Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13)
gp: K = bnfinit(y^16 - 2*y^15 - 23*y^14 + 36*y^13 + 122*y^12 + 112*y^11 - 825*y^10 - 1766*y^9 + 6976*y^8 - 2594*y^7 - 13113*y^6 + 21674*y^5 - 14124*y^4 + 3262*y^3 + 342*y^2 - 78*y + 13, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13)
\( x^{16} - 2 x^{15} - 23 x^{14} + 36 x^{13} + 122 x^{12} + 112 x^{11} - 825 x^{10} - 1766 x^{9} + \cdots + 13 \)
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: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(26428481046229354497662569\)
\(\medspace = 13^{10}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.80\) | 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: | $13^{3/4}61^{1/2}\approx 53.471507940612106$ | ||
| Ramified primes: |
\(13\), \(61\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{366}a^{14}+\frac{7}{122}a^{13}+\frac{19}{183}a^{12}-\frac{83}{366}a^{11}-\frac{12}{61}a^{10}-\frac{7}{366}a^{9}-\frac{65}{366}a^{8}+\frac{59}{366}a^{7}-\frac{35}{122}a^{6}+\frac{35}{122}a^{5}-\frac{10}{61}a^{4}-\frac{43}{366}a^{3}+\frac{71}{183}a^{2}+\frac{76}{183}a+\frac{95}{366}$, $\frac{1}{89\!\cdots\!50}a^{15}-\frac{10\!\cdots\!23}{14\!\cdots\!75}a^{14}-\frac{17\!\cdots\!53}{89\!\cdots\!05}a^{13}+\frac{13\!\cdots\!91}{89\!\cdots\!50}a^{12}+\frac{19\!\cdots\!07}{29\!\cdots\!50}a^{11}-\frac{66\!\cdots\!69}{89\!\cdots\!50}a^{10}-\frac{52\!\cdots\!41}{89\!\cdots\!50}a^{9}-\frac{73\!\cdots\!24}{89\!\cdots\!05}a^{8}-\frac{77\!\cdots\!53}{29\!\cdots\!50}a^{7}-\frac{31\!\cdots\!63}{59\!\cdots\!70}a^{6}+\frac{58\!\cdots\!72}{14\!\cdots\!75}a^{5}-\frac{20\!\cdots\!14}{44\!\cdots\!25}a^{4}-\frac{76\!\cdots\!83}{44\!\cdots\!25}a^{3}+\frac{58\!\cdots\!19}{44\!\cdots\!25}a^{2}+\frac{30\!\cdots\!49}{89\!\cdots\!50}a+\frac{28\!\cdots\!11}{29\!\cdots\!50}$
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 $1$ (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: | $11$ | 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: |
$\frac{50\!\cdots\!34}{14\!\cdots\!75}a^{15}-\frac{25\!\cdots\!92}{14\!\cdots\!75}a^{14}-\frac{24\!\cdots\!44}{29\!\cdots\!35}a^{13}+\frac{45\!\cdots\!44}{14\!\cdots\!75}a^{12}+\frac{69\!\cdots\!64}{14\!\cdots\!75}a^{11}+\frac{14\!\cdots\!29}{14\!\cdots\!75}a^{10}-\frac{23\!\cdots\!94}{14\!\cdots\!75}a^{9}-\frac{25\!\cdots\!92}{29\!\cdots\!35}a^{8}+\frac{19\!\cdots\!94}{14\!\cdots\!75}a^{7}+\frac{42\!\cdots\!79}{29\!\cdots\!35}a^{6}-\frac{57\!\cdots\!12}{14\!\cdots\!75}a^{5}+\frac{28\!\cdots\!98}{14\!\cdots\!75}a^{4}+\frac{18\!\cdots\!56}{14\!\cdots\!75}a^{3}-\frac{22\!\cdots\!83}{14\!\cdots\!75}a^{2}+\frac{31\!\cdots\!66}{14\!\cdots\!75}a+\frac{93\!\cdots\!22}{14\!\cdots\!75}$, $\frac{27\!\cdots\!83}{16\!\cdots\!50}a^{15}-\frac{13\!\cdots\!79}{16\!\cdots\!50}a^{14}-\frac{11\!\cdots\!63}{32\!\cdots\!90}a^{13}-\frac{15\!\cdots\!86}{81\!\cdots\!25}a^{12}+\frac{13\!\cdots\!43}{16\!\cdots\!50}a^{11}+\frac{40\!\cdots\!74}{81\!\cdots\!25}a^{10}-\frac{34\!\cdots\!39}{81\!\cdots\!25}a^{9}-\frac{39\!\cdots\!32}{16\!\cdots\!45}a^{8}+\frac{64\!\cdots\!53}{16\!\cdots\!50}a^{7}-\frac{12\!\cdots\!11}{16\!\cdots\!45}a^{6}+\frac{25\!\cdots\!31}{16\!\cdots\!50}a^{5}+\frac{47\!\cdots\!01}{16\!\cdots\!50}a^{4}-\frac{41\!\cdots\!89}{81\!\cdots\!25}a^{3}+\frac{55\!\cdots\!29}{16\!\cdots\!50}a^{2}-\frac{74\!\cdots\!33}{16\!\cdots\!50}a-\frac{50\!\cdots\!11}{16\!\cdots\!50}$, $\frac{33\!\cdots\!59}{16\!\cdots\!50}a^{15}+\frac{13\!\cdots\!83}{16\!\cdots\!50}a^{14}-\frac{16\!\cdots\!99}{32\!\cdots\!90}a^{13}-\frac{17\!\cdots\!03}{81\!\cdots\!25}a^{12}+\frac{27\!\cdots\!39}{16\!\cdots\!50}a^{11}+\frac{13\!\cdots\!52}{81\!\cdots\!25}a^{10}+\frac{19\!\cdots\!53}{81\!\cdots\!25}a^{9}-\frac{11\!\cdots\!36}{16\!\cdots\!45}a^{8}-\frac{25\!\cdots\!31}{16\!\cdots\!50}a^{7}+\frac{45\!\cdots\!37}{16\!\cdots\!45}a^{6}+\frac{21\!\cdots\!63}{16\!\cdots\!50}a^{5}-\frac{87\!\cdots\!77}{16\!\cdots\!50}a^{4}+\frac{34\!\cdots\!53}{81\!\cdots\!25}a^{3}-\frac{15\!\cdots\!33}{16\!\cdots\!50}a^{2}+\frac{18\!\cdots\!91}{16\!\cdots\!50}a-\frac{21\!\cdots\!03}{16\!\cdots\!50}$, $\frac{69\!\cdots\!76}{14\!\cdots\!75}a^{15}+\frac{19\!\cdots\!37}{14\!\cdots\!75}a^{14}-\frac{31\!\cdots\!56}{29\!\cdots\!35}a^{13}-\frac{47\!\cdots\!59}{14\!\cdots\!75}a^{12}-\frac{16\!\cdots\!04}{14\!\cdots\!75}a^{11}+\frac{25\!\cdots\!81}{14\!\cdots\!75}a^{10}+\frac{70\!\cdots\!34}{14\!\cdots\!75}a^{9}-\frac{10\!\cdots\!13}{29\!\cdots\!35}a^{8}-\frac{46\!\cdots\!84}{14\!\cdots\!75}a^{7}+\frac{84\!\cdots\!16}{29\!\cdots\!35}a^{6}+\frac{66\!\cdots\!32}{14\!\cdots\!75}a^{5}-\frac{12\!\cdots\!03}{14\!\cdots\!75}a^{4}+\frac{83\!\cdots\!84}{14\!\cdots\!75}a^{3}-\frac{12\!\cdots\!37}{14\!\cdots\!75}a^{2}+\frac{12\!\cdots\!24}{14\!\cdots\!75}a+\frac{67\!\cdots\!33}{14\!\cdots\!75}$, $\frac{27\!\cdots\!83}{16\!\cdots\!50}a^{15}-\frac{13\!\cdots\!79}{16\!\cdots\!50}a^{14}-\frac{11\!\cdots\!63}{32\!\cdots\!90}a^{13}-\frac{15\!\cdots\!86}{81\!\cdots\!25}a^{12}+\frac{13\!\cdots\!43}{16\!\cdots\!50}a^{11}+\frac{40\!\cdots\!74}{81\!\cdots\!25}a^{10}-\frac{34\!\cdots\!39}{81\!\cdots\!25}a^{9}-\frac{39\!\cdots\!32}{16\!\cdots\!45}a^{8}+\frac{64\!\cdots\!53}{16\!\cdots\!50}a^{7}-\frac{12\!\cdots\!11}{16\!\cdots\!45}a^{6}+\frac{25\!\cdots\!31}{16\!\cdots\!50}a^{5}+\frac{47\!\cdots\!01}{16\!\cdots\!50}a^{4}-\frac{41\!\cdots\!89}{81\!\cdots\!25}a^{3}+\frac{55\!\cdots\!29}{16\!\cdots\!50}a^{2}-\frac{74\!\cdots\!33}{16\!\cdots\!50}a-\frac{34\!\cdots\!61}{16\!\cdots\!50}$, $\frac{11\!\cdots\!91}{89\!\cdots\!05}a^{15}+\frac{16\!\cdots\!82}{14\!\cdots\!05}a^{14}-\frac{10\!\cdots\!99}{35\!\cdots\!22}a^{13}-\frac{26\!\cdots\!53}{17\!\cdots\!10}a^{12}+\frac{13\!\cdots\!56}{89\!\cdots\!05}a^{11}+\frac{80\!\cdots\!87}{17\!\cdots\!10}a^{10}-\frac{48\!\cdots\!97}{17\!\cdots\!10}a^{9}-\frac{55\!\cdots\!42}{17\!\cdots\!61}a^{8}+\frac{26\!\cdots\!36}{89\!\cdots\!05}a^{7}+\frac{59\!\cdots\!97}{11\!\cdots\!74}a^{6}-\frac{60\!\cdots\!47}{59\!\cdots\!70}a^{5}+\frac{92\!\cdots\!29}{17\!\cdots\!10}a^{4}+\frac{12\!\cdots\!94}{89\!\cdots\!05}a^{3}-\frac{16\!\cdots\!87}{89\!\cdots\!05}a^{2}+\frac{10\!\cdots\!11}{59\!\cdots\!70}a+\frac{68\!\cdots\!63}{89\!\cdots\!05}$, $\frac{29\!\cdots\!91}{89\!\cdots\!50}a^{15}+\frac{39\!\cdots\!17}{89\!\cdots\!50}a^{14}-\frac{74\!\cdots\!53}{89\!\cdots\!05}a^{13}-\frac{21\!\cdots\!74}{14\!\cdots\!75}a^{12}+\frac{19\!\cdots\!18}{44\!\cdots\!25}a^{11}+\frac{79\!\cdots\!48}{44\!\cdots\!25}a^{10}+\frac{12\!\cdots\!73}{29\!\cdots\!50}a^{9}-\frac{63\!\cdots\!01}{59\!\cdots\!70}a^{8}-\frac{13\!\cdots\!72}{44\!\cdots\!25}a^{7}+\frac{19\!\cdots\!37}{59\!\cdots\!70}a^{6}-\frac{19\!\cdots\!48}{14\!\cdots\!75}a^{5}-\frac{32\!\cdots\!73}{89\!\cdots\!50}a^{4}+\frac{20\!\cdots\!93}{48\!\cdots\!50}a^{3}-\frac{14\!\cdots\!82}{14\!\cdots\!75}a^{2}-\frac{17\!\cdots\!33}{44\!\cdots\!25}a-\frac{22\!\cdots\!11}{44\!\cdots\!25}$, $\frac{17\!\cdots\!71}{29\!\cdots\!50}a^{15}-\frac{28\!\cdots\!22}{44\!\cdots\!25}a^{14}-\frac{41\!\cdots\!18}{29\!\cdots\!35}a^{13}+\frac{39\!\cdots\!29}{44\!\cdots\!25}a^{12}+\frac{34\!\cdots\!99}{44\!\cdots\!25}a^{11}+\frac{37\!\cdots\!51}{29\!\cdots\!50}a^{10}-\frac{16\!\cdots\!54}{44\!\cdots\!25}a^{9}-\frac{23\!\cdots\!69}{17\!\cdots\!10}a^{8}+\frac{26\!\cdots\!83}{89\!\cdots\!50}a^{7}+\frac{59\!\cdots\!81}{59\!\cdots\!70}a^{6}-\frac{21\!\cdots\!03}{29\!\cdots\!50}a^{5}+\frac{20\!\cdots\!37}{29\!\cdots\!50}a^{4}-\frac{13\!\cdots\!33}{89\!\cdots\!50}a^{3}-\frac{44\!\cdots\!03}{44\!\cdots\!25}a^{2}+\frac{42\!\cdots\!87}{89\!\cdots\!50}a-\frac{25\!\cdots\!98}{44\!\cdots\!25}$, $\frac{26\!\cdots\!48}{17\!\cdots\!61}a^{15}-\frac{38\!\cdots\!21}{35\!\cdots\!22}a^{14}-\frac{63\!\cdots\!39}{17\!\cdots\!61}a^{13}+\frac{10\!\cdots\!49}{11\!\cdots\!74}a^{12}+\frac{31\!\cdots\!32}{17\!\cdots\!61}a^{11}+\frac{14\!\cdots\!69}{35\!\cdots\!22}a^{10}-\frac{78\!\cdots\!89}{11\!\cdots\!74}a^{9}-\frac{19\!\cdots\!02}{59\!\cdots\!87}a^{8}+\frac{10\!\cdots\!37}{17\!\cdots\!61}a^{7}+\frac{28\!\cdots\!69}{11\!\cdots\!74}a^{6}-\frac{17\!\cdots\!87}{11\!\cdots\!74}a^{5}+\frac{55\!\cdots\!85}{35\!\cdots\!22}a^{4}-\frac{43\!\cdots\!65}{59\!\cdots\!87}a^{3}+\frac{17\!\cdots\!03}{11\!\cdots\!74}a^{2}-\frac{10\!\cdots\!61}{35\!\cdots\!22}a-\frac{21\!\cdots\!41}{35\!\cdots\!22}$, $\frac{35\!\cdots\!09}{11\!\cdots\!74}a^{15}-\frac{64\!\cdots\!01}{17\!\cdots\!61}a^{14}-\frac{41\!\cdots\!13}{59\!\cdots\!87}a^{13}+\frac{90\!\cdots\!85}{17\!\cdots\!61}a^{12}+\frac{68\!\cdots\!16}{17\!\cdots\!61}a^{11}+\frac{38\!\cdots\!35}{59\!\cdots\!87}a^{10}-\frac{53\!\cdots\!70}{29\!\cdots\!01}a^{9}-\frac{11\!\cdots\!24}{17\!\cdots\!61}a^{8}+\frac{27\!\cdots\!25}{17\!\cdots\!61}a^{7}+\frac{14\!\cdots\!81}{59\!\cdots\!87}a^{6}-\frac{20\!\cdots\!24}{59\!\cdots\!87}a^{5}+\frac{38\!\cdots\!73}{97\!\cdots\!67}a^{4}-\frac{65\!\cdots\!49}{35\!\cdots\!22}a^{3}+\frac{41\!\cdots\!73}{35\!\cdots\!22}a^{2}+\frac{20\!\cdots\!08}{17\!\cdots\!61}a+\frac{12\!\cdots\!12}{17\!\cdots\!61}$, $\frac{37\!\cdots\!13}{11\!\cdots\!74}a^{15}-\frac{15\!\cdots\!67}{35\!\cdots\!22}a^{14}-\frac{45\!\cdots\!73}{59\!\cdots\!87}a^{13}+\frac{24\!\cdots\!99}{35\!\cdots\!22}a^{12}+\frac{15\!\cdots\!91}{35\!\cdots\!22}a^{11}+\frac{73\!\cdots\!05}{11\!\cdots\!74}a^{10}-\frac{39\!\cdots\!67}{17\!\cdots\!61}a^{9}-\frac{24\!\cdots\!11}{35\!\cdots\!22}a^{8}+\frac{63\!\cdots\!07}{35\!\cdots\!22}a^{7}+\frac{81\!\cdots\!44}{59\!\cdots\!87}a^{6}-\frac{23\!\cdots\!27}{59\!\cdots\!87}a^{5}+\frac{55\!\cdots\!69}{11\!\cdots\!74}a^{4}-\frac{39\!\cdots\!65}{17\!\cdots\!61}a^{3}+\frac{59\!\cdots\!51}{35\!\cdots\!22}a^{2}+\frac{38\!\cdots\!01}{35\!\cdots\!22}a-\frac{77\!\cdots\!17}{35\!\cdots\!22}$
| 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: | \( 6419363.9793 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 6419363.9793 \cdot 1}{2\cdot\sqrt{26428481046229354497662569}}\cr\approx \mathstrut & 0.24910666212 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13)
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 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13, 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 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13);
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 - 2*x^15 - 23*x^14 + 36*x^13 + 122*x^12 + 112*x^11 - 825*x^10 - 1766*x^9 + 6976*x^8 - 2594*x^7 - 13113*x^6 + 21674*x^5 - 14124*x^4 + 3262*x^3 + 342*x^2 - 78*x + 13);
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
$C_4\wr C_2$ (as 16T42):
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 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{61}) \), \(\Q(\sqrt{793}) \), 4.4.10309.1 x2, 4.4.48373.1 x2, \(\Q(\sqrt{13}, \sqrt{61})\), 8.4.5140863842413.1, 8.4.30419312677.1, 8.8.395451064801.1 |
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 8 siblings: | 8.4.30419312677.1, 8.4.5140863842413.1 |
| Degree 16 sibling: | 16.0.1200326067404665657109641.8 |
| Minimal sibling: | 8.4.30419312677.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(61\)
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |