# Properties

 Label 17.1.604...088.1 Degree $17$ Signature $[1, 8]$ Discriminant $6.045\times 10^{23}$ Root discriminant $$25.06$$ Ramified prime $2$ Class number $1$ Class group trivial Galois group $F_{17}$ (as 17T5)

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)

gp: K = bnfinit(y^17 - 2*y^16 + 8*y^13 + 16*y^12 - 16*y^11 + 64*y^9 - 32*y^8 - 80*y^7 + 32*y^6 + 40*y^5 + 80*y^4 + 16*y^3 - 128*y^2 - 2*y + 68, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)

$$x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + 40 x^{5} + 80 x^{4} + 16 x^{3} - 128 x^{2} - 2 x + 68$$

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: $$604462909807314587353088$$ 604462909807314587353088 $$\medspace = 2^{79}$$ sage: K.disc()  gp: K.disc  magma: OK := Integers(K); Discriminant(OK);  oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$25.06$$ 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)) Ramified primes: $$2$$ 2 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(OK));  oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q(\sqrt{2})$$ $\card{ \Aut(K/\Q) }$: $1$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23\!\cdots\!82}a^{16}+\frac{76599019108737}{11\!\cdots\!91}a^{15}-\frac{31\!\cdots\!69}{11\!\cdots\!91}a^{14}-\frac{37\!\cdots\!82}{11\!\cdots\!91}a^{13}+\frac{54\!\cdots\!14}{11\!\cdots\!91}a^{12}+\frac{45\!\cdots\!51}{11\!\cdots\!91}a^{11}+\frac{44\!\cdots\!93}{11\!\cdots\!91}a^{10}+\frac{51\!\cdots\!16}{11\!\cdots\!91}a^{9}-\frac{29\!\cdots\!81}{11\!\cdots\!91}a^{8}-\frac{54\!\cdots\!87}{11\!\cdots\!91}a^{7}+\frac{17\!\cdots\!44}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{5}+\frac{33\!\cdots\!28}{11\!\cdots\!91}a^{4}-\frac{40\!\cdots\!18}{11\!\cdots\!91}a^{3}+\frac{57\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!51}{11\!\cdots\!91}a+\frac{24\!\cdots\!97}{11\!\cdots\!91}$

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: $$-1$$ -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: $\frac{10\!\cdots\!32}{11\!\cdots\!91}a^{16}-\frac{13\!\cdots\!38}{11\!\cdots\!91}a^{15}+\frac{153497539627694}{11\!\cdots\!91}a^{14}-\frac{20\!\cdots\!47}{11\!\cdots\!91}a^{13}+\frac{73\!\cdots\!74}{11\!\cdots\!91}a^{12}+\frac{19\!\cdots\!19}{11\!\cdots\!91}a^{11}+\frac{70\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{19\!\cdots\!26}{11\!\cdots\!91}a^{9}+\frac{66\!\cdots\!91}{11\!\cdots\!91}a^{8}+\frac{15\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{16\!\cdots\!82}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!83}{11\!\cdots\!91}a^{5}-\frac{35\!\cdots\!48}{11\!\cdots\!91}a^{4}+\frac{42\!\cdots\!32}{11\!\cdots\!91}a^{3}+\frac{77\!\cdots\!97}{11\!\cdots\!91}a^{2}+\frac{35\!\cdots\!26}{11\!\cdots\!91}a-\frac{76\!\cdots\!89}{11\!\cdots\!91}$, $\frac{238227155182776}{11\!\cdots\!91}a^{16}-\frac{820973558914092}{11\!\cdots\!91}a^{15}+\frac{437201946462560}{11\!\cdots\!91}a^{14}+\frac{214894272296728}{11\!\cdots\!91}a^{13}+\frac{19\!\cdots\!89}{11\!\cdots\!91}a^{12}+\frac{16\!\cdots\!49}{11\!\cdots\!91}a^{11}-\frac{10\!\cdots\!58}{11\!\cdots\!91}a^{10}-\frac{68944875781624}{11\!\cdots\!91}a^{9}+\frac{95\!\cdots\!74}{11\!\cdots\!91}a^{8}-\frac{35\!\cdots\!73}{11\!\cdots\!91}a^{7}-\frac{25\!\cdots\!83}{11\!\cdots\!91}a^{6}+\frac{29\!\cdots\!46}{11\!\cdots\!91}a^{5}-\frac{21\!\cdots\!73}{11\!\cdots\!91}a^{4}-\frac{12\!\cdots\!16}{11\!\cdots\!91}a^{3}+\frac{86\!\cdots\!54}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!39}{11\!\cdots\!91}a-\frac{19\!\cdots\!55}{11\!\cdots\!91}$, $\frac{361331936520056}{11\!\cdots\!91}a^{16}-\frac{395500407207728}{11\!\cdots\!91}a^{15}-\frac{404488405846192}{11\!\cdots\!91}a^{14}-\frac{477299961985816}{11\!\cdots\!91}a^{13}+\frac{31\!\cdots\!50}{11\!\cdots\!91}a^{12}+\frac{75\!\cdots\!71}{11\!\cdots\!91}a^{11}+\frac{19\!\cdots\!99}{11\!\cdots\!91}a^{10}-\frac{14\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{22\!\cdots\!02}{11\!\cdots\!91}a^{8}+\frac{11\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{33\!\cdots\!78}{11\!\cdots\!91}a^{6}-\frac{18\!\cdots\!13}{11\!\cdots\!91}a^{5}+\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{4}+\frac{29\!\cdots\!40}{11\!\cdots\!91}a^{3}+\frac{21\!\cdots\!84}{11\!\cdots\!91}a^{2}-\frac{14\!\cdots\!61}{11\!\cdots\!91}a-\frac{10\!\cdots\!97}{11\!\cdots\!91}$, $\frac{541474688268459}{11\!\cdots\!91}a^{16}-\frac{503105829785995}{11\!\cdots\!91}a^{15}-\frac{759647397584390}{11\!\cdots\!91}a^{14}-\frac{700905168343220}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!64}{11\!\cdots\!91}a^{12}+\frac{12\!\cdots\!98}{11\!\cdots\!91}a^{11}+\frac{40\!\cdots\!57}{11\!\cdots\!91}a^{10}-\frac{22\!\cdots\!00}{11\!\cdots\!91}a^{9}+\frac{31\!\cdots\!97}{11\!\cdots\!91}a^{8}+\frac{19\!\cdots\!09}{11\!\cdots\!91}a^{7}-\frac{42\!\cdots\!75}{11\!\cdots\!91}a^{6}-\frac{40\!\cdots\!01}{11\!\cdots\!91}a^{5}+\frac{57\!\cdots\!01}{11\!\cdots\!91}a^{4}+\frac{49\!\cdots\!83}{11\!\cdots\!91}a^{3}+\frac{49\!\cdots\!17}{11\!\cdots\!91}a^{2}-\frac{18\!\cdots\!89}{11\!\cdots\!91}a-\frac{37\!\cdots\!43}{11\!\cdots\!91}$, $\frac{49211265331281}{11\!\cdots\!91}a^{16}-\frac{455679016301548}{11\!\cdots\!91}a^{15}+\frac{604764560884272}{11\!\cdots\!91}a^{14}-\frac{220167784432132}{11\!\cdots\!91}a^{13}+\frac{11\!\cdots\!62}{11\!\cdots\!91}a^{12}-\frac{21\!\cdots\!68}{11\!\cdots\!91}a^{11}-\frac{66\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{345221016246508}{11\!\cdots\!91}a^{9}-\frac{26\!\cdots\!13}{11\!\cdots\!91}a^{8}-\frac{23\!\cdots\!71}{11\!\cdots\!91}a^{7}-\frac{21\!\cdots\!05}{11\!\cdots\!91}a^{6}+\frac{12\!\cdots\!43}{11\!\cdots\!91}a^{5}+\frac{73\!\cdots\!29}{11\!\cdots\!91}a^{4}+\frac{10\!\cdots\!93}{11\!\cdots\!91}a^{3}-\frac{18\!\cdots\!40}{11\!\cdots\!91}a^{2}-\frac{20\!\cdots\!23}{11\!\cdots\!91}a+\frac{78\!\cdots\!63}{11\!\cdots\!91}$, $\frac{13\!\cdots\!49}{11\!\cdots\!91}a^{16}-\frac{19\!\cdots\!23}{11\!\cdots\!91}a^{15}-\frac{286237517021309}{11\!