Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800)
gp: K = bnfinit(y^17 - 5*y^16 + 40*y^15 - 140*y^14 + 610*y^13 - 1622*y^12 + 4870*y^11 - 10220*y^10 + 22720*y^9 - 38080*y^8 + 63500*y^7 - 84100*y^6 + 102200*y^5 - 102400*y^4 + 83000*y^3 - 55288*y^2 + 23360*y - 5800, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800)
\( x^{17} - 5 x^{16} + 40 x^{15} - 140 x^{14} + 610 x^{13} - 1622 x^{12} + 4870 x^{11} - 10220 x^{10} + \cdots - 5800 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $17$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2058911320946490000000000000000\)
\(\medspace = 2^{16}\cdot 3^{30}\cdot 5^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.70\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}$, $\frac{1}{6}a^{8}+\frac{1}{6}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{42}a^{9}-\frac{1}{14}a^{8}-\frac{1}{14}a^{7}-\frac{5}{14}a^{6}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{1}{21}$, $\frac{1}{42}a^{10}+\frac{1}{21}a^{8}-\frac{5}{21}a^{7}+\frac{11}{42}a^{6}+\frac{4}{21}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{2}{21}a^{2}-\frac{1}{3}a+\frac{10}{21}$, $\frac{1}{42}a^{11}+\frac{1}{14}a^{8}+\frac{1}{14}a^{7}+\frac{1}{14}a^{6}+\frac{2}{7}a^{4}-\frac{2}{7}a^{3}-\frac{2}{21}a^{2}+\frac{2}{7}a-\frac{3}{7}$, $\frac{1}{84}a^{12}-\frac{1}{84}a^{11}+\frac{1}{42}a^{8}+\frac{1}{42}a^{7}+\frac{1}{6}a^{6}-\frac{4}{21}a^{5}+\frac{2}{21}a^{4}-\frac{8}{21}a^{3}+\frac{2}{7}a^{2}-\frac{1}{3}a-\frac{4}{21}$, $\frac{1}{84}a^{13}-\frac{1}{84}a^{11}-\frac{1}{21}a^{8}+\frac{2}{21}a^{7}-\frac{1}{3}a^{6}+\frac{5}{21}a^{5}+\frac{4}{21}a^{4}-\frac{4}{21}a^{3}-\frac{3}{7}a^{2}+\frac{8}{21}a+\frac{2}{21}$, $\frac{1}{84}a^{14}-\frac{1}{84}a^{11}-\frac{1}{42}a^{8}+\frac{1}{21}a^{7}+\frac{4}{21}a^{6}-\frac{8}{21}a^{4}+\frac{1}{21}a^{3}+\frac{2}{21}a^{2}-\frac{8}{21}a-\frac{2}{21}$, $\frac{1}{252}a^{15}+\frac{1}{252}a^{12}-\frac{1}{126}a^{9}+\frac{3}{14}a^{7}-\frac{16}{63}a^{6}-\frac{1}{7}a^{5}+\frac{2}{7}a^{4}+\frac{29}{63}a^{3}+\frac{1}{7}a^{2}+\frac{2}{7}a-\frac{31}{63}$, $\frac{1}{1260}a^{16}-\frac{1}{252}a^{13}-\frac{1}{105}a^{11}+\frac{1}{126}a^{10}+\frac{1}{14}a^{8}+\frac{13}{63}a^{7}-\frac{1}{2}a^{6}-\frac{3}{7}a^{5}+\frac{1}{9}a^{4}+\frac{1}{7}a^{3}+\frac{8}{21}a^{2}-\frac{1}{45}a+\frac{1}{7}$
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 $1$ (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: | $8$ | 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: |
$\frac{3}{140}a^{16}-\frac{23}{126}a^{15}+\frac{13}{12}a^{14}-\frac{106}{21}a^{13}+\frac{2197}{126}a^{12}-\frac{1954}{35}a^{11}+\frac{2882}{21}a^{10}-\frac{20530}{63}a^{9}+\frac{4238}{7}a^{8}-\frac{15139}{14}a^{7}+\frac{193127}{126}a^{6}-\frac{41857}{21}a^{5}+\frac{43879}{21}a^{4}-\frac{112496}{63}a^{3}+\frac{8536}{7}a^{2}-\frac{54793}{105}a+\frac{7957}{63}$, $\frac{1}{140}a^{16}-\frac{5}{252}a^{15}+\frac{1}{7}a^{14}-\frac{5}{21}a^{13}+\frac{181}{252}a^{12}-\frac{41}{420}a^{11}-\frac{53}{21}a^{10}+\frac{1487}{126}a^{9}-\frac{239}{6}a^{8}+\frac{3313}{42}a^{7}-\frac{10453}{63}a^{6}+\frac{4835}{21}a^{5}-\frac{6578}{21}a^{4}+\frac{2876}{9}a^{3}-\frac{1677}{7}a^{2}+\frac{16979}{105}a-\frac{1777}{63}$, $\frac{1}{36}a^{16}-\frac{1}{9}a^{15}+\frac{5}{6}a^{14}-\frac{611}{252}a^{13}+\frac{2423}{252}a^{12}-\frac{611}{28}a^{11}+\frac{3637}{63}a^{10}-\frac{13289}{126}a^{9}+\frac{2771}{14}a^{8}-\frac{36755}{126}a^{7}+\frac{24442}{63}a^{6}-\frac{9367}{21}a^{5}+\frac{25328}{63}a^{4}-\frac{20291}{63}a^{3}+\frac{1244}{7}a^{2}-\frac{4447}{63}a+\frac{1411}{63}$, $\frac{1}{315}a^{16}-\frac{1}{18}a^{15}+\frac{2}{7}a^{14}-\frac{421}{252}a^{13}+\frac{1381}{252}a^{12}-\frac{4153}{210}a^{11}+\frac{6061}{126}a^{10}-\frac{7769}{63}a^{9}+\frac{3259}{14}a^{8}-\frac{55525}{126}a^{7}+\frac{40769}{63}a^{6}-\frac{6231}{7}a^{5}+\frac{61771}{63}a^{4}-\frac{56611}{63}a^{3}+\frac{13840}{21}a^{2}-\frac{13279}{45}a+\frac{665}{9}$, $\frac{71}{1260}a^{16}-\frac{5}{36}a^{15}+\frac{19}{14}a^{14}-\frac{101}{63}a^{13}+\frac{331}{36}a^{12}+\frac{2431}{420}a^{11}-\frac{709}{126}a^{10}+\frac{20807}{126}a^{9}-\frac{2325}{7}a^{8}+\frac{61142}{63}a^{7}-\frac{99583}{63}a^{6}+\frac{17915}{7}a^{5}-\frac{195793}{63}a^{4}+\frac{186146}{63}a^{3}-\frac{48392}{21}a^{2}+\frac{332473}{315}a-\frac{18553}{63}$, $\frac{71}{252}a^{16}-\frac{107}{84}a^{15}+\frac{769}{84}a^{14}-\frac{7363}{252}a^{13}+\frac{4733}{42}a^{12}-\frac{5792}{21}a^{11}+\frac{45305}{63}a^{10}-\frac{29287}{21}a^{9}+\frac{110039}{42}a^{8}-\frac{506131}{126}a^{7}+\frac{38246}{7}a^{6}-\frac{44495}{7}a^{5}+\frac{54085}{9}a^{4}-\frac{32946}{7}a^{3}+\frac{18964}{7}a^{2}-\frac{9029}{9}a+\frac{3917}{21}$, $\frac{19}{1260}a^{16}-\frac{4}{63}a^{15}+\frac{11}{21}a^{14}-\frac{391}{252}a^{13}+\frac{440}{63}a^{12}-\frac{1091}{70}a^{11}+\frac{6133}{126}a^{10}-\frac{5290}{63}a^{9}+\frac{2733}{14}a^{8}-\frac{16349}{63}a^{7}+\frac{28300}{63}a^{6}-\frac{9386}{21}a^{5}+\frac{33310}{63}a^{4}-\frac{23192}{63}a^{3}+\frac{1321}{7}a^{2}-\frac{3694}{45}a-\frac{4997}{63}$, $\frac{13}{630}a^{16}-\frac{1}{7}a^{15}+\frac{83}{84}a^{14}-\frac{1037}{252}a^{13}+\frac{1321}{84}a^{12}-\frac{6393}{140}a^{11}+\frac{14993}{126}a^{10}-\frac{5351}{21}a^{9}+\frac{20201}{42}a^{8}-\frac{48502}{63}a^{7}+\frac{22186}{21}a^{6}-\frac{25855}{21}a^{5}+\frac{73781}{63}a^{4}-\frac{6319}{7}a^{3}+\frac{10876}{21}a^{2}-\frac{60197}{315}a+37$
| 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: | \( 1704688667.94 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 1704688667.94 \cdot 1}{2\cdot\sqrt{2058911320946490000000000000000}}\cr\approx \mathstrut & 2.88579316349 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800)
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^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800, 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^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800);
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^17 - 5*x^16 + 40*x^15 - 140*x^14 + 610*x^13 - 1622*x^12 + 4870*x^11 - 10220*x^10 + 22720*x^9 - 38080*x^8 + 63500*x^7 - 84100*x^6 + 102200*x^5 - 102400*x^4 + 83000*x^3 - 55288*x^2 + 23360*x - 5800);
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
$\PSL(2,16).C_4$ (as 17T8):
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 16320 |
| The 17 conjugacy class representatives for $\PSL(2,16):C_4$ |
| Character table for $\PSL(2,16):C_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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]
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 | R | ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | $17$ | ${\href{/padicField/37.8.0.1}{8} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $16$ | $16$ | $1$ | $16$ | ||||
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.15.29.83 | $x^{15} + 6 x^{9} + 9 x^{7} + 18 x^{6} + 9 x^{5} + 12$ | $15$ | $1$ | $29$ | $F_5 \times S_3$ | $[5/2]_{10}^{4}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.15.15.18 | $x^{15} - 60 x^{12} + 60 x^{11} + 15 x^{10} + 4425 x^{9} - 2400 x^{8} + 600 x^{7} + 17725 x^{6} + 88575 x^{5} - 1875 x^{4} - 4000 x^{3} + 4500 x^{2} + 1500 x + 125$ | $5$ | $3$ | $15$ | $F_5\times C_3$ | $[5/4]_{4}^{3}$ |