Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225)
gp: K = bnfinit(y^17 - 2*y^16 + 9*y^15 - 23*y^14 + 59*y^13 - 112*y^12 + 156*y^11 - 49*y^10 + 221*y^9 - 25*y^8 - 162*y^7 - 4*y^6 + 553*y^5 + 1435*y^4 + 1716*y^3 + 1462*y^2 + 615*y + 225, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225)
\( x^{17} - 2 x^{16} + 9 x^{15} - 23 x^{14} + 59 x^{13} - 112 x^{12} + 156 x^{11} - 49 x^{10} + 221 x^{9} + \cdots + 225 \)
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: |
\(58497592625273225470725121\)
\(\medspace = 1663^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.79\) | 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: | $1663^{1/2}\approx 40.779897008207364$ | ||
| Ramified primes: |
\(1663\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{21}a^{11}+\frac{2}{21}a^{10}-\frac{2}{21}a^{9}+\frac{3}{7}a^{8}-\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}-\frac{3}{7}a^{4}-\frac{4}{21}a^{3}-\frac{5}{21}a^{2}-\frac{1}{21}a+\frac{3}{7}$, $\frac{1}{21}a^{12}+\frac{1}{21}a^{10}-\frac{1}{21}a^{9}+\frac{3}{7}a^{7}-\frac{1}{3}a^{4}+\frac{1}{7}a^{3}+\frac{2}{21}a^{2}+\frac{4}{21}a+\frac{1}{7}$, $\frac{1}{21}a^{13}-\frac{1}{7}a^{10}+\frac{2}{21}a^{9}+\frac{1}{7}a^{7}-\frac{1}{7}a^{6}+\frac{8}{21}a^{5}-\frac{3}{7}a^{4}+\frac{2}{7}a^{3}+\frac{3}{7}a^{2}+\frac{4}{21}a-\frac{3}{7}$, $\frac{1}{105}a^{14}+\frac{2}{105}a^{13}-\frac{1}{105}a^{12}+\frac{2}{105}a^{11}+\frac{1}{21}a^{10}+\frac{2}{105}a^{9}-\frac{12}{35}a^{8}+\frac{1}{5}a^{7}+\frac{38}{105}a^{6}-\frac{26}{105}a^{5}-\frac{10}{21}a^{4}+\frac{19}{105}a^{3}-\frac{26}{105}a^{2}+\frac{4}{105}a+\frac{3}{7}$, $\frac{1}{315}a^{15}-\frac{1}{315}a^{14}-\frac{1}{45}a^{13}-\frac{1}{315}a^{11}+\frac{52}{315}a^{10}-\frac{2}{315}a^{9}+\frac{8}{105}a^{8}+\frac{1}{9}a^{7}-\frac{1}{9}a^{6}+\frac{19}{45}a^{5}+\frac{11}{35}a^{4}-\frac{14}{45}a^{3}+\frac{2}{315}a^{2}+\frac{83}{315}a-\frac{10}{21}$, $\frac{1}{621500650885485}a^{16}+\frac{7512315506}{88785807269355}a^{15}+\frac{130614479333}{621500650885485}a^{14}-\frac{446711379343}{41433376725699}a^{13}+\frac{279502235714}{12683686752765}a^{12}+\frac{4266351704}{183063520143}a^{11}+\frac{40188942280804}{621500650885485}a^{10}-\frac{1216593563974}{69055627876165}a^{9}+\frac{75376096973189}{621500650885485}a^{8}+\frac{196803201298633}{621500650885485}a^{7}-\frac{208227430263023}{621500650885485}a^{6}+\frac{16245006096862}{41433376725699}a^{5}-\frac{35586115088516}{621500650885485}a^{4}+\frac{57728372863951}{124300130177097}a^{3}-\frac{232377545045389}{621500650885485}a^{2}-\frac{28320475462726}{69055627876165}a-\frac{805141343915}{13811125575233}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{807961821524}{207166883628495}a^{16}+\frac{916876629}{9865089696595}a^{15}+\frac{917770481157}{69055627876165}a^{14}-\frac{4534937749397}{207166883628495}a^{13}+\frac{181073112196}{4227895584255}a^{12}+\frac{3391295776}{305105866905}a^{11}-\frac{38677989850562}{207166883628495}a^{10}+\frac{127275608118094}{207166883628495}a^{9}+\frac{69897254790647}{41433376725699}a^{8}-\frac{59684250766426}{69055627876165}a^{7}-\frac{126565786744219}{69055627876165}a^{6}-\frac{144729808258012}{207166883628495}a^{5}+\frac{221483990195431}{41433376725699}a^{4}+\frac{22\!\cdots\!52}{207166883628495}a^{3}+\frac{19\!\cdots\!01}{207166883628495}a^{2}+\frac{862591223479007}{207166883628495}a+\frac{16550539453012}{13811125575233}$, $\frac{8245455661}{2135741068335}a^{16}-\frac{8378762269}{915317600715}a^{15}+\frac{205521916717}{6407223205005}a^{14}-\frac{549871563089}{6407223205005}a^{13}+\frac{2999236894}{14528850805}a^{12}-\frac{336809164214}{915317600715}a^{11}+\frac{2507150164139}{6407223205005}a^{10}+\frac{1884685876328}{6407223205005}a^{9}+\frac{20592971953}{711913689445}a^{8}-\frac{2758439959127}{6407223205005}a^{7}-\frac{2989636884217}{6407223205005}a^{6}+\frac{3635726371679}{6407223205005}a^{5}+\frac{2012577033834}{711913689445}a^{4}+\frac{17800456495253}{6407223205005}a^{3}+\frac{12563393671306}{6407223205005}a^{2}+\frac{4701202642267}{6407223205005}a+\frac{41544525481}{427148213667}$, $\frac{1368680141438}{621500650885485}a^{16}-\frac{168354253078}{88785807269355}a^{15}+\frac{445256807984}{41433376725699}a^{14}-\frac{10864605267913}{621500650885485}a^{13}+\frac{429972332416}{12683686752765}a^{12}+\frac{1457612792}{183063520143}a^{11}-\frac{41802244180484}{207166883628495}a^{10}+\frac{491209730910364}{621500650885485}a^{9}-\frac{41490351119650}{124300130177097}a^{8}+\frac{379332013837153}{621500650885485}a^{7}-\frac{177548000522299}{207166883628495}a^{6}+\frac{6507268148078}{124300130177097}a^{5}+\frac{11\!\cdots\!14}{621500650885485}a^{4}+\frac{25\!\cdots\!57}{621500650885485}a^{3}+\frac{148243861708885}{41433376725699}a^{2}+\frac{17\!\cdots\!43}{621500650885485}a+\frac{16570609651403}{41433376725699}$, $\frac{4651056026}{356160831453}a^{16}-\frac{42128246}{16960039593}a^{15}+\frac{83233254814}{1780804157265}a^{14}-\frac{106076404102}{1780804157265}a^{13}+\frac{487797977}{5191848855}a^{12}+\frac{196316674}{874228845}a^{11}-\frac{447098981978}{356160831453}a^{10}+\frac{6762686039773}{1780804157265}a^{9}+\frac{3561019546151}{1780804157265}a^{8}+\frac{56812949623}{593601385755}a^{7}-\frac{7873557816958}{1780804157265}a^{6}-\frac{4660242560114}{1780804157265}a^{5}+\frac{4338536480195}{356160831453}a^{4}+\frac{20404585540277}{593601385755}a^{3}+\frac{61841913187051}{1780804157265}a^{2}+\frac{30980277347771}{1780804157265}a+\frac{772638489596}{118720277151}$, $\frac{13995327571007}{621500650885485}a^{16}-\frac{1989067538557}{29595269089785}a^{15}+\frac{140856591344453}{621500650885485}a^{14}-\frac{399572367224396}{621500650885485}a^{13}+\frac{20252947439824}{12683686752765}a^{12}-\frac{317235222662}{101701955635}a^{11}+\frac{26\!\cdots\!91}{621500650885485}a^{10}-\frac{540931489885168}{621500650885485}a^{9}+\frac{393343755992098}{621500650885485}a^{8}-\frac{547838034833441}{207166883628495}a^{7}-\frac{21\!\cdots\!08}{621500650885485}a^{6}+\frac{36\!\cdots\!26}{621500650885485}a^{5}+\frac{92\!\cdots\!79}{621500650885485}a^{4}+\frac{891978653025473}{69055627876165}a^{3}-\frac{24\!\cdots\!61}{621500650885485}a^{2}-\frac{45\!\cdots\!02}{621500650885485}a-\frac{235193918529566}{41433376725699}$, $\frac{1479887124538}{621500650885485}a^{16}-\frac{67476819679}{29595269089785}a^{15}+\frac{10529198536249}{621500650885485}a^{14}-\frac{3682960179602}{124300130177097}a^{13}+\frac{1050679615397}{12683686752765}a^{12}-\frac{2383700153}{20340391127}a^{11}+\frac{30963749710202}{621500650885485}a^{10}+\frac{151610375207902}{621500650885485}a^{9}+\frac{128716946166332}{621500650885485}a^{8}+\frac{28900534245283}{207166883628495}a^{7}-\frac{28417238511499}{621500650885485}a^{6}-\frac{44477211390340}{124300130177097}a^{5}+\frac{473281201727107}{621500650885485}a^{4}+\frac{17754354222421}{13811125575233}a^{3}+\frac{11\!\cdots\!28}{621500650885485}a^{2}+\frac{942032492219498}{621500650885485}a+\frac{38764962935291}{41433376725699}$, $\frac{246477011933}{69055627876165}a^{16}-\frac{275572771513}{29595269089785}a^{15}+\frac{8207599869401}{207166883628495}a^{14}-\frac{21485645476727}{207166883628495}a^{13}+\frac{1240540728611}{4227895584255}a^{12}-\frac{178652950159}{305105866905}a^{11}+\frac{209810777135773}{207166883628495}a^{10}-\frac{191523398414281}{207166883628495}a^{9}+\frac{23253521447700}{13811125575233}a^{8}-\frac{363259091304383}{207166883628495}a^{7}+\frac{432855310659118}{207166883628495}a^{6}-\frac{153241064726767}{207166883628495}a^{5}+\frac{198537605710013}{41433376725699}a^{4}+\frac{609716827932592}{207166883628495}a^{3}+\frac{930731483610206}{207166883628495}a^{2}+\frac{479204329247737}{207166883628495}a+\frac{22350040989608}{13811125575233}$, $\frac{3654057582592}{621500650885485}a^{16}-\frac{23060860826}{1973017939319}a^{15}+\frac{5794924960268}{124300130177097}a^{14}-\frac{73007223780322}{621500650885485}a^{13}+\frac{3570953715548}{12683686752765}a^{12}-\frac{146242707473}{305105866905}a^{11}+\frac{58085301587743}{124300130177097}a^{10}+\frac{376294519849201}{621500650885485}a^{9}+\frac{41194534077017}{621500650885485}a^{8}+\frac{24964799375549}{69055627876165}a^{7}-\frac{927965111674711}{621500650885485}a^{6}+\frac{113386352669578}{621500650885485}a^{5}+\frac{26\!\cdots\!82}{621500650885485}a^{4}+\frac{15\!\cdots\!99}{207166883628495}a^{3}+\frac{50\!\cdots\!69}{621500650885485}a^{2}+\frac{660324619173838}{124300130177097}a+\frac{72913392949199}{41433376725699}$
| 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: | \( 3798575.81941 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 3798575.81941 \cdot 1}{2\cdot\sqrt{58497592625273225470725121}}\cr\approx \mathstrut & 1.20639839104 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225)
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 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225, 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 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225);
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 + 9*x^15 - 23*x^14 + 59*x^13 - 112*x^12 + 156*x^11 - 49*x^10 + 221*x^9 - 25*x^8 - 162*x^7 - 4*x^6 + 553*x^5 + 1435*x^4 + 1716*x^3 + 1462*x^2 + 615*x + 225);
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 34 |
| The 10 conjugacy class representatives for $D_{17}$ |
| Character table for $D_{17}$ |
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]
Sibling fields
| Galois closure: | data not computed |
| 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 | $17$ | ${\href{/padicField/3.2.0.1}{2} }^{8}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.2.0.1}{2} }^{8}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $17$ | $17$ | ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $17$ | $17$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $17$ | $17$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $17$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1663\)
| $\Q_{1663}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |