Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103)
gp: K = bnfinit(y^17 - y^16 - 208*y^15 - 17*y^14 + 15287*y^13 + 13881*y^12 - 487578*y^11 - 703261*y^10 + 6754359*y^9 + 10540902*y^8 - 41136753*y^7 - 57683825*y^6 + 92010954*y^5 + 95287840*y^4 - 17501435*y^3 - 25026156*y^2 - 563260*y + 1246103, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103)
\( x^{17} - x^{16} - 208 x^{15} - 17 x^{14} + 15287 x^{13} + 13881 x^{12} - 487578 x^{11} - 703261 x^{10} + \cdots + 1246103 \)
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: | $[17, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2200187128095499475530336818113367454680001\)
\(\medspace = 443^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(309.55\) | 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: | $443^{16/17}\approx 309.5515042584762$ | ||
| Ramified primes: |
\(443\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $17$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(443\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{443}(1,·)$, $\chi_{443}(67,·)$, $\chi_{443}(324,·)$, $\chi_{443}(267,·)$, $\chi_{443}(13,·)$, $\chi_{443}(270,·)$, $\chi_{443}(209,·)$, $\chi_{443}(409,·)$, $\chi_{443}(225,·)$, $\chi_{443}(123,·)$, $\chi_{443}(425,·)$, $\chi_{443}(428,·)$, $\chi_{443}(370,·)$, $\chi_{443}(169,·)$, $\chi_{443}(248,·)$, $\chi_{443}(59,·)$, $\chi_{443}(380,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{40\!\cdots\!61}a^{16}+\frac{30\!\cdots\!51}{40\!\cdots\!61}a^{15}-\frac{23\!\cdots\!23}{40\!\cdots\!61}a^{14}-\frac{15\!\cdots\!62}{40\!\cdots\!61}a^{13}+\frac{55\!\cdots\!16}{40\!\cdots\!61}a^{12}+\frac{75\!\cdots\!35}{40\!\cdots\!61}a^{11}+\frac{44\!\cdots\!44}{40\!\cdots\!61}a^{10}+\frac{13\!\cdots\!35}{40\!\cdots\!61}a^{9}-\frac{25\!\cdots\!89}{40\!\cdots\!61}a^{8}-\frac{14\!\cdots\!79}{40\!\cdots\!61}a^{7}-\frac{26\!\cdots\!97}{40\!\cdots\!61}a^{6}-\frac{18\!\cdots\!13}{40\!\cdots\!61}a^{5}+\frac{78\!\cdots\!20}{40\!\cdots\!61}a^{4}-\frac{42\!\cdots\!25}{40\!\cdots\!61}a^{3}+\frac{15\!\cdots\!39}{40\!\cdots\!61}a^{2}+\frac{11\!\cdots\!06}{40\!\cdots\!61}a-\frac{12\!\cdots\!69}{40\!\cdots\!61}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| 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: | $16$ | 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: |
$\frac{10\!\cdots\!17}{40\!\cdots\!61}a^{16}-\frac{15\!\cdots\!95}{40\!\cdots\!61}a^{15}-\frac{21\!\cdots\!57}{40\!\cdots\!61}a^{14}+\frac{84\!\cdots\!67}{40\!\cdots\!61}a^{13}+\frac{15\!\cdots\!21}{40\!\cdots\!61}a^{12}+\frac{69\!\cdots\!72}{40\!\cdots\!61}a^{11}-\frac{51\!\cdots\!17}{40\!\cdots\!61}a^{10}-\frac{49\!\cdots\!99}{40\!\cdots\!61}a^{9}+\frac{72\!\cdots\!26}{40\!\cdots\!61}a^{8}+\frac{75\!\cdots\!44}{40\!\cdots\!61}a^{7}-\frac{46\!\cdots\!63}{40\!\cdots\!61}a^{6}-\frac{38\!\cdots\!50}{40\!\cdots\!61}a^{5}+\frac{11\!\cdots\!05}{40\!\cdots\!61}a^{4}+\frac{45\!\cdots\!69}{40\!\cdots\!61}a^{3}-\frac{34\!\cdots\!35}{40\!\cdots\!61}a^{2}-\frac{83\!\cdots\!34}{40\!\cdots\!61}a+\frac{22\!\cdots\!21}{40\!\cdots\!61}$, $\frac{40\!\cdots\!28}{40\!\cdots\!61}a^{16}-\frac{54\!\cdots\!57}{40\!\cdots\!61}a^{15}-\frac{83\!\cdots\!64}{40\!\cdots\!61}a^{14}+\frac{22\!\cdots\!56}{40\!\cdots\!61}a^{13}+\frac{61\!\cdots\!53}{40\!\cdots\!61}a^{12}+\frac{34\!\cdots\!19}{40\!\cdots\!61}a^{11}-\frac{19\!\cdots\!06}{40\!\cdots\!61}a^{10}-\frac{21\!\cdots\!71}{40\!\cdots\!61}a^{9}+\frac{28\!\cdots\!07}{40\!\cdots\!61}a^{8}+\frac{32\!\cdots\!08}{40\!\cdots\!61}a^{7}-\frac{17\!\cdots\!96}{40\!\cdots\!61}a^{6}-\frac{17\!\cdots\!09}{40\!\cdots\!61}a^{5}+\frac{43\!\cdots\!30}{40\!\cdots\!61}a^{4}+\frac{23\!\cdots\!01}{40\!\cdots\!61}a^{3}-\frac{15\!\cdots\!60}{40\!\cdots\!61}a^{2}-\frac{47\!\cdots\!67}{40\!\cdots\!61}a+\frac{14\!\cdots\!90}{40\!\cdots\!61}$, $\frac{94\!\cdots\!94}{40\!\cdots\!61}a^{16}-\frac{10\!\cdots\!91}{40\!\cdots\!61}a^{15}-\frac{19\!\cdots\!00}{40\!\cdots\!61}a^{14}+\frac{43\!\cdots\!87}{40\!\cdots\!61}a^{13}+\frac{14\!\cdots\!79}{40\!\cdots\!61}a^{12}+\frac{12\!\cdots\!35}{40\!\cdots\!61}a^{11}-\frac{46\!\cdots\!14}{40\!\cdots\!61}a^{10}-\frac{63\!\cdots\!27}{40\!\cdots\!61}a^{9}+\frac{64\!\cdots\!01}{40\!\cdots\!61}a^{8}+\frac{98\!\cdots\!74}{40\!\cdots\!61}a^{7}-\frac{39\!\cdots\!31}{40\!\cdots\!61}a^{6}-\frac{55\!\cdots\!19}{40\!\cdots\!61}a^{5}+\frac{86\!\cdots\!51}{40\!\cdots\!61}a^{4}+\frac{93\!\cdots\!63}{40\!\cdots\!61}a^{3}-\frac{53\!\cdots\!85}{40\!\cdots\!61}a^{2}-\frac{21\!\cdots\!81}{40\!\cdots\!61}a-\frac{43\!\cdots\!80}{40\!\cdots\!61}$, $\frac{18\!\cdots\!88}{40\!\cdots\!61}a^{16}+\frac{25\!\cdots\!63}{40\!\cdots\!61}a^{15}-\frac{39\!\cdots\!92}{40\!\cdots\!61}a^{14}-\frac{95\!\cdots\!41}{40\!\cdots\!61}a^{13}+\frac{28\!\cdots\!73}{40\!\cdots\!61}a^{12}+\frac{93\!\cdots\!80}{40\!\cdots\!61}a^{11}-\frac{85\!\cdots\!30}{40\!\cdots\!61}a^{10}-\frac{34\!\cdots\!22}{40\!\cdots\!61}a^{9}+\frac{95\!\cdots\!00}{40\!\cdots\!61}a^{8}+\frac{48\!\cdots\!96}{40\!\cdots\!61}a^{7}-\frac{30\!\cdots\!10}{40\!\cdots\!61}a^{6}-\frac{28\!\cdots\!61}{40\!\cdots\!61}a^{5}-\frac{80\!\cdots\!12}{40\!\cdots\!61}a^{4}+\frac{53\!\cdots\!35}{40\!\cdots\!61}a^{3}+\frac{35\!\cdots\!40}{40\!\cdots\!61}a^{2}-\frac{18\!\cdots\!84}{40\!\cdots\!61}a-\frac{27\!\cdots\!81}{40\!\cdots\!61}$, $\frac{34\!\cdots\!95}{40\!\cdots\!61}a^{16}-\frac{35\!\cdots\!08}{40\!\cdots\!61}a^{15}-\frac{71\!\cdots\!99}{40\!\cdots\!61}a^{14}-\frac{31\!\cdots\!31}{40\!\cdots\!61}a^{13}+\frac{52\!\cdots\!76}{40\!\cdots\!61}a^{12}+\frac{45\!\cdots\!10}{40\!\cdots\!61}a^{11}-\frac{16\!\cdots\!45}{40\!\cdots\!61}a^{10}-\frac{23\!\cdots\!86}{40\!\cdots\!61}a^{9}+\frac{23\!\cdots\!28}{40\!\cdots\!61}a^{8}+\frac{35\!\cdots\!88}{40\!\cdots\!61}a^{7}-\frac{14\!\cdots\!55}{40\!\cdots\!61}a^{6}-\frac{18\!\cdots\!88}{40\!\cdots\!61}a^{5}+\frac{30\!\cdots\!41}{40\!\cdots\!61}a^{4}+\frac{29\!\cdots\!14}{40\!\cdots\!61}a^{3}-\frac{42\!\cdots\!19}{40\!\cdots\!61}a^{2}-\frac{50\!\cdots\!08}{40\!\cdots\!61}a-\frac{45\!\cdots\!81}{40\!\cdots\!61}$, $\frac{12\!\cdots\!47}{40\!\cdots\!61}a^{16}-\frac{20\!\cdots\!17}{40\!\cdots\!61}a^{15}-\frac{26\!\cdots\!62}{40\!\cdots\!61}a^{14}+\frac{13\!\cdots\!65}{40\!\cdots\!61}a^{13}+\frac{19\!\cdots\!45}{40\!\cdots\!61}a^{12}+\frac{64\!\cdots\!06}{40\!\cdots\!61}a^{11}-\frac{64\!\cdots\!11}{40\!\cdots\!61}a^{10}-\frac{53\!\cdots\!12}{40\!\cdots\!61}a^{9}+\frac{92\!\cdots\!92}{40\!\cdots\!61}a^{8}+\frac{81\!\cdots\!84}{40\!\cdots\!61}a^{7}-\frac{60\!\cdots\!01}{40\!\cdots\!61}a^{6}-\frac{38\!\cdots\!04}{40\!\cdots\!61}a^{5}+\frac{15\!\cdots\!32}{40\!\cdots\!61}a^{4}+\frac{24\!\cdots\!20}{40\!\cdots\!61}a^{3}-\frac{40\!\cdots\!85}{40\!\cdots\!61}a^{2}-\frac{42\!\cdots\!39}{40\!\cdots\!61}a+\frac{22\!\cdots\!38}{40\!\cdots\!61}$, $\frac{61\!\cdots\!70}{40\!\cdots\!61}a^{16}-\frac{91\!\cdots\!97}{40\!\cdots\!61}a^{15}-\frac{12\!\cdots\!26}{40\!\cdots\!61}a^{14}+\frac{51\!\cdots\!79}{40\!\cdots\!61}a^{13}+\frac{94\!\cdots\!38}{40\!\cdots\!61}a^{12}+\frac{39\!\cdots\!79}{40\!\cdots\!61}a^{11}-\frac{30\!\cdots\!24}{40\!\cdots\!61}a^{10}-\frac{28\!\cdots\!57}{40\!\cdots\!61}a^{9}+\frac{43\!\cdots\!41}{40\!\cdots\!61}a^{8}+\frac{44\!\cdots\!11}{40\!\cdots\!61}a^{7}-\frac{27\!\cdots\!24}{40\!\cdots\!61}a^{6}-\frac{22\!\cdots\!64}{40\!\cdots\!61}a^{5}+\frac{67\!\cdots\!38}{40\!\cdots\!61}a^{4}+\frac{26\!\cdots\!63}{40\!\cdots\!61}a^{3}-\frac{23\!\cdots\!21}{40\!\cdots\!61}a^{2}-\frac{57\!\cdots\!56}{40\!\cdots\!61}a+\frac{19\!\cdots\!47}{40\!\cdots\!61}$, $\frac{79\!\cdots\!99}{40\!\cdots\!61}a^{16}-\frac{71\!\cdots\!69}{40\!\cdots\!61}a^{15}-\frac{14\!\cdots\!54}{40\!\cdots\!61}a^{14}+\frac{12\!\cdots\!49}{40\!\cdots\!61}a^{13}+\frac{95\!\cdots\!42}{40\!\cdots\!61}a^{12}-\frac{76\!\cdots\!66}{40\!\cdots\!61}a^{11}-\frac{29\!\cdots\!00}{40\!\cdots\!61}a^{10}+\frac{20\!\cdots\!95}{40\!\cdots\!61}a^{9}+\frac{45\!\cdots\!53}{40\!\cdots\!61}a^{8}-\frac{26\!\cdots\!10}{40\!\cdots\!61}a^{7}-\frac{35\!\cdots\!78}{40\!\cdots\!61}a^{6}+\frac{13\!\cdots\!11}{40\!\cdots\!61}a^{5}+\frac{13\!\cdots\!69}{40\!\cdots\!61}a^{4}-\frac{24\!\cdots\!37}{40\!\cdots\!61}a^{3}-\frac{19\!\cdots\!67}{40\!\cdots\!61}a^{2}-\frac{20\!\cdots\!62}{40\!\cdots\!61}a+\frac{11\!\cdots\!91}{40\!\cdots\!61}$, $\frac{58\!\cdots\!53}{40\!\cdots\!61}a^{16}-\frac{86\!\cdots\!04}{40\!\cdots\!61}a^{15}-\frac{12\!\cdots\!82}{40\!\cdots\!61}a^{14}+\frac{48\!\cdots\!24}{40\!\cdots\!61}a^{13}+\frac{89\!\cdots\!13}{40\!\cdots\!61}a^{12}+\frac{38\!\cdots\!21}{40\!\cdots\!61}a^{11}-\frac{28\!\cdots\!74}{40\!\cdots\!61}a^{10}-\frac{27\!\cdots\!25}{40\!\cdots\!61}a^{9}+\frac{40\!\cdots\!43}{40\!\cdots\!61}a^{8}+\frac{41\!\cdots\!95}{40\!\cdots\!61}a^{7}-\frac{25\!\cdots\!88}{40\!\cdots\!61}a^{6}-\frac{21\!\cdots\!67}{40\!\cdots\!61}a^{5}+\frac{63\!\cdots\!79}{40\!\cdots\!61}a^{4}+\frac{24\!\cdots\!16}{40\!\cdots\!61}a^{3}-\frac{21\!\cdots\!77}{40\!\cdots\!61}a^{2}-\frac{38\!\cdots\!03}{40\!\cdots\!61}a+\frac{12\!\cdots\!50}{40\!\cdots\!61}$, $\frac{15\!\cdots\!55}{40\!\cdots\!61}a^{16}-\frac{26\!\cdots\!30}{40\!\cdots\!61}a^{15}-\frac{31\!\cdots\!47}{40\!\cdots\!61}a^{14}+\frac{20\!\cdots\!30}{40\!\cdots\!61}a^{13}+\frac{23\!\cdots\!58}{40\!\cdots\!61}a^{12}+\frac{43\!\cdots\!65}{40\!\cdots\!61}a^{11}-\frac{75\!\cdots\!39}{40\!\cdots\!61}a^{10}-\frac{53\!\cdots\!95}{40\!\cdots\!61}a^{9}+\frac{10\!\cdots\!18}{40\!\cdots\!61}a^{8}+\frac{84\!\cdots\!53}{40\!\cdots\!61}a^{7}-\frac{73\!\cdots\!59}{40\!\cdots\!61}a^{6}-\frac{40\!\cdots\!88}{40\!\cdots\!61}a^{5}+\frac{19\!\cdots\!10}{40\!\cdots\!61}a^{4}+\frac{41\!\cdots\!03}{40\!\cdots\!61}a^{3}-\frac{11\!\cdots\!47}{40\!\cdots\!61}a^{2}-\frac{25\!\cdots\!69}{40\!\cdots\!61}a+\frac{96\!\cdots\!63}{40\!\cdots\!61}$, $\frac{26\!\cdots\!65}{40\!\cdots\!61}a^{16}-\frac{41\!\cdots\!37}{40\!\cdots\!61}a^{15}-\frac{55\!\cdots\!08}{40\!\cdots\!61}a^{14}+\frac{25\!\cdots\!35}{40\!\cdots\!61}a^{13}+\frac{40\!\cdots\!33}{40\!\cdots\!61}a^{12}+\frac{15\!\cdots\!14}{40\!\cdots\!61}a^{11}-\frac{13\!\cdots\!10}{40\!\cdots\!61}a^{10}-\frac{11\!\cdots\!90}{40\!\cdots\!61}a^{9}+\frac{18\!\cdots\!36}{40\!\cdots\!61}a^{8}+\frac{18\!\cdots\!74}{40\!\cdots\!61}a^{7}-\frac{12\!\cdots\!95}{40\!\cdots\!61}a^{6}-\frac{91\!\cdots\!11}{40\!\cdots\!61}a^{5}+\frac{29\!\cdots\!59}{40\!\cdots\!61}a^{4}+\frac{10\!\cdots\!24}{40\!\cdots\!61}a^{3}-\frac{10\!\cdots\!96}{40\!\cdots\!61}a^{2}-\frac{15\!\cdots\!08}{40\!\cdots\!61}a+\frac{68\!\cdots\!29}{40\!\cdots\!61}$, $\frac{35\!\cdots\!11}{40\!\cdots\!61}a^{16}-\frac{47\!\cdots\!04}{40\!\cdots\!61}a^{15}-\frac{72\!\cdots\!97}{40\!\cdots\!61}a^{14}+\frac{19\!\cdots\!81}{40\!\cdots\!61}a^{13}+\frac{53\!\cdots\!40}{40\!\cdots\!61}a^{12}+\frac{29\!\cdots\!79}{40\!\cdots\!61}a^{11}-\frac{17\!\cdots\!20}{40\!\cdots\!61}a^{10}-\frac{18\!\cdots\!99}{40\!\cdots\!61}a^{9}+\frac{24\!\cdots\!23}{40\!\cdots\!61}a^{8}+\frac{28\!\cdots\!93}{40\!\cdots\!61}a^{7}-\frac{15\!\cdots\!96}{40\!\cdots\!61}a^{6}-\frac{14\!\cdots\!67}{40\!\cdots\!61}a^{5}+\frac{37\!\cdots\!73}{40\!\cdots\!61}a^{4}+\frac{19\!\cdots\!28}{40\!\cdots\!61}a^{3}-\frac{13\!\cdots\!75}{40\!\cdots\!61}a^{2}-\frac{40\!\cdots\!36}{40\!\cdots\!61}a+\frac{12\!\cdots\!32}{40\!\cdots\!61}$, $\frac{37\!\cdots\!52}{40\!\cdots\!61}a^{16}-\frac{54\!\cdots\!19}{40\!\cdots\!61}a^{15}-\frac{77\!\cdots\!48}{40\!\cdots\!61}a^{14}+\frac{29\!\cdots\!51}{40\!\cdots\!61}a^{13}+\frac{56\!\cdots\!73}{40\!\cdots\!61}a^{12}+\frac{25\!\cdots\!82}{40\!\cdots\!61}a^{11}-\frac{18\!\cdots\!13}{40\!\cdots\!61}a^{10}-\frac{17\!\cdots\!65}{40\!\cdots\!61}a^{9}+\frac{25\!\cdots\!51}{40\!\cdots\!61}a^{8}+\frac{27\!\cdots\!80}{40\!\cdots\!61}a^{7}-\frac{16\!\cdots\!11}{40\!\cdots\!61}a^{6}-\frac{13\!\cdots\!34}{40\!\cdots\!61}a^{5}+\frac{40\!\cdots\!86}{40\!\cdots\!61}a^{4}+\frac{16\!\cdots\!50}{40\!\cdots\!61}a^{3}-\frac{14\!\cdots\!68}{40\!\cdots\!61}a^{2}-\frac{22\!\cdots\!89}{40\!\cdots\!61}a+\frac{95\!\cdots\!39}{40\!\cdots\!61}$, $\frac{18\!\cdots\!45}{40\!\cdots\!61}a^{16}-\frac{21\!\cdots\!75}{40\!\cdots\!61}a^{15}-\frac{38\!\cdots\!68}{40\!\cdots\!61}a^{14}+\frac{27\!\cdots\!57}{40\!\cdots\!61}a^{13}+\frac{28\!\cdots\!23}{40\!\cdots\!61}a^{12}+\frac{21\!\cdots\!79}{40\!\cdots\!61}a^{11}-\frac{90\!\cdots\!81}{40\!\cdots\!61}a^{10}-\frac{11\!\cdots\!01}{40\!\cdots\!61}a^{9}+\frac{12\!\cdots\!97}{40\!\cdots\!61}a^{8}+\frac{17\!\cdots\!56}{40\!\cdots\!61}a^{7}-\frac{77\!\cdots\!94}{40\!\cdots\!61}a^{6}-\frac{91\!\cdots\!40}{40\!\cdots\!61}a^{5}+\frac{17\!\cdots\!78}{40\!\cdots\!61}a^{4}+\frac{13\!\cdots\!24}{40\!\cdots\!61}a^{3}-\frac{15\!\cdots\!78}{40\!\cdots\!61}a^{2}-\frac{26\!\cdots\!39}{40\!\cdots\!61}a-\frac{51\!\cdots\!71}{40\!\cdots\!61}$, $\frac{73\!\cdots\!17}{40\!\cdots\!61}a^{16}-\frac{11\!\cdots\!16}{40\!