Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 1)
gp: K = bnfinit(y^16 - 8*y^14 + 4*y^12 + 120*y^10 + 198*y^8 - 120*y^6 + 4*y^4 + 8*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 1)
\( x^{16} - 8x^{14} + 4x^{12} + 120x^{10} + 198x^{8} - 120x^{6} + 4x^{4} + 8x^{2} + 1 \)
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: |
\(4668406261161555656704\)
\(\medspace = 2^{38}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(22.61\) | 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: | $2^{19/8}19^{2/3}\approx 36.93589613868019$ | ||
| Ramified primes: |
\(2\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{16}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}+\frac{1}{16}$, $\frac{1}{16}a^{9}-\frac{1}{8}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}+\frac{5}{16}a+\frac{1}{4}$, $\frac{1}{16}a^{10}-\frac{1}{8}a^{6}-\frac{1}{4}a^{4}-\frac{3}{16}a^{2}+\frac{1}{4}$, $\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{1}{32}a^{9}-\frac{1}{32}a^{8}+\frac{1}{16}a^{7}-\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}+\frac{9}{32}a^{3}+\frac{7}{32}a^{2}+\frac{7}{32}a-\frac{9}{32}$, $\frac{1}{64}a^{12}-\frac{1}{32}a^{10}+\frac{1}{64}a^{8}+\frac{1}{16}a^{6}+\frac{15}{64}a^{4}-\frac{5}{32}a^{2}+\frac{31}{64}$, $\frac{1}{64}a^{13}-\frac{1}{32}a^{10}-\frac{1}{64}a^{9}-\frac{1}{32}a^{8}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}-\frac{5}{64}a^{5}-\frac{1}{16}a^{4}-\frac{1}{8}a^{3}+\frac{7}{32}a^{2}+\frac{29}{64}a-\frac{9}{32}$, $\frac{1}{320}a^{14}+\frac{9}{320}a^{10}+\frac{1}{160}a^{8}-\frac{1}{320}a^{6}-\frac{17}{80}a^{4}+\frac{27}{64}a^{2}+\frac{69}{160}$, $\frac{1}{640}a^{15}-\frac{1}{640}a^{14}-\frac{1}{128}a^{13}-\frac{1}{128}a^{12}-\frac{1}{640}a^{11}+\frac{1}{640}a^{10}+\frac{17}{640}a^{9}-\frac{7}{640}a^{8}+\frac{19}{640}a^{7}+\frac{61}{640}a^{6}-\frac{23}{640}a^{5}-\frac{87}{640}a^{4}-\frac{47}{128}a^{3}-\frac{1}{128}a^{2}-\frac{77}{640}a-\frac{53}{640}$
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: |
\( \frac{19}{80} a^{14} - \frac{61}{32} a^{12} + \frac{81}{80} a^{10} + \frac{4541}{160} a^{8} + \frac{3701}{80} a^{6} - \frac{4499}{160} a^{4} + \frac{75}{16} a^{2} + \frac{159}{160} \)
(order $4$)
| 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: |
$a$, $\frac{1}{128}a^{15}+\frac{33}{640}a^{14}-\frac{1}{128}a^{13}-\frac{49}{128}a^{12}-\frac{49}{128}a^{11}-\frac{13}{640}a^{10}+\frac{125}{128}a^{9}+\frac{3981}{640}a^{8}+\frac{1043}{128}a^{7}+\frac{8827}{640}a^{6}+\frac{1621}{128}a^{5}+\frac{721}{640}a^{4}-\frac{123}{128}a^{3}-\frac{139}{128}a^{2}-\frac{9}{128}a-\frac{561}{640}$, $\frac{117}{640}a^{15}+\frac{89}{640}a^{14}-\frac{183}{128}a^{13}-\frac{143}{128}a^{12}+\frac{303}{640}a^{11}+\frac{391}{640}a^{10}+\frac{14099}{640}a^{9}+\frac{10583}{640}a^{8}+\frac{25703}{640}a^{7}+\frac{17291}{640}a^{6}-\frac{9481}{640}a^{5}-\frac{9977}{640}a^{4}-\frac{343}{128}a^{3}+\frac{633}{128}a^{2}+\frac{81}{640}a+\frac{517}{640}$, $\frac{239}{640}a^{15}+\frac{127}{640}a^{14}-\frac{391}{128}a^{13}-\frac{207}{128}a^{12}+\frac{1321}{640}a^{11}+\frac{653}{640}a^{10}+\frac{28343}{640}a^{9}+\frac{15219}{640}a^{8}+\frac{42221}{640}a^{7}+\frac{22853}{640}a^{6}-\frac{34657}{640}a^{5}-\frac{19921}{640}a^{4}+\frac{2135}{128}a^{3}+\frac{299}{128}a^{2}-\frac{603}{640}a+\frac{1521}{640}$, $\frac{139}{320}a^{15}+\frac{3}{4}a^{14}-\frac{29}{8}a^{13}-\frac{97}{16}a^{12}+\frac{951}{320