Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49)
gp: K = bnfinit(y^16 - 8*y^14 + 41*y^12 - 144*y^10 + 320*y^8 - 438*y^6 + 365*y^4 - 182*y^2 + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49)
\( x^{16} - 8x^{14} + 41x^{12} - 144x^{10} + 320x^{8} - 438x^{6} + 365x^{4} - 182x^{2} + 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(162447943996702457856\)
\(\medspace = 2^{32}\cdot 3^{8}\cdot 7^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.33\) | 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: | $2^{9/4}3^{1/2}7^{1/2}\approx 21.798526485920096$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{6}+\frac{1}{3}a^{2}+\frac{1}{6}$, $\frac{1}{6}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{7}+\frac{1}{3}a^{3}+\frac{1}{6}a$, $\frac{1}{5502}a^{14}+\frac{419}{5502}a^{12}+\frac{176}{917}a^{10}+\frac{719}{2751}a^{8}-\frac{313}{917}a^{6}-\frac{443}{2751}a^{4}+\frac{1681}{5502}a^{2}-\frac{63}{262}$, $\frac{1}{38514}a^{15}+\frac{1585}{19257}a^{13}-\frac{1658}{6419}a^{11}+\frac{8972}{19257}a^{9}-\frac{2147}{6419}a^{7}-\frac{5945}{19257}a^{5}-\frac{3821}{38514}a^{3}+\frac{34}{917}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
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: |
\( \frac{83}{786} a^{14} - \frac{77}{131} a^{12} + \frac{1118}{393} a^{10} - \frac{3203}{393} a^{8} + \frac{5248}{393} a^{6} - \frac{5329}{393} a^{4} + \frac{2317}{262} a^{2} - \frac{998}{393} \)
(order $6$)
| 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: |
$\frac{110}{917}a^{14}-\frac{4979}{5502}a^{12}+\frac{11941}{2751}a^{10}-\frac{13299}{917}a^{8}+\frac{77180}{2751}a^{6}-\frac{27768}{917}a^{4}+\frac{50380}{2751}a^{2}-\frac{4613}{786}$, $\frac{1625}{38514}a^{15}-\frac{4813}{19257}a^{13}+\frac{8149}{6419}a^{11}-\frac{75077}{19257}a^{9}+\frac{47994}{6419}a^{7}-\frac{186181}{19257}a^{5}+\frac{261227}{38514}a^{3}-\frac{687}{917}a$, $\frac{403}{6419}a^{15}-\frac{6161}{12838}a^{13}+\frac{15669}{6419}a^{11}-\frac{54133}{6419}a^{9}+\frac{117067}{6419}a^{7}-\frac{157152}{6419}a^{5}+\frac{122658}{6419}a^{3}-\frac{12557}{1834}a$, $\frac{3611}{12838}a^{15}-\frac{23883}{12838}a^{13}+\frac{57019}{6419}a^{11}-\frac{178533}{6419}a^{9}+\frac{318536}{6419}a^{7}-\frac{316790}{6419}a^{5}+\frac{337007}{12838}a^{3}-\frac{12545}{1834}a+1$, $\frac{81}{12838}a^{15}-\frac{635}{5502}a^{14}-\frac{6389}{38514}a^{13}+\frac{3533}{5502}a^{12}+\frac{17347}{19257}a^{11}-\frac{2637}{917}a^{10}-\frac{24291}{6419}a^{9}+\frac{22109}{2751}a^{8}+\frac{193643}{19257}a^{7}-\frac{9404}{917}a^{6}-\frac{83567}{6419}a^{5}+\frac{17209}{2751}a^{4}+\frac{329621}{38514}a^{3}-\frac{16553}{5502}a^{2}-\frac{22871}{5502}a+\frac{181}{262}$, $\frac{1199}{38514}a^{15}+\frac{90}{917}a^{14}-\frac{1879}{12838}a^{13}-\frac{691}{1834}a^{12}+\frac{12263}{19257}a^{11}+\frac{1506}{917}a^{10}-\frac{26492}{19257}a^{9}-\frac{2628}{917}a^{8}+\frac{12136}{19257}a^{7}-\frac{1209}{917}a^{6}+\frac{54806}{19257}a^{5}+\frac{8292}{917}a^{4}-\frac{46477}{12838}a^{3}-\frac{8268}{917}a^{2}+\frac{3425}{5502}a+\frac{957}{262}$, $\frac{1175}{38514}a^{15}+\frac{367}{5502}a^{14}-\frac{5554}{19257}a^{13}-\frac{2117}{5502}a^{12}+\frac{9645}{6419}a^{11}+\frac{4874}{2751}a^{10}-\frac{107021}{19257}a^{9}-\frac{13978}{2751}a^{8}+\frac{83389}{6419}a^{7}+\frac{20353}{2751}a^{6}-\frac{341710}{19257}a^{5}-\frac{14027}{2751}a^{4}+\frac{517145}{38514}a^{3}+\frac{8039}{5502}a^{2}-\frac{4983}{917}a+\frac{329}{786}$
| 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: | \( 11233.812692 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 11233.812692 \cdot 2}{6\cdot\sqrt{162447943996702457856}}\cr\approx \mathstrut & 0.71365409301 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49)
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^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49, 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^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49);
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^16 - 8*x^14 + 41*x^12 - 144*x^10 + 320*x^8 - 438*x^6 + 365*x^4 - 182*x^2 + 49);
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
$C_2^2\times D_4$ (as 16T25):
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 solvable group of order 32 |
| The 20 conjugacy class representatives for $C_2^2 \times D_4$ |
| Character table for $C_2^2 \times D_4$ |
Intermediate fields
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{8}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.13 | $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ |
| 2.8.16.13 | $x^{8} + 8 x^{7} + 18 x^{6} + 4 x^{5} - 8 x^{4} + 24 x^{3} + 20 x^{2} - 8 x + 4$ | $4$ | $2$ | $16$ | $D_4\times C_2$ | $[2, 2, 3]^{2}$ | |
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |