Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4)
gp: K = bnfinit(y^17 - 34*y^12 + 17*y^11 + 85*y^10 + 102*y^9 - 170*y^8 - 272*y^7 - 68*y^6 + 255*y^5 + 255*y^4 + 68*y^3 + 17*y^2 + 17*y + 4, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4)
\( x^{17} - 34 x^{12} + 17 x^{11} + 85 x^{10} + 102 x^{9} - 170 x^{8} - 272 x^{7} - 68 x^{6} + 255 x^{5} + \cdots + 4 \)
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: |
\(239072435685151324847153\)
\(\medspace = 17^{19}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(23.73\) | 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: | $17^{319/272}\approx 27.737240813400426$ | ||
| Ramified primes: |
\(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{88\!\cdots\!69}a^{16}+\frac{701881911652902}{88\!\cdots\!69}a^{15}+\frac{30\!\cdots\!40}{88\!\cdots\!69}a^{14}+\frac{33\!\cdots\!49}{88\!\cdots\!69}a^{13}+\frac{18\!\cdots\!01}{88\!\cdots\!69}a^{12}+\frac{40\!\cdots\!13}{88\!\cdots\!69}a^{11}-\frac{28\!\cdots\!30}{88\!\cdots\!69}a^{10}+\frac{66\!\cdots\!70}{88\!\cdots\!69}a^{9}-\frac{32\!\cdots\!33}{88\!\cdots\!69}a^{8}+\frac{21\!\cdots\!49}{88\!\cdots\!69}a^{7}+\frac{28\!\cdots\!75}{88\!\cdots\!69}a^{6}+\frac{14\!\cdots\!75}{88\!\cdots\!69}a^{5}+\frac{42\!\cdots\!78}{88\!\cdots\!69}a^{4}+\frac{10\!\cdots\!80}{88\!\cdots\!69}a^{3}-\frac{22\!\cdots\!18}{88\!\cdots\!69}a^{2}+\frac{21\!\cdots\!20}{88\!\cdots\!69}a+\frac{31\!\cdots\!59}{88\!\cdots\!69}$
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: | $8$ | 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: |
$\frac{16\!\cdots\!03}{88\!\cdots\!69}a^{16}-\frac{62\!\cdots\!97}{88\!\cdots\!69}a^{15}+\frac{11\!\cdots\!65}{88\!\cdots\!69}a^{14}-\frac{158617993707356}{88\!\cdots\!69}a^{13}+\frac{657429256058771}{88\!\cdots\!69}a^{12}-\frac{57\!\cdots\!78}{88\!\cdots\!69}a^{11}+\frac{49\!\cdots\!43}{88\!\cdots\!69}a^{10}+\frac{12\!\cdots\!70}{88\!\cdots\!69}a^{9}+\frac{12\!\cdots\!01}{88\!\cdots\!69}a^{8}-\frac{34\!\cdots\!33}{88\!\cdots\!69}a^{7}-\frac{33\!\cdots\!50}{88\!\cdots\!69}a^{6}+\frac{45\!\cdots\!30}{88\!\cdots\!69}a^{5}+\frac{43\!\cdots\!87}{88\!\cdots\!69}a^{4}+\frac{24\!\cdots\!19}{88\!\cdots\!69}a^{3}-\frac{31\!\cdots\!09}{88\!\cdots\!69}a^{2}+\frac{34\!\cdots\!72}{88\!\cdots\!69}a+\frac{39\!\cdots\!85}{88\!\cdots\!69}$, $\frac{37\!\cdots\!76}{88\!\cdots\!69}a^{16}-\frac{26\!\cdots\!31}{88\!\cdots\!69}a^{15}+\frac{19\!\cdots\!12}{88\!\cdots\!69}a^{14}-\frac{15\!\cdots\!62}{88\!\cdots\!69}a^{13}+\frac{98\!\cdots\!73}{88\!\cdots\!69}a^{12}-\frac{12\!\cdots\!43}{88\!\cdots\!69}a^{11}+\frac{15\!\cdots\!59}{88\!\cdots\!69}a^{10}+\frac{20\!\cdots\!63}{88\!\cdots\!69}a^{9}+\frac{24\!\cdots\!57}{88\!\cdots\!69}a^{8}-\frac{80\!\cdots\!90}{88\!\cdots\!69}a^{7}-\frac{46\!\cdots\!60}{88\!\cdots\!69}a^{6}+\frac{60\!\cdots\!77}{88\!\cdots\!69}a^{5}+\frac{92\!\cdots\!52}{88\!\cdots\!69}a^{4}+\frac{34\!\cdots\!46}{88\!\cdots\!69}a^{3}+\frac{11\!\cdots\!42}{88\!\cdots\!69}a^{2}+\frac{53\!\cdots\!84}{88\!\cdots\!69}a+\frac{14\!\cdots\!69}{88\!\cdots\!69}$, $\frac{11\!\cdots\!48}{88\!\cdots\!69}a^{16}-\frac{93\!\cdots\!99}{88\!\cdots\!69}a^{15}+\frac{84\!\cdots\!55}{88\!\cdots\!69}a^{14}-\frac{60\!\cdots\!03}{88\!\cdots\!69}a^{13}+\frac{39\!\cdots\!36}{88\!\cdots\!69}a^{12}-\frac{41\!\cdots\!66}{88\!\cdots\!69}a^{11}+\frac{33\!\cdots\!75}{88\!\cdots\!69}a^{10}-\frac{34\!\cdots\!54}{88\!\cdots\!69}a^{9}-\frac{32\!\cdots\!07}{88\!\cdots\!69}a^{8}-\frac{66\!\cdots\!51}{88\!\cdots\!69}a^{7}+\frac{17\!\cdots\!17}{88\!\cdots\!69}a^{6}+\frac{66\!\cdots\!65}{88\!\cdots\!69}a^{5}-\frac{10\!\cdots\!90}{88\!\cdots\!69}a^{4}-\frac{18\!\cdots\!01}{88\!\cdots\!69}a^{3}-\frac{30\!\cdots\!68}{88\!\cdots\!69}a^{2}+\frac{94\!\cdots\!83}{88\!\cdots\!69}a-\frac{17\!\cdots\!41}{88\!\cdots\!69}$, $\frac{28\!\cdots\!51}{88\!\cdots\!69}a^{16}+\frac{458898470848348}{88\!\cdots\!69}a^{15}-\frac{30\!\cdots\!12}{88\!\cdots\!69}a^{14}-\frac{29\!\cdots\!84}{88\!\cdots\!69}a^{13}+\frac{37\!\cdots\!47}{88\!\cdots\!69}a^{12}-\frac{98\!\cdots\!74}{88\!\cdots\!69}a^{11}+\frac{32\!\cdots\!84}{88\!\cdots\!69}a^{10}+\frac{35\!\cdots\!11}{88\!\cdots\!69}a^{9}+\frac{36\!\cdots\!21}{88\!\cdots\!69}a^{8}-\frac{86\!\cdots\!18}{88\!\cdots\!69}a^{7}-\frac{12\!\cdots\!82}{88\!\cdots\!69}a^{6}+\frac{15\!\cdots\!94}{88\!\cdots\!69}a^{5}+\frac{20\!\cdots\!49}{88\!\cdots\!69}a^{4}+\frac{11\!\cdots\!10}{88\!\cdots\!69}a^{3}-\frac{59\!\cdots\!88}{88\!\cdots\!69}a^{2}-\frac{11\!\cdots\!32}{88\!\cdots\!69}a-\frac{40\!\cdots\!79}{88\!\cdots\!69}$, $\frac{11\!\cdots\!12}{88\!\cdots\!69}a^{16}-\frac{88\!\cdots\!42}{88\!\cdots\!69}a^{15}+\frac{26\!\cdots\!46}{88\!\cdots\!69}a^{14}+\frac{14\!\cdots\!24}{88\!\cdots\!69}a^{13}-\frac{10\!\cdots\!43}{88\!\cdots\!69}a^{12}-\frac{42\!\cdots\!74}{88\!\cdots\!69}a^{11}+\frac{32\!\cdots\!10}{88\!\cdots\!69}a^{10}-\frac{15\!\cdots\!61}{88\!\cdots\!69}a^{9}-\frac{63\!\cdots\!38}{88\!\cdots\!69}a^{8}-\frac{79\!\cdots\!43}{88\!\cdots\!69}a^{7}+\frac{16\!\cdots\!84}{88\!\cdots\!69}a^{6}+\frac{15\!\cdots\!61}{88\!\cdots\!69}a^{5}+\frac{22\!\cdots\!85}{88\!\cdots\!69}a^{4}-\frac{22\!\cdots\!17}{88\!\cdots\!69}a^{3}-\frac{10\!\cdots\!58}{88\!\cdots\!69}a^{2}+\frac{41\!\cdots\!69}{88\!\cdots\!69}a-\frac{34\!\cdots\!39}{88\!\cdots\!69}$, $\frac{12\!\cdots\!18}{88\!\cdots\!69}a^{16}-\frac{31\!\cdots\!25}{88\!\cdots\!69}a^{15}+\frac{11\!\cdots\!59}{88\!\cdots\!69}a^{14}-\frac{10\!\cdots\!12}{88\!\cdots\!69}a^{13}+\frac{909519314251164}{88\!\cdots\!69}a^{12}-\frac{40\!\cdots\!35}{88\!\cdots\!69}a^{11}+\frac{30\!\cdots\!73}{88\!\cdots\!69}a^{10}+\frac{93\!\cdots\!52}{88\!\cdots\!69}a^{9}+\frac{10\!\cdots\!31}{88\!\cdots\!69}a^{8}-\frac{23\!\cdots\!08}{88\!\cdots\!69}a^{7}-\frac{27\!\cdots\!64}{88\!\cdots\!69}a^{6}-\frac{16\!\cdots\!64}{88\!\cdots\!69}a^{5}+\frac{31\!\cdots\!94}{88\!\cdots\!69}a^{4}+\frac{23\!\cdots\!22}{88\!\cdots\!69}a^{3}+\frac{21\!\cdots\!93}{88\!\cdots\!69}a^{2}+\frac{85\!\cdots\!42}{88\!\cdots\!69}a+\frac{94\!\cdots\!19}{88\!\cdots\!69}$, $\frac{48\!\cdots\!04}{88\!\cdots\!69}a^{16}-\frac{63\!\cdots\!77}{88\!\cdots\!69}a^{15}+\frac{95\!\cdots\!24}{88\!\cdots\!69}a^{14}-\frac{83\!\cdots\!55}{88\!\cdots\!69}a^{13}+\frac{636193307694728}{88\!\cdots\!69}a^{12}-\frac{15\!\cdots\!69}{88\!\cdots\!69}a^{11}+\frac{28\!\cdots\!29}{88\!\cdots\!69}a^{10}-\frac{18\!\cdots\!34}{88\!\cdots\!69}a^{9}+\frac{41\!\cdots\!35}{88\!\cdots\!69}a^{8}-\frac{84\!\cdots\!62}{88\!\cdots\!69}a^{7}-\frac{25\!\cdots\!10}{88\!\cdots\!69}a^{6}-\frac{49\!\cdots\!41}{88\!\cdots\!69}a^{5}+\frac{10\!\cdots\!79}{88\!\cdots\!69}a^{4}+\frac{71\!\cdots\!17}{88\!\cdots\!69}a^{3}-\frac{56\!\cdots\!15}{88\!\cdots\!69}a^{2}+\frac{30\!\cdots\!72}{88\!\cdots\!69}a+\frac{74\!\cdots\!29}{88\!\cdots\!69}$, $\frac{15\!\cdots\!82}{88\!\cdots\!69}a^{16}-\frac{35\!\cdots\!02}{88\!\cdots\!69}a^{15}-\frac{632123160510473}{88\!\cdots\!69}a^{14}+\frac{14\!\cdots\!57}{88\!\cdots\!69}a^{13}-\frac{22\!\cdots\!33}{88\!\cdots\!69}a^{12}-\frac{52\!\cdots\!96}{88\!\cdots\!69}a^{11}+\frac{38\!\cdots\!69}{88\!\cdots\!69}a^{10}+\frac{12\!\cdots\!22}{88\!\cdots\!69}a^{9}+\frac{12\!\cdots\!70}{88\!\cdots\!69}a^{8}-\frac{29\!\cdots\!26}{88\!\cdots\!69}a^{7}-\frac{36\!\cdots\!25}{88\!\cdots\!69}a^{6}+\frac{55\!\cdots\!10}{88\!\cdots\!69}a^{5}+\frac{39\!\cdots\!95}{88\!\cdots\!69}a^{4}+\frac{31\!\cdots\!67}{88\!\cdots\!69}a^{3}+\frac{12\!\cdots\!38}{88\!\cdots\!69}a^{2}+\frac{25\!\cdots\!67}{88\!\cdots\!69}a+\frac{83\!\cdots\!67}{88\!\cdots\!69}$
| 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: | \( 195995.113671 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 195995.113671 \cdot 1}{2\cdot\sqrt{239072435685151324847153}}\cr\approx \mathstrut & 0.973687148598 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4)
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 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4, 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 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4);
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 - 34*x^12 + 17*x^11 + 85*x^10 + 102*x^9 - 170*x^8 - 272*x^7 - 68*x^6 + 255*x^5 + 255*x^4 + 68*x^3 + 17*x^2 + 17*x + 4);
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
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 272 |
| The 17 conjugacy class representatives for $F_{17}$ |
| Character table for $F_{17}$ |
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 | ${\href{/padicField/2.8.0.1}{8} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.8.0.1}{8} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.17.19.3 | $x^{17} + 102 x^{3} + 17$ | $17$ | $1$ | $19$ | $F_{17}$ | $[19/16]_{16}$ |