}a^{11}+\frac{7}{2}a^{10}+\frac{8199}{160}a^{9}+\frac{359}{4}a^{8}+\frac{21821}{320}a^{7}+141a^{6}-\frac{6213}{80}a^{5}-\frac{1637}{16}a^{4}+\frac{1645}{64}a^{3}+\frac{41}{4}a^{2}-\frac{509}{160}a+\frac{43}{8}$, $\frac{3}{40}a^{15}-\frac{7}{64}a^{14}-\frac{37}{64}a^{13}+\frac{53}{64}a^{12}+\frac{23}{160}a^{11}-\frac{5}{64}a^{10}+\frac{2863}{320}a^{9}-\frac{845}{64}a^{8}+\frac{1399}{80}a^{7}-\frac{1741}{64}a^{6}-\frac{807}{320}a^{5}+\frac{111}{64}a^{4}+\frac{95}{32}a^{3}+\frac{49}{64}a^{2}+\frac{417}{320}a+\frac{17}{64}$, $\frac{51}{160}a^{15}+\frac{1}{80}a^{14}-\frac{161}{64}a^{13}-\frac{3}{32}a^{12}+\frac{41}{40}a^{11}-\frac{1}{80}a^{10}+\frac{12219}{320}a^{9}+\frac{259}{160}a^{8}+\frac{10779}{160}a^{7}+\frac{259}{80}a^{6}-\frac{9131}{320}a^{5}-\frac{301}{160}a^{4}+\frac{29}{16}a^{3}-\frac{55}{16}a^{2}+\frac{101}{320}a-\frac{79}{160}$
| 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: | \( 60925.785768593894 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 60925.785768593894 \cdot 1}{4\cdot\sqrt{4668406261161555656704}}\cr\approx \mathstrut & 0.541496648727078 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 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^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 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^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 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^16 - 8*x^14 + 4*x^12 + 120*x^10 + 198*x^8 - 120*x^6 + 4*x^4 + 8*x^2 + 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
$\SL(2,3):C_2$ (as 16T60):
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 48 |
| The 14 conjugacy class representatives for $\SL(2,3):C_2$ |
| Character table for $\SL(2,3):C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.23104.1, 8.0.2135179264.1 |
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 24 sibling: | deg 24 |
| 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 | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.3.0.1}{3} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $16$ | $16$ | $1$ | $38$ | |||
|
\(19\)
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 19.12.8.1 | $x^{12} + 386 x^{10} + 109 x^{9} + 55308 x^{8} + 21792 x^{7} + 3500499 x^{6} + 2034936 x^{5} + 84821873 x^{4} + 99877907 x^{3} + 174885148 x^{2} + 920938017 x + 335157671$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.76.6t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.76.6t1.a.b | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 2.23104.24t21.a.a | $2$ | $ 2^{6} \cdot 19^{2}$ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
| 2.23104.24t21.a.b | $2$ | $ 2^{6} \cdot 19^{2}$ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ | |
| * | 2.1216.16t60.a.a | $2$ | $ 2^{6} \cdot 19 $ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.1216.16t60.a.b | $2$ | $ 2^{6} \cdot 19 $ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.1216.16t60.a.c | $2$ | $ 2^{6} \cdot 19 $ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 2.1216.16t60.a.d | $2$ | $ 2^{6} \cdot 19 $ | 16.0.4668406261161555656704.1 | $\SL(2,3):C_2$ (as 16T60) | $0$ | $0$ |
| * | 3.23104.6t6.a.a | $3$ | $ 2^{6} \cdot 19^{2}$ | 6.4.8340544.1 | $A_4\times C_2$ (as 6T6) | $1$ | $1$ |
| * | 3.23104.4t4.b.a | $3$ | $ 2^{6} \cdot 19^{2}$ | 4.0.23104.1 | $A_4$ (as 4T4) | $1$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.