\cdots\!91}a^{14}-\frac{23\!\cdots\!39}{11\!\cdots\!91}a^{13}+\frac{10\!\cdots\!77}{11\!\cdots\!91}a^{12}+\frac{27\!\cdots\!63}{11\!\cdots\!91}a^{11}+\frac{14\!\cdots\!32}{11\!\cdots\!91}a^{10}+\frac{98\!\cdots\!34}{11\!\cdots\!91}a^{9}+\frac{75\!\cdots\!04}{11\!\cdots\!91}a^{8}+\frac{38\!\cdots\!35}{11\!\cdots\!91}a^{7}-\frac{72\!\cdots\!01}{11\!\cdots\!91}a^{6}-\frac{55\!\cdots\!00}{11\!\cdots\!91}a^{5}-\frac{28\!\cdots\!26}{11\!\cdots\!91}a^{4}+\frac{11\!\cdots\!10}{11\!\cdots\!91}a^{3}+\frac{90\!\cdots\!75}{11\!\cdots\!91}a^{2}-\frac{48\!\cdots\!36}{11\!\cdots\!91}a-\frac{39\!\cdots\!07}{11\!\cdots\!91}$, $\frac{133348952174370}{11\!\cdots\!91}a^{16}-\frac{621949317058588}{11\!\cdots\!91}a^{15}-\frac{12\!\cdots\!06}{11\!\cdots\!91}a^{14}+\frac{21\!\cdots\!88}{11\!\cdots\!91}a^{13}+\frac{14\!\cdots\!21}{11\!\cdots\!91}a^{12}+\frac{34\!\cdots\!20}{11\!\cdots\!91}a^{11}-\frac{22\!\cdots\!06}{11\!\cdots\!91}a^{10}-\frac{34\!\cdots\!84}{11\!\cdots\!91}a^{9}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{8}-\frac{59\!\cdots\!82}{11\!\cdots\!91}a^{7}-\frac{12\!\cdots\!55}{11\!\cdots\!91}a^{6}-\frac{28\!\cdots\!18}{11\!\cdots\!91}a^{5}+\frac{35\!\cdots\!94}{11\!\cdots\!91}a^{4}+\frac{54\!\cdots\!04}{11\!\cdots\!91}a^{3}+\frac{23\!\cdots\!64}{11\!\cdots\!91}a^{2}-\frac{13\!\cdots\!15}{11\!\cdots\!91}a-\frac{12\!\cdots\!23}{11\!\cdots\!91}$, $\frac{636029286616479}{11\!\cdots\!91}a^{16}-\frac{20\!\cdots\!81}{11\!\cdots\!91}a^{15}+\frac{19\!\cdots\!57}{11\!\cdots\!91}a^{14}-\frac{466211932185150}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!59}{11\!\cdots\!91}a^{12}+\frac{54\!\cdots\!40}{11\!\cdots\!91}a^{11}-\frac{21\!\cdots\!29}{11\!\cdots\!91}a^{10}+\frac{25\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{37\!\cdots\!47}{11\!\cdots\!91}a^{8}-\frac{58\!\cdots\!17}{11\!\cdots\!91}a^{7}+\frac{15\!\cdots\!46}{11\!\cdots\!91}a^{6}+\frac{76\!\cdots\!77}{11\!\cdots\!91}a^{5}-\frac{11\!\cdots\!17}{11\!\cdots\!91}a^{4}+\frac{58\!\cdots\!54}{11\!\cdots\!91}a^{3}-\frac{83\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{63\!\cdots\!59}{11\!\cdots\!91}a+\frac{13\!\cdots\!79}{11\!\cdots\!91}$ 1016308550883132/11563101439788991*a^16 - 1350954411228338/11563101439788991*a^15 + 153497539627694/11563101439788991*a^14 - 2072988722130147/11563101439788991*a^13 + 7388973309154674/11563101439788991*a^12 + 19476315316497919/11563101439788991*a^11 + 7019562308341754/11563101439788991*a^10 + 19303010737718626/11563101439788991*a^9 + 66611212487402091/11563101439788991*a^8 + 15080126570669303/11563101439788991*a^7 - 16329658256399982/11563101439788991*a^6 - 10646154606155983/11563101439788991*a^5 - 35605588873454948/11563101439788991*a^4 + 42342922038177332/11563101439788991*a^3 + 77872336470040197/11563101439788991*a^2 + 3558557265357826/11563101439788991*a - 7640146742881689/11563101439788991, 238227155182776/11563101439788991*a^16 - 820973558914092/11563101439788991*a^15 + 437201946462560/11563101439788991*a^14 + 214894272296728/11563101439788991*a^13 + 1959403877011389/11563101439788991*a^12 + 1640692096435049/11563101439788991*a^11 - 10696874792022558/11563101439788991*a^10 - 68944875781624/11563101439788991*a^9 + 9594937239194574/11563101439788991*a^8 - 35910232567942773/11563101439788991*a^7 - 25914904628667983/11563101439788991*a^6 + 29545550711247946/11563101439788991*a^5 - 2196853801234773/11563101439788991*a^4 - 1214870764616616/11563101439788991*a^3 + 8666273589087854/11563101439788991*a^2 - 26666010649193639/11563101439788991*a - 19291425163555355/11563101439788991, 361331936520056/11563101439788991*a^16 - 395500407207728/11563101439788991*a^15 - 404488405846192/11563101439788991*a^14 - 477299961985816/11563101439788991*a^13 + 3111360543555550/11563101439788991*a^12 + 7592129260614271/11563101439788991*a^11 + 1956375094295199/11563101439788991*a^10 - 1411857007597616/11563101439788991*a^9 + 22301870170984202/11563101439788991*a^8 + 11617701433946203/11563101439788991*a^7 - 33570269642292378/11563101439788991*a^6 - 18423708156355113/11563101439788991*a^5 + 10258934756362565/11563101439788991*a^4 + 29837286208701840/11563101439788991*a^3 + 21440286719322684/11563101439788991*a^2 - 14862195421859061/11563101439788991*a - 10170490422114997/11563101439788991, 541474688268459/11563101439788991*a^16 - 503105829785995/11563101439788991*a^15 - 759647397584390/11563101439788991*a^14 - 700905168343220/11563101439788991*a^13 + 4302585631665464/11563101439788991*a^12 + 12732954226412998/11563101439788991*a^11 + 4060732285614557/11563101439788991*a^10 - 2258883277305300/11563101439788991*a^9 + 31994095646277897/11563101439788991*a^8 + 19174141561779309/11563101439788991*a^7 - 42911514751751875/11563101439788991*a^6 - 40360016532305201/11563101439788991*a^5 + 5715113845966201/11563101439788991*a^4 + 49098698914763983/11563101439788991*a^3 + 49622976242570617/11563101439788991*a^2 - 18052935220074989/11563101439788991*a - 37490878285670643/11563101439788991, 49211265331281/11563101439788991*a^16 - 455679016301548/11563101439788991*a^15 + 604764560884272/11563101439788991*a^14 - 220167784432132/11563101439788991*a^13 + 1145091129068362/11563101439788991*a^12 - 2157113822635368/11563101439788991*a^11 - 6637227301539954/11563101439788991*a^10 + 345221016246508/11563101439788991*a^9 - 2600114669223013/11563101439788991*a^8 - 23318616329941871/11563101439788991*a^7 - 2134331323321005/11563101439788991*a^6 + 12101248109058743/11563101439788991*a^5 + 7356456768293929/11563101439788991*a^4 + 10073175370275893/11563101439788991*a^3 - 18691101113706040/11563101439788991*a^2 - 20462655130707423/11563101439788991*a + 7833181402291463/11563101439788991, 1385692473067449/11563101439788991*a^16 - 1982480452881123/11563101439788991*a^15 - 286237517021309/11563101439788991*a^14 - 2361818740037339/11563101439788991*a^13 + 10949487649020577/11563101439788991*a^12 + 27213368039687763/11563101439788991*a^11 + 1448610037894232/11563101439788991*a^10 + 9818807058087934/11563101439788991*a^9 + 75386909926020104/11563101439788991*a^8 + 3862760191174035/11563101439788991*a^7 - 72916162505070001/11563101439788991*a^6 - 55859704602387600/11563101439788991*a^5 - 28358914454352926/11563101439788991*a^4 + 111614166009605210/11563101439788991*a^3 + 90970473474340475/11563101439788991*a^2 - 48927387615789536/11563101439788991*a - 39951316006146907/11563101439788991, 133348952174370/11563101439788991*a^16 - 621949317058588/11563101439788991*a^15 - 1224371564896806/11563101439788991*a^14 + 2170599959385488/11563101439788991*a^13 + 1488447797551621/11563101439788991*a^12 + 3453807891379020/11563101439788991*a^11 - 22006263289611406/11563101439788991*a^10 - 34794740616935184/11563101439788991*a^9 - 10995367635361765/11563101439788991*a^8 - 59793417734359582/11563101439788991*a^7 - 122870294238485755/11563101439788991*a^6 - 28242472164877318/11563101439788991*a^5 + 35094044211441994/11563101439788991*a^4 + 54068385995464204/11563101439788991*a^3 + 23283414484110164/11563101439788991*a^2 - 134512822541303115/11563101439788991*a - 120705748106072123/11563101439788991, 636029286616479/11563101439788991*a^16 - 2012262997197081/11563101439788991*a^15 + 1931992420179357/11563101439788991*a^14 - 466211932185150/11563101439788991*a^13 + 4365963986455559/11563101439788991*a^12 + 5410864828505640/11563101439788991*a^11 - 21812935647401529/11563101439788991*a^10 + 25155660065830316/11563101439788991*a^9 + 37679987267676447/11563101439788991*a^8 - 58106369766033117/11563101439788991*a^7 + 15970800951006946/11563101439788991*a^6 + 76663822213201677/11563101439788991*a^5 - 11951912971421317/11563101439788991*a^4 + 58667708512084354/11563101439788991*a^3 - 83498378163845713/11563101439788991*a^2 - 63358615618596859/11563101439788991*a + 136878997551937179/11563101439788991 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: $$600164.932841$$ 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 600164.932841 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 1.87510129816 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)

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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68, 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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);

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 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);

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

$F_{17}$ (as 17T5):

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 R $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $16{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ $17$ $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $16{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ $16{,}\,{\href{/padicField/59.1.0.1}{1} }$

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$LabelPolynomial$efc$Galois group Slope content $$2$$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$2.16.79.2$x^{16} + 16 x^{14} + 56 x^{12} + 48 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 2$$16$$1$$79$$C_{16}[3, 4, 5, 6]$## Artin representations Label Dimension Conductor Artin stem field$G$Ind$\chi(c)$* 1.1.1t1.a.a$11$$$\Q$$$C_111$1.8.2t1.a.a$1 2^{3}$$$\Q(\sqrt{2})$$$C_2$(as 2T1)$11$1.16.4t1.a.a$1 2^{4}$$$\Q(\zeta_{16})^+$$$C_4$(as 4T1)$01$1.16.4t1.a.b$1 2^{4}$$$\Q(\zeta_{16})^+$$$C_4$(as 4T1)$01$1.32.8t1.a.a$1 2^{5}$$$\Q(\zeta_{32})^+$$$C_8$(as 8T1)$01$1.32.8t1.a.b$1 2^{5}$$$\Q(\zeta_{32})^+$$$C_8$(as 8T1)$01$1.32.8t1.a.c$1 2^{5}$$$\Q(\zeta_{32})^+$$$C_8$(as 8T1)$01$1.32.8t1.a.d$1 2^{5}$$$\Q(\zeta_{32})^+$$$C_8$(as 8T1)$01$1.64.16t1.a.a$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.b$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.c$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.d$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.e$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.f$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.g$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$1.64.16t1.a.h$1 2^{6}$16.0.604462909807314587353088.1$C_{16}$(as 16T1)$0-1$* 16.604...088.17t5.a.a$16 2^{79}$17.1.604462909807314587353088.1$F_{17}$(as 17T5)$10\$

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 *.