Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16)
gp: K = bnfinit(y^16 - y^15 - 36*y^14 + 8*y^13 + 481*y^12 + 140*y^11 - 2935*y^10 - 1689*y^9 + 8381*y^8 + 5551*y^7 - 10875*y^6 - 6618*y^5 + 5375*y^4 + 2400*y^3 - 936*y^2 - 216*y + 16, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16)
\( x^{16} - x^{15} - 36 x^{14} + 8 x^{13} + 481 x^{12} + 140 x^{11} - 2935 x^{10} - 1689 x^{9} + 8381 x^{8} + \cdots + 16 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1328582041288248401656849\)
\(\medspace = 17^{14}\cdot 53^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(32.19\) | 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: | $17^{7/8}53^{1/2}\approx 86.8521834484431$ | ||
| Ramified primes: |
\(17\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{40\!\cdots\!36}a^{15}+\frac{118991280337037}{40\!\cdots\!36}a^{14}-\frac{37\!\cdots\!97}{20\!\cdots\!68}a^{13}-\frac{11\!\cdots\!73}{10\!\cdots\!34}a^{12}-\frac{88\!\cdots\!03}{40\!\cdots\!36}a^{11}-\frac{27\!\cdots\!39}{20\!\cdots\!68}a^{10}+\frac{206514959415577}{29\!\cdots\!28}a^{9}+\frac{33\!\cdots\!85}{40\!\cdots\!36}a^{8}-\frac{13\!\cdots\!17}{40\!\cdots\!36}a^{7}+\frac{112897890080593}{40\!\cdots\!36}a^{6}+\frac{61\!\cdots\!47}{40\!\cdots\!36}a^{5}-\frac{11\!\cdots\!05}{10\!\cdots\!34}a^{4}+\frac{63\!\cdots\!23}{40\!\cdots\!36}a^{3}+\frac{96\!\cdots\!71}{20\!\cdots\!68}a^{2}-\frac{13\!\cdots\!09}{10\!\cdots\!34}a+\frac{71\!\cdots\!74}{50\!\cdots\!67}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $15$ | 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: |
$\frac{891914562613137}{10\!\cdots\!34}a^{15}-\frac{472840282001043}{10\!\cdots\!34}a^{14}-\frac{16\!\cdots\!53}{50\!\cdots\!67}a^{13}-\frac{76\!\cdots\!73}{10\!\cdots\!34}a^{12}+\frac{21\!\cdots\!79}{50\!\cdots\!67}a^{11}+\frac{16\!\cdots\!38}{50\!\cdots\!67}a^{10}-\frac{18\!\cdots\!01}{743368415621182}a^{9}-\frac{26\!\cdots\!65}{10\!\cdots\!34}a^{8}+\frac{33\!\cdots\!78}{50\!\cdots\!67}a^{7}+\frac{41\!\cdots\!13}{50\!\cdots\!67}a^{6}-\frac{69\!\cdots\!73}{10\!\cdots\!34}a^{5}-\frac{47\!\cdots\!87}{50\!\cdots\!67}a^{4}+\frac{95\!\cdots\!25}{10\!\cdots\!34}a^{3}+\frac{29\!\cdots\!33}{10\!\cdots\!34}a^{2}+\frac{41\!\cdots\!45}{10\!\cdots\!34}a-\frac{11\!\cdots\!57}{50\!\cdots\!67}$, $\frac{891339036823457}{10\!\cdots\!34}a^{15}-\frac{24\!\cdots\!31}{20\!\cdots\!68}a^{14}-\frac{62\!\cdots\!17}{20\!\cdots\!68}a^{13}+\frac{97\!\cdots\!43}{50\!\cdots\!67}a^{12}+\frac{20\!\cdots\!69}{50\!\cdots\!67}a^{11}-\frac{97\!\cdots\!31}{20\!\cdots\!68}a^{10}-\frac{18\!\cdots\!69}{743368415621182}a^{9}-\frac{68\!\cdots\!23}{20\!\cdots\!68}a^{8}+\frac{13\!\cdots\!85}{20\!\cdots\!68}a^{7}+\frac{21\!\cdots\!23}{20\!\cdots\!68}a^{6}-\frac{16\!\cdots\!73}{20\!\cdots\!68}a^{5}+\frac{29\!\cdots\!85}{20\!\cdots\!68}a^{4}+\frac{16\!\cdots\!62}{50\!\cdots\!67}a^{3}-\frac{23\!\cdots\!91}{20\!\cdots\!68}a^{2}-\frac{17\!\cdots\!50}{50\!\cdots\!67}a+\frac{11\!\cdots\!26}{50\!\cdots\!67}$, $\frac{35\!\cdots\!99}{40\!\cdots\!36}a^{15}-\frac{65\!\cdots\!55}{40\!\cdots\!36}a^{14}-\frac{31\!\cdots\!41}{10\!\cdots\!34}a^{13}+\frac{34\!\cdots\!07}{10\!\cdots\!34}a^{12}+\frac{16\!\cdots\!43}{40\!\cdots\!36}a^{11}-\frac{24\!\cdots\!43}{10\!\cdots\!34}a^{10}-\frac{77\!\cdots\!13}{29\!\cdots\!28}a^{9}+\frac{31\!\cdots\!73}{40\!\cdots\!36}a^{8}+\frac{32\!\cdots\!23}{40\!\cdots\!36}a^{7}-\frac{70\!\cdots\!43}{40\!\cdots\!36}a^{6}-\frac{46\!\cdots\!29}{40\!\cdots\!36}a^{5}+\frac{56\!\cdots\!19}{20\!\cdots\!68}a^{4}+\frac{23\!\cdots\!81}{40\!\cdots\!36}a^{3}-\frac{14\!\cdots\!47}{10\!\cdots\!34}a^{2}-\frac{48\!\cdots\!77}{10\!\cdots\!34}a-\frac{17\!\cdots\!81}{50\!\cdots\!67}$, $\frac{21\!\cdots\!01}{40\!\cdots\!36}a^{15}-\frac{80\!\cdots\!69}{40\!\cdots\!36}a^{14}-\frac{15\!\cdots\!85}{10\!\cdots\!34}a^{13}+\frac{24\!\cdots\!79}{50\!\cdots\!67}a^{12}+\frac{67\!\cdots\!93}{40\!\cdots\!36}a^{11}-\frac{23\!\cdots\!30}{50\!\cdots\!67}a^{10}-\frac{25\!\cdots\!71}{29\!\cdots\!28}a^{9}+\frac{91\!\cdots\!27}{40\!\cdots\!36}a^{8}+\frac{89\!\cdots\!29}{40\!\cdots\!36}a^{7}-\frac{23\!\cdots\!45}{40\!\cdots\!36}a^{6}-\frac{12\!\cdots\!23}{40\!\cdots\!36}a^{5}+\frac{12\!\cdots\!55}{20\!\cdots\!68}a^{4}+\frac{10\!\cdots\!27}{40\!\cdots\!36}a^{3}-\frac{89\!\cdots\!01}{50\!\cdots\!67}a^{2}-\frac{37\!\cdots\!89}{10\!\cdots\!34}a+\frac{37\!\cdots\!11}{50\!\cdots\!67}$, $\frac{150487822041575}{40\!\cdots\!36}a^{15}-\frac{328905443271411}{40\!\cdots\!36}a^{14}-\frac{13\!\cdots\!67}{10\!\cdots\!34}a^{13}+\frac{12\!\cdots\!33}{50\!\cdots\!67}a^{12}+\frac{82\!\cdots\!31}{40\!\cdots\!36}a^{11}-\frac{35\!\cdots\!45}{10\!\cdots\!34}a^{10}-\frac{46\!\cdots\!05}{29\!\cdots\!28}a^{9}+\frac{12\!\cdots\!25}{40\!\cdots\!36}a^{8}+\frac{27\!\cdots\!99}{40\!\cdots\!36}a^{7}-\frac{53\!\cdots\!87}{40\!\cdots\!36}a^{6}-\frac{69\!\cdots\!89}{40\!\cdots\!36}a^{5}+\frac{47\!\cdots\!37}{20\!\cdots\!68}a^{4}+\frac{67\!\cdots\!65}{40\!\cdots\!36}a^{3}-\frac{11\!\cdots\!39}{10\!\cdots\!34}a^{2}-\frac{13\!\cdots\!10}{50\!\cdots\!67}a+\frac{45\!\cdots\!27}{50\!\cdots\!67}$, $\frac{520663250807825}{20\!\cdots\!68}a^{15}+\frac{452080275319029}{20\!\cdots\!68}a^{14}-\frac{56\!\cdots\!25}{50\!\cdots\!67}a^{13}-\frac{10\!\cdots\!19}{10\!\cdots\!34}a^{12}+\frac{33\!\cdots\!55}{20\!\cdots\!68}a^{11}+\frac{87\!\cdots\!64}{50\!\cdots\!67}a^{10}-\frac{16\!\cdots\!33}{14\!\cdots\!64}a^{9}-\frac{22\!\cdots\!33}{20\!\cdots\!68}a^{8}+\frac{65\!\cdots\!59}{20\!\cdots\!68}a^{7}+\frac{60\!\cdots\!85}{20\!\cdots\!68}a^{6}-\frac{79\!\cdots\!75}{20\!\cdots\!68}a^{5}-\frac{13\!\cdots\!48}{50\!\cdots\!67}a^{4}+\frac{20\!\cdots\!63}{20\!\cdots\!68}a^{3}+\frac{20\!\cdots\!67}{10\!\cdots\!34}a^{2}-\frac{32\!\cdots\!11}{10\!\cdots\!34}a+\frac{35\!\cdots\!70}{50\!\cdots\!67}$, $\frac{17\!\cdots\!95}{40\!\cdots\!36}a^{15}-\frac{30\!\cdots\!53}{40\!\cdots\!36}a^{14}-\frac{29\!\cdots\!33}{20\!\cdots\!68}a^{13}+\frac{77\!\cdots\!69}{50\!\cdots\!67}a^{12}+\frac{76\!\cdots\!87}{40\!\cdots\!36}a^{11}-\frac{22\!\cdots\!41}{20\!\cdots\!68}a^{10}-\frac{32\!\cdots\!25}{29\!\cdots\!28}a^{9}+\frac{21\!\cdots\!55}{40\!\cdots\!36}a^{8}+\frac{11\!\cdots\!41}{40\!\cdots\!36}a^{7}-\frac{79\!\cdots\!53}{40\!\cdots\!36}a^{6}-\frac{15\!\cdots\!87}{40\!\cdots\!36}a^{5}+\frac{33\!\cdots\!45}{10\!\cdots\!34}a^{4}+\frac{88\!\cdots\!93}{40\!\cdots\!36}a^{3}-\frac{19\!\cdots\!49}{20\!\cdots\!68}a^{2}-\frac{11\!\cdots\!08}{50\!\cdots\!67}a-\frac{11\!\cdots\!95}{50\!\cdots\!67}$, $\frac{12\!\cdots\!13}{40\!\cdots\!36}a^{15}-\frac{17\!\cdots\!61}{40\!\cdots\!36}a^{14}-\frac{45\!\cdots\!62}{50\!\cdots\!67}a^{13}+\frac{42\!\cdots\!77}{10\!\cdots\!34}a^{12}+\frac{34\!\cdots\!61}{40\!\cdots\!36}a^{11}+\frac{19\!\cdots\!97}{10\!\cdots\!34}a^{10}-\frac{69\!\cdots\!03}{29\!\cdots\!28}a^{9}-\frac{64\!\cdots\!65}{40\!\cdots\!36}a^{8}-\frac{22\!\cdots\!79}{40\!\cdots\!36}a^{7}-\frac{20\!\cdots\!25}{40\!\cdots\!36}a^{6}+\frac{95\!\cdots\!45}{40\!\cdots\!36}a^{5}+\frac{43\!\cdots\!71}{20\!\cdots\!68}a^{4}-\frac{65\!\cdots\!13}{40\!\cdots\!36}a^{3}-\frac{72\!\cdots\!67}{50\!\cdots\!67}a^{2}+\frac{27\!\cdots\!75}{10\!\cdots\!34}a+\frac{85\!\cdots\!14}{50\!\cdots\!67}$, $\frac{17\!\cdots\!35}{40\!\cdots\!36}a^{15}-\frac{32\!\cdots\!21}{40\!\cdots\!36}a^{14}-\frac{27\!\cdots\!03}{20\!\cdots\!68}a^{13}+\frac{72\!\cdots\!34}{50\!\cdots\!67}a^{12}+\frac{64\!\cdots\!91}{40\!\cdots\!36}a^{11}-\frac{12\!\cdots\!09}{20\!\cdots\!68}a^{10}-\frac{22\!\cdots\!97}{29\!\cdots\!28}a^{9}-\frac{21\!\cdots\!93}{40\!\cdots\!36}a^{8}+\frac{50\!\cdots\!33}{40\!\cdots\!36}a^{7}+\frac{21\!\cdots\!27}{40\!\cdots\!36}a^{6}+\frac{38\!\cdots\!41}{40\!\cdots\!36}a^{5}-\frac{48\!\cdots\!33}{10\!\cdots\!34}a^{4}-\frac{13\!\cdots\!79}{40\!\cdots\!36}a^{3}-\frac{57\!\cdots\!27}{20\!\cdots\!68}a^{2}+\frac{45\!\cdots\!21}{50\!\cdots\!67}a+\frac{31\!\cdots\!46}{50\!\cdots\!67}$, $\frac{81\!\cdots\!43}{40\!\cdots\!36}a^{15}-\frac{20\!\cdots\!01}{40\!\cdots\!36}a^{14}-\frac{12\!\cdots\!05}{20\!\cdots\!68}a^{13}+\frac{55\!\cdots\!67}{50\!\cdots\!67}a^{12}+\frac{31\!\cdots\!51}{40\!\cdots\!36}a^{11}-\frac{17\!\cdots\!71}{20\!\cdots\!68}a^{10}-\frac{12\!\cdots\!61}{29\!\cdots\!28}a^{9}+\frac{11\!\cdots\!47}{40\!\cdots\!36}a^{8}+\frac{45\!\cdots\!97}{40\!\cdots\!36}a^{7}-\frac{22\!\cdots\!45}{40\!\cdots\!36}a^{6}-\frac{49\!\cdots\!31}{40\!\cdots\!36}a^{5}+\frac{58\!\cdots\!79}{10\!\cdots\!34}a^{4}+\frac{13\!\cdots\!41}{40\!\cdots\!36}a^{3}-\frac{54\!\cdots\!23}{20\!\cdots\!68}a^{2}+\frac{22\!\cdots\!23}{10\!\cdots\!34}a+\frac{94\!\cdots\!00}{50\!\cdots\!67}$, $\frac{22401114655435}{40\!\cdots\!36}a^{15}-\frac{71\!\cdots\!47}{40\!\cdots\!36}a^{14}+\frac{43\!\cdots\!31}{10\!\cdots\!34}a^{13}+\frac{53\!\cdots\!17}{10\!\cdots\!34}a^{12}-\frac{35\!\cdots\!93}{40\!\cdots\!36}a^{11}-\frac{29\!\cdots\!61}{50\!\cdots\!67}a^{10}+\frac{17\!\cdots\!99}{29\!\cdots\!28}a^{9}+\frac{11\!\cdots\!73}{40\!\cdots\!36}a^{8}-\frac{66\!\cdots\!45}{40\!\cdots\!36}a^{7}-\frac{21\!\cdots\!19}{40\!\cdots\!36}a^{6}+\frac{10\!\cdots\!83}{40\!\cdots\!36}a^{5}+\frac{72\!\cdots\!15}{20\!\cdots\!68}a^{4}-\frac{86\!\cdots\!87}{40\!\cdots\!36}a^{3}-\frac{67\!\cdots\!61}{50\!\cdots\!67}a^{2}+\frac{44\!\cdots\!43}{10\!\cdots\!34}a+\frac{14\!\cdots\!52}{50\!\cdots\!67}$, $\frac{38\!\cdots\!77}{40\!\cdots\!36}a^{15}-\frac{10\!\cdots\!65}{40\!\cdots\!36}a^{14}-\frac{15\!\cdots\!27}{50\!\cdots\!67}a^{13}+\frac{56\!\cdots\!19}{10\!\cdots\!34}a^{12}+\frac{14\!\cdots\!77}{40\!\cdots\!36}a^{11}-\frac{22\!\cdots\!51}{50\!\cdots\!67}a^{10}-\frac{60\!\cdots\!95}{29\!\cdots\!28}a^{9}+\frac{63\!\cdots\!63}{40\!\cdots\!36}a^{8}+\frac{22\!\cdots\!85}{40\!\cdots\!36}a^{7}-\frac{12\!\cdots\!37}{40\!\cdots\!36}a^{6}-\frac{27\!\cdots\!55}{40\!\cdots\!36}a^{5}+\frac{52\!\cdots\!55}{20\!\cdots\!68}a^{4}+\frac{13\!\cdots\!59}{40\!\cdots\!36}a^{3}-\frac{54\!\cdots\!25}{10\!\cdots\!34}a^{2}-\frac{50\!\cdots\!45}{10\!\cdots\!34}a-\frac{20\!\cdots\!52}{50\!\cdots\!67}$, $\frac{10\!\cdots\!11}{40\!\cdots\!36}a^{15}-\frac{13\!\cdots\!93}{40\!\cdots\!36}a^{14}-\frac{17\!\cdots\!93}{20\!\cdots\!68}a^{13}+\frac{37\!\cdots\!57}{10\!\cdots\!34}a^{12}+\frac{44\!\cdots\!19}{40\!\cdots\!36}a^{11}+\frac{39\!\cdots\!67}{20\!\cdots\!68}a^{10}-\frac{18\!\cdots\!97}{29\!\cdots\!28}a^{9}-\frac{14\!\cdots\!73}{40\!\cdots\!36}a^{8}+\frac{66\!\cdots\!49}{40\!\cdots\!36}a^{7}+\frac{49\!\cdots\!11}{40\!\cdots\!36}a^{6}-\frac{71\!\cdots\!91}{40\!\cdots\!36}a^{5}-\frac{14\!\cdots\!97}{10\!\cdots\!34}a^{4}+\frac{20\!\cdots\!81}{40\!\cdots\!36}a^{3}+\frac{85\!\cdots\!65}{20\!\cdots\!68}a^{2}-\frac{17\!\cdots\!64}{50\!\cdots\!67}a-\frac{16\!\cdots\!05}{50\!\cdots\!67}$, $\frac{62\!\cdots\!79}{40\!\cdots\!36}a^{15}-\frac{893526174124175}{40\!\cdots\!36}a^{14}-\frac{60\!\cdots\!13}{10\!\cdots\!34}a^{13}-\frac{25\!\cdots\!29}{10\!\cdots\!34}a^{12}+\frac{33\!\cdots\!99}{40\!\cdots\!36}a^{11}+\frac{31\!\cdots\!02}{50\!\cdots\!67}a^{10}-\frac{15\!\cdots\!65}{29\!\cdots\!28}a^{9}-\frac{17\!\cdots\!55}{40\!\cdots\!36}a^{8}+\frac{61\!\cdots\!63}{40\!\cdots\!36}a^{7}+\frac{45\!\cdots\!17}{40\!\cdots\!36}a^{6}-\frac{82\!\cdots\!53}{40\!\cdots\!36}a^{5}-\frac{19\!\cdots\!91}{20\!\cdots\!68}a^{4}+\frac{39\!\cdots\!25}{40\!\cdots\!36}a^{3}+\frac{93\!\cdots\!65}{50\!\cdots\!67}a^{2}-\frac{83\!\cdots\!92}{50\!\cdots\!67}a+\frac{56\!\cdots\!62}{50\!\cdots\!67}$, $\frac{11\!\cdots\!84}{50\!\cdots\!67}a^{15}-\frac{88\!\cdots\!85}{20\!\cdots\!68}a^{14}-\frac{14\!\cdots\!41}{20\!\cdots\!68}a^{13}+\frac{41\!\cdots\!33}{50\!\cdots\!67}a^{12}+\frac{45\!\cdots\!12}{50\!\cdots\!67}a^{11}-\frac{81\!\cdots\!45}{20\!\cdots\!68}a^{10}-\frac{19\!\cdots\!02}{371684207810591}a^{9}-\frac{39\!\cdots\!71}{20\!\cdots\!68}a^{8}+\frac{28\!\cdots\!29}{20\!\cdots\!68}a^{7}+\frac{96\!\cdots\!51}{20\!\cdots\!68}a^{6}-\frac{33\!\cdots\!75}{20\!\cdots\!68}a^{5}-\frac{17\!\cdots\!21}{20\!\cdots\!68}a^{4}+\frac{33\!\cdots\!90}{50\!\cdots\!67}a^{3}+\frac{67\!\cdots\!33}{20\!\cdots\!68}a^{2}-\frac{39\!\cdots\!74}{50\!\cdots\!67}a-\frac{13\!\cdots\!72}{50\!\cdots\!67}$
| 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: | \( 6471571.45455 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 6471571.45455 \cdot 1}{2\cdot\sqrt{1328582041288248401656849}}\cr\approx \mathstrut & 0.183977832003 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16)
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^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16, 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^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16);
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^16 - x^15 - 36*x^14 + 8*x^13 + 481*x^12 + 140*x^11 - 2935*x^10 - 1689*x^9 + 8381*x^8 + 5551*x^7 - 10875*x^6 - 6618*x^5 + 5375*x^4 + 2400*x^3 - 936*x^2 - 216*x + 16);
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
$C_2^2:C_8$ (as 16T24):
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 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.4.15317.1, 4.4.260389.1, 8.8.1152641332457.1, \(\Q(\zeta_{17})^+\), 8.8.67802431321.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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: | deg 32 |
| Degree 16 sibling: | 16.16.10483151353726139536553735554369.2 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.14.1 | $x^{16} + 128 x^{15} + 7192 x^{14} + 232064 x^{13} + 4716796 x^{12} + 62185088 x^{11} + 525781480 x^{10} + 2696730752 x^{9} + 7365142088 x^{8} + 8090194432 x^{7} + 4732152320 x^{6} + 1682759680 x^{5} + 456414056 x^{4} + 996830464 x^{3} + 7439529968 x^{2} + 33582546688 x + 66368009604$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |
|
\(53\)
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 53.4.0.1 | $x^{4} + 9 x^{2} + 38 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 53.8.4.1 | $x^{8} + 9752 x^{7} + 35663294 x^{6} + 57966048984 x^{5} + 35333436724405 x^{4} + 3595233169984 x^{3} + 320278224174124 x^{2} + 1356456509257952 x + 99990743929156$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |