}{173}a^{5}-\frac{319}{346}a^{4}+\frac{130}{173}a^{3}-\frac{37}{173}a^{2}+\frac{253}{346}a-\frac{121}{173}$, $\frac{22}{173}a^{13}-\frac{28}{173}a^{12}+\frac{75}{346}a^{11}+\frac{143}{346}a^{10}-\frac{171}{346}a^{9}+\frac{97}{346}a^{8}+\frac{199}{173}a^{7}-\frac{180}{173}a^{6}+\frac{112}{173}a^{5}+\frac{155}{173}a^{4}-\frac{58}{173}a^{3}-\frac{66}{173}a^{2}+\frac{62}{173}a-\frac{127}{173}$, $\frac{177}{692}a^{13}-\frac{17}{173}a^{12}-\frac{119}{692}a^{11}+\frac{133}{173}a^{10}-\frac{47}{692}a^{9}-\frac{332}{173}a^{8}+\frac{681}{692}a^{7}+\frac{212}{173}a^{6}-\frac{278}{173}a^{5}+\frac{77}{346}a^{4}+\frac{671}{346}a^{3}-\frac{176}{173}a^{2}-\frac{73}{346}a-\frac{43}{346}$, $\frac{3}{346}a^{13}+\frac{61}{173}a^{12}-\frac{23}{173}a^{11}+\frac{53}{346}a^{10}+\frac{177}{173}a^{9}-\frac{131}{346}a^{8}-\frac{311}{346}a^{7}+\frac{145}{173}a^{6}-\frac{71}{173}a^{5}+\frac{101}{173}a^{4}+\frac{114}{173}a^{3}+\frac{82}{173}a^{2}+\frac{248}{173}a+\frac{184}{173}$, $\frac{67}{346}a^{13}-\frac{101}{346}a^{12}+\frac{63}{173}a^{11}+\frac{44}{173}a^{10}-\frac{225}{346}a^{9}+\frac{73}{346}a^{8}+\frac{16}{173}a^{7}-\frac{164}{173}a^{6}+\frac{202}{173}a^{5}-\frac{224}{173}a^{4}+\frac{124}{173}a^{3}+\frac{159}{173}a^{2}-\frac{55}{173}a+\frac{15}{173}$, $\frac{157}{692}a^{13}-\frac{247}{692}a^{12}+\frac{65}{346}a^{11}+\frac{147}{346}a^{10}-\frac{677}{692}a^{9}-\frac{397}{692}a^{8}+\frac{184}{173}a^{7}-\frac{156}{173}a^{6}+\frac{74}{173}a^{5}+\frac{211}{346}a^{4}+\frac{42}{173}a^{3}+\frac{12}{173}a^{2}+\frac{119}{346}a-\frac{87}{173}$
| 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: | \( 114.877857314 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 114.877857314 \cdot 1}{2\cdot\sqrt{951316069511168}}\cr\approx \mathstrut & 0.719951089045 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2)
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 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2, 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 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2);
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 + x^12 + 3*x^11 - 3*x^10 - x^9 + 5*x^8 - 5*x^7 + 2*x^5 + 2*x^3 + 2*x^2 + 2);
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 $C_2^7.\GL(3,2)$ is not computed |
Intermediate fields
| 7.3.6431296.1 |
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 | ${\href{/padicField/3.7.0.1}{7} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }$ | ${\href{/padicField/7.14.0.1}{14} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.14.0.1}{14} }$ | R | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.7.0.1}{7} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(23\)
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.0.1 | $x^{2} + 21 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 23.4.0.1 | $x^{4} + 3 x^{2} + 19 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(317\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |