Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328)
gp: K = bnfinit(y^16 - y^15 - 521*y^14 - 352*y^13 + 97262*y^12 + 111457*y^11 - 8578319*y^10 - 7737475*y^9 + 391096418*y^8 + 106474606*y^7 - 9220425289*y^6 + 3780858515*y^5 + 102878217925*y^4 - 87419452512*y^3 - 457661368284*y^2 + 433370758320*y + 437343265328, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328)
\( x^{16} - x^{15} - 521 x^{14} - 352 x^{13} + 97262 x^{12} + 111457 x^{11} - 8578319 x^{10} + \cdots + 437343265328 \)
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: |
\(462732306245995722656474121747020143758680081\)
\(\medspace = 29^{14}\cdot 41^{15}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(618.85\) | 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: | $29^{7/8}41^{15/16}\approx 618.8467490579351$ | ||
| Ramified primes: |
\(29\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{41}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{236}a^{13}-\frac{13}{118}a^{12}-\frac{7}{118}a^{11}+\frac{25}{118}a^{10}+\frac{23}{118}a^{9}-\frac{99}{236}a^{8}+\frac{13}{118}a^{7}-\frac{14}{59}a^{6}+\frac{29}{118}a^{5}+\frac{29}{59}a^{4}-\frac{51}{236}a^{3}+\frac{9}{118}a^{2}+\frac{17}{59}a+\frac{10}{59}$, $\frac{1}{65608}a^{14}-\frac{135}{65608}a^{13}+\frac{1109}{65608}a^{12}-\frac{4817}{32804}a^{11}+\frac{10455}{32804}a^{10}+\frac{29461}{65608}a^{9}-\frac{12901}{65608}a^{8}-\frac{12153}{65608}a^{7}+\frac{6821}{16402}a^{6}-\frac{7587}{32804}a^{5}-\frac{32165}{65608}a^{4}+\frac{23985}{65608}a^{3}+\frac{31323}{65608}a^{2}+\frac{16197}{32804}a+\frac{6875}{16402}$, $\frac{1}{32\!\cdots\!08}a^{15}-\frac{11\!\cdots\!81}{32\!\cdots\!08}a^{14}+\frac{30\!\cdots\!75}{32\!\cdots\!08}a^{13}-\frac{40\!\cdots\!27}{80\!\cdots\!52}a^{12}-\frac{31\!\cdots\!75}{16\!\cdots\!04}a^{11}-\frac{30\!\cdots\!03}{32\!\cdots\!08}a^{10}+\frac{82\!\cdots\!53}{32\!\cdots\!08}a^{9}+\frac{30\!\cdots\!93}{32\!\cdots\!08}a^{8}-\frac{37\!\cdots\!77}{16\!\cdots\!04}a^{7}+\frac{32\!\cdots\!45}{16\!\cdots\!04}a^{6}-\frac{43\!\cdots\!93}{32\!\cdots\!08}a^{5}+\frac{29\!\cdots\!03}{32\!\cdots\!08}a^{4}-\frac{13\!\cdots\!83}{32\!\cdots\!08}a^{3}-\frac{37\!\cdots\!85}{80\!\cdots\!52}a^{2}+\frac{36\!\cdots\!14}{20\!\cdots\!13}a+\frac{25\!\cdots\!67}{40\!\cdots\!26}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{89\!\cdots\!90}{28\!\cdots\!77}a^{15}+\frac{21\!\cdots\!60}{28\!\cdots\!77}a^{14}-\frac{45\!\cdots\!40}{28\!\cdots\!77}a^{13}-\frac{18\!\cdots\!90}{28\!\cdots\!77}a^{12}+\frac{80\!\cdots\!10}{28\!\cdots\!77}a^{11}+\frac{37\!\cdots\!00}{28\!\cdots\!77}a^{10}-\frac{64\!\cdots\!00}{28\!\cdots\!77}a^{9}-\frac{28\!\cdots\!70}{28\!\cdots\!77}a^{8}+\frac{25\!\cdots\!10}{28\!\cdots\!77}a^{7}+\frac{94\!\cdots\!10}{28\!\cdots\!77}a^{6}-\frac{50\!\cdots\!80}{28\!\cdots\!77}a^{5}-\frac{13\!\cdots\!10}{28\!\cdots\!77}a^{4}+\frac{47\!\cdots\!00}{28\!\cdots\!77}a^{3}+\frac{79\!\cdots\!00}{28\!\cdots\!77}a^{2}-\frac{15\!\cdots\!40}{28\!\cdots\!77}a-\frac{14\!\cdots\!09}{28\!\cdots\!77}$, $\frac{10\!\cdots\!57}{24\!\cdots\!11}a^{15}-\frac{73\!\cdots\!55}{96\!\cdots\!44}a^{14}-\frac{21\!\cdots\!37}{96\!\cdots\!44}a^{13}-\frac{14\!\cdots\!17}{96\!\cdots\!44}a^{12}+\frac{98\!\cdots\!31}{24\!\cdots\!11}a^{11}+\frac{11\!\cdots\!03}{48\!\cdots\!22}a^{10}-\frac{32\!\cdots\!43}{96\!\cdots\!44}a^{9}-\frac{91\!\cdots\!23}{96\!\cdots\!44}a^{8}+\frac{13\!\cdots\!17}{96\!\cdots\!44}a^{7}-\frac{31\!\cdots\!51}{48\!\cdots\!22}a^{6}-\frac{12\!\cdots\!35}{48\!\cdots\!22}a^{5}+\frac{34\!\cdots\!15}{96\!\cdots\!44}a^{4}+\frac{17\!\cdots\!71}{96\!\cdots\!44}a^{3}-\frac{35\!\cdots\!67}{96\!\cdots\!44}a^{2}-\frac{49\!\cdots\!32}{24\!\cdots\!11}a+\frac{42\!\cdots\!54}{24\!\cdots\!11}$, $\frac{19\!\cdots\!57}{24\!\cdots\!11}a^{15}+\frac{97\!\cdots\!39}{96\!\cdots\!44}a^{14}-\frac{39\!\cdots\!11}{96\!\cdots\!44}a^{13}-\frac{11\!\cdots\!27}{96\!\cdots\!44}a^{12}+\frac{17\!\cdots\!37}{24\!\cdots\!11}a^{11}+\frac{12\!\cdots\!85}{48\!\cdots\!22}a^{10}-\frac{56\!\cdots\!21}{96\!\cdots\!44}a^{9}-\frac{19\!\cdots\!89}{96\!\cdots\!44}a^{8}+\frac{22\!\cdots\!23}{96\!\cdots\!44}a^{7}+\frac{30\!\cdots\!25}{48\!\cdots\!22}a^{6}-\frac{22\!\cdots\!61}{48\!\cdots\!22}a^{5}-\frac{83\!\cdots\!15}{96\!\cdots\!44}a^{4}+\frac{39\!\cdots\!41}{96\!\cdots\!44}a^{3}+\frac{48\!\cdots\!63}{96\!\cdots\!44}a^{2}-\frac{29\!\cdots\!00}{24\!\cdots\!11}a-\frac{21\!\cdots\!57}{24\!\cdots\!11}$, $\frac{30\!\cdots\!12}{24\!\cdots\!11}a^{15}+\frac{32\!\cdots\!57}{96\!\cdots\!44}a^{14}-\frac{61\!\cdots\!89}{96\!\cdots\!44}a^{13}-\frac{27\!\cdots\!17}{96\!\cdots\!44}a^{12}+\frac{26\!\cdots\!72}{24\!\cdots\!11}a^{11}+\frac{26\!\cdots\!25}{48\!\cdots\!22}a^{10}-\frac{82\!\cdots\!51}{96\!\cdots\!44}a^{9}-\frac{40\!\cdots\!91}{96\!\cdots\!44}a^{8}+\frac{31\!\cdots\!69}{96\!\cdots\!44}a^{7}+\frac{65\!\cdots\!47}{48\!\cdots\!22}a^{6}-\frac{28\!\cdots\!11}{48\!\cdots\!22}a^{5}-\frac{17\!\cdots\!89}{96\!\cdots\!44}a^{4}+\frac{46\!\cdots\!63}{96\!\cdots\!44}a^{3}+\frac{75\!\cdots\!21}{96\!\cdots\!44}a^{2}-\frac{28\!\cdots\!84}{24\!\cdots\!11}a+\frac{16\!\cdots\!49}{24\!\cdots\!11}$, $\frac{31\!\cdots\!84}{24\!\cdots\!11}a^{15}+\frac{10\!\cdots\!73}{24\!\cdots\!11}a^{14}-\frac{15\!\cdots\!39}{24\!\cdots\!11}a^{13}-\frac{82\!\cdots\!56}{24\!\cdots\!11}a^{12}+\frac{26\!\cdots\!11}{24\!\cdots\!11}a^{11}+\frac{15\!\cdots\!62}{24\!\cdots\!11}a^{10}-\frac{19\!\cdots\!78}{24\!\cdots\!11}a^{9}-\frac{11\!\cdots\!98}{24\!\cdots\!11}a^{8}+\frac{70\!\cdots\!33}{24\!\cdots\!11}a^{7}+\frac{35\!\cdots\!52}{24\!\cdots\!11}a^{6}-\frac{12\!\cdots\!27}{24\!\cdots\!11}a^{5}-\frac{46\!\cdots\!42}{24\!\cdots\!11}a^{4}+\frac{10\!\cdots\!30}{24\!\cdots\!11}a^{3}+\frac{22\!\cdots\!46}{24\!\cdots\!11}a^{2}-\frac{32\!\cdots\!24}{24\!\cdots\!11}a-\frac{23\!\cdots\!17}{24\!\cdots\!11}$, $\frac{40\!\cdots\!47}{16\!\cdots\!04}a^{15}+\frac{41\!\cdots\!13}{80\!\cdots\!52}a^{14}-\frac{10\!\cdots\!57}{80\!\cdots\!52}a^{13}-\frac{77\!\cdots\!29}{16\!\cdots\!04}a^{12}+\frac{91\!\cdots\!11}{40\!\cdots\!26}a^{11}+\frac{15\!\cdots\!13}{16\!\cdots\!04}a^{10}-\frac{36\!\cdots\!94}{20\!\cdots\!13}a^{9}-\frac{60\!\cdots\!97}{80\!\cdots\!52}a^{8}+\frac{11\!\cdots\!11}{16\!\cdots\!04}a^{7}+\frac{20\!\cdots\!53}{80\!\cdots\!52}a^{6}-\frac{24\!\cdots\!81}{16\!\cdots\!04}a^{5}-\frac{14\!\cdots\!53}{40\!\cdots\!26}a^{4}+\frac{28\!\cdots\!77}{20\!\cdots\!13}a^{3}+\frac{35\!\cdots\!25}{16\!\cdots\!04}a^{2}-\frac{36\!\cdots\!63}{80\!\cdots\!52}a-\frac{14\!\cdots\!49}{40\!\cdots\!26}$, $\frac{62\!\cdots\!91}{32\!\cdots\!08}a^{15}+\frac{36\!\cdots\!19}{32\!\cdots\!08}a^{14}-\frac{31\!\cdots\!17}{32\!\cdots\!08}a^{13}-\frac{12\!\cdots\!11}{16\!\cdots\!04}a^{12}+\frac{26\!\cdots\!05}{16\!\cdots\!04}a^{11}+\frac{45\!\cdots\!19}{32\!\cdots\!08}a^{10}-\frac{39\!\cdots\!19}{32\!\cdots\!08}a^{9}-\frac{35\!\cdots\!99}{32\!\cdots\!08}a^{8}+\frac{35\!\cdots\!81}{80\!\cdots\!52}a^{7}+\frac{63\!\cdots\!79}{16\!\cdots\!04}a^{6}-\frac{28\!\cdots\!95}{32\!\cdots\!08}a^{5}-\frac{20\!\cdots\!49}{32\!\cdots\!08}a^{4}+\frac{33\!\cdots\!77}{32\!\cdots\!08}a^{3}+\frac{67\!\cdots\!79}{16\!\cdots\!04}a^{2}-\frac{44\!\cdots\!89}{80\!\cdots\!52}a-\frac{97\!\cdots\!47}{20\!\cdots\!13}$, $\frac{16\!\cdots\!35}{80\!\cdots\!52}a^{15}+\frac{40\!\cdots\!87}{16\!\cdots\!04}a^{14}-\frac{16\!\cdots\!19}{16\!\cdots\!04}a^{13}-\frac{49\!\cdots\!85}{16\!\cdots\!04}a^{12}+\frac{15\!\cdots\!37}{80\!\cdots\!52}a^{11}+\frac{26\!\cdots\!85}{40\!\cdots\!26}a^{10}-\frac{25\!\cdots\!65}{16\!\cdots\!04}a^{9}-\frac{82\!\cdots\!29}{16\!\cdots\!04}a^{8}+\frac{10\!\cdots\!77}{16\!\cdots\!04}a^{7}+\frac{69\!\cdots\!11}{40\!\cdots\!26}a^{6}-\frac{29\!\cdots\!41}{20\!\cdots\!13}a^{5}-\frac{40\!\cdots\!03}{16\!\cdots\!04}a^{4}+\frac{23\!\cdots\!53}{16\!\cdots\!04}a^{3}+\frac{26\!\cdots\!05}{16\!\cdots\!04}a^{2}-\frac{43\!\cdots\!57}{80\!\cdots\!52}a-\frac{15\!\cdots\!47}{40\!\cdots\!26}$, $\frac{23\!\cdots\!51}{32\!\cdots\!08}a^{15}+\frac{60\!\cdots\!29}{32\!\cdots\!08}a^{14}-\frac{11\!\cdots\!19}{32\!\cdots\!08}a^{13}-\frac{12\!\cdots\!73}{80\!\cdots\!52}a^{12}+\frac{10\!\cdots\!39}{16\!\cdots\!04}a^{11}+\frac{99\!\cdots\!95}{32\!\cdots\!08}a^{10}-\frac{16\!\cdots\!65}{32\!\cdots\!08}a^{9}-\frac{75\!\cdots\!37}{32\!\cdots\!08}a^{8}+\frac{32\!\cdots\!41}{16\!\cdots\!04}a^{7}+\frac{12\!\cdots\!19}{16\!\cdots\!04}a^{6}-\frac{12\!\cdots\!03}{32\!\cdots\!08}a^{5}-\frac{34\!\cdots\!79}{32\!\cdots\!08}a^{4}+\frac{11\!\cdots\!43}{32\!\cdots\!08}a^{3}+\frac{49\!\cdots\!69}{80\!\cdots\!52}a^{2}-\frac{49\!\cdots\!09}{40\!\cdots\!26}a-\frac{38\!\cdots\!77}{40\!\cdots\!26}$, $\frac{26\!\cdots\!87}{32\!\cdots\!08}a^{15}+\frac{48\!\cdots\!25}{32\!\cdots\!08}a^{14}-\frac{13\!\cdots\!27}{32\!\cdots\!08}a^{13}-\frac{60\!\cdots\!71}{40\!\cdots\!26}a^{12}+\frac{12\!\cdots\!83}{16\!\cdots\!04}a^{11}+\frac{99\!\cdots\!11}{32\!\cdots\!08}a^{10}-\frac{20\!\cdots\!73}{32\!\cdots\!08}a^{9}-\frac{77\!\cdots\!69}{32\!\cdots\!08}a^{8}+\frac{41\!\cdots\!55}{16\!\cdots\!04}a^{7}+\frac{13\!\cdots\!67}{16\!\cdots\!04}a^{6}-\frac{17\!\cdots\!47}{32\!\cdots\!08}a^{5}-\frac{39\!\cdots\!39}{32\!\cdots\!08}a^{4}+\frac{16\!\cdots\!35}{32\!\cdots\!08}a^{3}+\frac{29\!\cdots\!71}{40\!\cdots\!26}a^{2}-\frac{33\!\cdots\!17}{20\!\cdots\!13}a-\frac{49\!\cdots\!59}{40\!\cdots\!26}$, $\frac{79\!\cdots\!49}{32\!\cdots\!08}a^{15}+\frac{26\!\cdots\!97}{32\!\cdots\!08}a^{14}-\frac{40\!\cdots\!23}{32\!\cdots\!08}a^{13}-\frac{10\!\cdots\!25}{16\!\cdots\!04}a^{12}+\frac{34\!\cdots\!87}{16\!\cdots\!04}a^{11}+\frac{38\!\cdots\!29}{32\!\cdots\!08}a^{10}-\frac{51\!\cdots\!81}{32\!\cdots\!08}a^{9}-\frac{28\!\cdots\!89}{32\!\cdots\!08}a^{8}+\frac{47\!\cdots\!05}{80\!\cdots\!52}a^{7}+\frac{45\!\cdots\!57}{16\!\cdots\!04}a^{6}-\frac{34\!\cdots\!77}{32\!\cdots\!08}a^{5}-\frac{12\!\cdots\!59}{32\!\cdots\!08}a^{4}+\frac{30\!\cdots\!71}{32\!\cdots\!08}a^{3}+\frac{31\!\cdots\!69}{16\!\cdots\!04}a^{2}-\frac{24\!\cdots\!23}{80\!\cdots\!52}a-\frac{55\!\cdots\!00}{20\!\cdots\!13}$, $\frac{15\!\cdots\!37}{80\!\cdots\!52}a^{15}+\frac{32\!\cdots\!83}{16\!\cdots\!04}a^{14}-\frac{16\!\cdots\!15}{16\!\cdots\!04}a^{13}-\frac{44\!\cdots\!77}{16\!\cdots\!04}a^{12}+\frac{15\!\cdots\!47}{80\!\cdots\!52}a^{11}+\frac{12\!\cdots\!82}{20\!\cdots\!13}a^{10}-\frac{25\!\cdots\!49}{16\!\cdots\!04}a^{9}-\frac{77\!\cdots\!49}{16\!\cdots\!04}a^{8}+\frac{10\!\cdots\!69}{16\!\cdots\!04}a^{7}+\frac{33\!\cdots\!80}{20\!\cdots\!13}a^{6}-\frac{60\!\cdots\!13}{40\!\cdots\!26}a^{5}-\frac{40\!\cdots\!67}{16\!\cdots\!04}a^{4}+\frac{24\!\cdots\!29}{16\!\cdots\!04}a^{3}+\frac{26\!\cdots\!33}{16\!\cdots\!04}a^{2}-\frac{45\!\cdots\!83}{80\!\cdots\!52}a-\frac{15\!\cdots\!65}{40\!\cdots\!26}$, $\frac{75\!\cdots\!35}{16\!\cdots\!04}a^{15}+\frac{11\!\cdots\!01}{16\!\cdots\!04}a^{14}-\frac{38\!\cdots\!95}{16\!\cdots\!04}a^{13}-\frac{63\!\cdots\!71}{80\!\cdots\!52}a^{12}+\frac{34\!\cdots\!45}{80\!\cdots\!52}a^{11}+\frac{27\!\cdots\!67}{16\!\cdots\!04}a^{10}-\frac{54\!\cdots\!45}{16\!\cdots\!04}a^{9}-\frac{21\!\cdots\!05}{16\!\cdots\!04}a^{8}+\frac{26\!\cdots\!90}{20\!\cdots\!13}a^{7}+\frac{38\!\cdots\!45}{80\!\cdots\!52}a^{6}-\frac{41\!\cdots\!47}{16\!\cdots\!04}a^{5}-\frac{11\!\cdots\!51}{16\!\cdots\!04}a^{4}+\frac{36\!\cdots\!11}{16\!\cdots\!04}a^{3}+\frac{33\!\cdots\!65}{80\!\cdots\!52}a^{2}-\frac{29\!\cdots\!55}{40\!\cdots\!26}a-\frac{14\!\cdots\!09}{20\!\cdots\!13}$, $\frac{23\!\cdots\!59}{16\!\cdots\!04}a^{15}+\frac{22\!\cdots\!72}{20\!\cdots\!13}a^{14}-\frac{15\!\cdots\!31}{20\!\cdots\!13}a^{13}-\frac{31\!\cdots\!69}{16\!\cdots\!04}a^{12}+\frac{27\!\cdots\!25}{20\!\cdots\!13}a^{11}+\frac{71\!\cdots\!97}{16\!\cdots\!04}a^{10}-\frac{91\!\cdots\!91}{80\!\cdots\!52}a^{9}-\frac{71\!\cdots\!76}{20\!\cdots\!13}a^{8}+\frac{76\!\cdots\!95}{16\!\cdots\!04}a^{7}+\frac{97\!\cdots\!37}{80\!\cdots\!52}a^{6}-\frac{16\!\cdots\!41}{16\!\cdots\!04}a^{5}-\frac{14\!\cdots\!95}{80\!\cdots\!52}a^{4}+\frac{83\!\cdots\!85}{80\!\cdots\!52}a^{3}+\frac{18\!\cdots\!85}{16\!\cdots\!04}a^{2}-\frac{30\!\cdots\!53}{80\!\cdots\!52}a-\frac{10\!\cdots\!43}{40\!\cdots\!26}$, $\frac{15\!\cdots\!55}{32\!\cdots\!08}a^{15}+\frac{70\!\cdots\!81}{32\!\cdots\!08}a^{14}-\frac{17\!\cdots\!71}{32\!\cdots\!08}a^{13}-\frac{71\!\cdots\!53}{80\!\cdots\!52}a^{12}+\frac{21\!\cdots\!15}{16\!\cdots\!04}a^{11}+\frac{40\!\cdots\!03}{32\!\cdots\!08}a^{10}-\frac{48\!\cdots\!85}{32\!\cdots\!08}a^{9}-\frac{23\!\cdots\!61}{32\!\cdots\!08}a^{8}+\frac{94\!\cdots\!39}{11\!\cdots\!36}a^{7}+\frac{27\!\cdots\!35}{16\!\cdots\!04}a^{6}-\frac{69\!\cdots\!39}{32\!\cdots\!08}a^{5}-\frac{14\!\cdots\!39}{32\!\cdots\!08}a^{4}+\frac{80\!\cdots\!95}{32\!\cdots\!08}a^{3}-\frac{16\!\cdots\!81}{80\!\cdots\!52}a^{2}-\frac{19\!\cdots\!10}{20\!\cdots\!13}a+\frac{53\!\cdots\!09}{40\!\cdots\!26}$
| 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: | \( 90185580600800000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 90185580600800000 \cdot 2}{2\cdot\sqrt{462732306245995722656474121747020143758680081}}\cr\approx \mathstrut & 0.274759125524566 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328)
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 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328, 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 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328);
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 - 521*x^14 - 352*x^13 + 97262*x^12 + 111457*x^11 - 8578319*x^10 - 7737475*x^9 + 391096418*x^8 + 106474606*x^7 - 9220425289*x^6 + 3780858515*x^5 + 102878217925*x^4 - 87419452512*x^3 - 457661368284*x^2 + 433370758320*x + 437343265328);
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
$\OD_{32}$ (as 16T22):
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 $C_{16} : C_2$ |
| Character table for $C_{16} : C_2$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.4.57962561.1, 8.8.115844383968839978801.1 |
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 |
| 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 | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | $16$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | $16$ | $16$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(29\)
| 29.16.14.6 | $x^{16} - 14906 x^{8} - 7627029$ | $8$ | $2$ | $14$ | $C_{16} : C_2$ | $[\ ]_{8}^{4}$ |
|
\(41\)
| 41.16.15.2 | $x^{16} + 205$ | $16$ | $1$ | $15$ | $C_{16} : C_2$ | $[\ ]_{16}^{2}$ |