\cdots\!61}a^{15}-\frac{15\!\cdots\!65}{40\!\cdots\!61}a^{14}+\frac{65\!\cdots\!42}{40\!\cdots\!61}a^{13}+\frac{11\!\cdots\!47}{40\!\cdots\!61}a^{12}+\frac{45\!\cdots\!58}{40\!\cdots\!61}a^{11}-\frac{36\!\cdots\!85}{40\!\cdots\!61}a^{10}-\frac{33\!\cdots\!66}{40\!\cdots\!61}a^{9}+\frac{51\!\cdots\!38}{40\!\cdots\!61}a^{8}+\frac{51\!\cdots\!63}{40\!\cdots\!61}a^{7}-\frac{33\!\cdots\!31}{40\!\cdots\!61}a^{6}-\frac{26\!\cdots\!19}{40\!\cdots\!61}a^{5}+\frac{82\!\cdots\!75}{40\!\cdots\!61}a^{4}+\frac{30\!\cdots\!34}{40\!\cdots\!61}a^{3}-\frac{31\!\cdots\!73}{40\!\cdots\!61}a^{2}-\frac{63\!\cdots\!57}{40\!\cdots\!61}a+\frac{24\!\cdots\!67}{40\!\cdots\!61}$, $\frac{11\!\cdots\!31}{40\!\cdots\!61}a^{16}-\frac{96\!\cdots\!16}{40\!\cdots\!61}a^{15}-\frac{24\!\cdots\!18}{40\!\cdots\!61}a^{14}-\frac{60\!\cdots\!33}{40\!\cdots\!61}a^{13}+\frac{17\!\cdots\!37}{40\!\cdots\!61}a^{12}+\frac{19\!\cdots\!37}{40\!\cdots\!61}a^{11}-\frac{56\!\cdots\!86}{40\!\cdots\!61}a^{10}-\frac{90\!\cdots\!02}{40\!\cdots\!61}a^{9}+\frac{76\!\cdots\!00}{40\!\cdots\!61}a^{8}+\frac{13\!\cdots\!16}{40\!\cdots\!61}a^{7}-\frac{45\!\cdots\!21}{40\!\cdots\!61}a^{6}-\frac{74\!\cdots\!67}{40\!\cdots\!61}a^{5}+\frac{93\!\cdots\!24}{40\!\cdots\!61}a^{4}+\frac{12\!\cdots\!43}{40\!\cdots\!61}a^{3}+\frac{33\!\cdots\!26}{40\!\cdots\!61}a^{2}-\frac{25\!\cdots\!92}{40\!\cdots\!61}a-\frac{56\!\cdots\!75}{40\!\cdots\!61}$
| 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: | \( 2360552434044002.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{0}\cdot 2360552434044002.5 \cdot 1}{2\cdot\sqrt{2200187128095499475530336818113367454680001}}\cr\approx \mathstrut & 0.104295068223382 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103)
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 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103, 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 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103);
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 - x^16 - 208*x^15 - 17*x^14 + 15287*x^13 + 13881*x^12 - 487578*x^11 - 703261*x^10 + 6754359*x^9 + 10540902*x^8 - 41136753*x^7 - 57683825*x^6 + 92010954*x^5 + 95287840*x^4 - 17501435*x^3 - 25026156*x^2 - 563260*x + 1246103);
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 cyclic group of order 17 |
| The 17 conjugacy class representatives for $C_{17}$ |
| Character table for $C_{17}$ |
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 | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(443\)
| Deg $17$ | $17$ | $1$ | $16$ |