Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1)
gp: K = bnfinit(y^21 - 7*y^20 + 21*y^19 - 42*y^18 + 77*y^17 - 126*y^16 + 168*y^15 - 213*y^14 + 266*y^13 - 280*y^12 + 259*y^11 - 217*y^10 + 133*y^9 - 42*y^8 - 53*y^7 + 126*y^6 - 112*y^5 + 7*y^4 + 63*y^3 - 49*y^2 + 14*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1)
\( x^{21} - 7 x^{20} + 21 x^{19} - 42 x^{18} + 77 x^{17} - 126 x^{16} + 168 x^{15} - 213 x^{14} + 266 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-7174552902718171733819392\)
\(\medspace = -\,2^{18}\cdot 7^{23}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.26\) | 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: | $2^{6/7}7^{47/42}\approx 15.985663254345006$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{13}a^{19}+\frac{3}{13}a^{18}-\frac{2}{13}a^{17}+\frac{1}{13}a^{15}+\frac{1}{13}a^{14}-\frac{5}{13}a^{13}-\frac{4}{13}a^{12}-\frac{3}{13}a^{11}+\frac{6}{13}a^{10}-\frac{3}{13}a^{9}-\frac{6}{13}a^{8}-\frac{2}{13}a^{7}-\frac{4}{13}a^{6}+\frac{5}{13}a^{3}+\frac{5}{13}a^{2}+\frac{4}{13}a-\frac{1}{13}$, $\frac{1}{17\!\cdots\!31}a^{20}+\frac{223953102682691}{17\!\cdots\!31}a^{19}+\frac{49\!\cdots\!59}{17\!\cdots\!31}a^{18}+\frac{41\!\cdots\!67}{17\!\cdots\!31}a^{17}+\frac{69\!\cdots\!01}{17\!\cdots\!31}a^{16}-\frac{74\!\cdots\!27}{17\!\cdots\!31}a^{15}+\frac{79\!\cdots\!90}{17\!\cdots\!31}a^{14}-\frac{16\!\cdots\!46}{17\!\cdots\!31}a^{13}-\frac{79\!\cdots\!16}{17\!\cdots\!31}a^{12}+\frac{267024020850163}{13\!\cdots\!87}a^{11}+\frac{44\!\cdots\!94}{17\!\cdots\!31}a^{10}+\frac{78\!\cdots\!35}{17\!\cdots\!31}a^{9}+\frac{750035580866610}{17\!\cdots\!31}a^{8}-\frac{44\!\cdots\!67}{17\!\cdots\!31}a^{7}+\frac{68\!\cdots\!92}{17\!\cdots\!31}a^{6}+\frac{137295108916514}{13\!\cdots\!87}a^{5}-\frac{32\!\cdots\!01}{17\!\cdots\!31}a^{4}+\frac{25\!\cdots\!25}{17\!\cdots\!31}a^{3}+\frac{33\!\cdots\!96}{17\!\cdots\!31}a^{2}-\frac{70\!\cdots\!88}{17\!\cdots\!31}a+\frac{77\!\cdots\!49}{17\!\cdots\!31}$
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: |
$\frac{18\!\cdots\!78}{17\!\cdots\!31}a^{20}-\frac{11\!\cdots\!23}{17\!\cdots\!31}a^{19}+\frac{29\!\cdots\!93}{17\!\cdots\!31}a^{18}-\frac{54\!\cdots\!82}{17\!\cdots\!31}a^{17}+\frac{10\!\cdots\!23}{17\!\cdots\!31}a^{16}-\frac{15\!\cdots\!40}{17\!\cdots\!31}a^{15}+\frac{19\!\cdots\!01}{17\!\cdots\!31}a^{14}-\frac{25\!\cdots\!96}{17\!\cdots\!31}a^{13}+\frac{30\!\cdots\!86}{17\!\cdots\!31}a^{12}-\frac{29\!\cdots\!14}{17\!\cdots\!31}a^{11}+\frac{27\!\cdots\!73}{17\!\cdots\!31}a^{10}-\frac{21\!\cdots\!44}{17\!\cdots\!31}a^{9}+\frac{99\!\cdots\!71}{17\!\cdots\!31}a^{8}-\frac{18\!\cdots\!87}{17\!\cdots\!31}a^{7}-\frac{99\!\cdots\!21}{17\!\cdots\!31}a^{6}+\frac{11\!\cdots\!04}{13\!\cdots\!87}a^{5}-\frac{90\!\cdots\!76}{17\!\cdots\!31}a^{4}-\frac{43\!\cdots\!56}{17\!\cdots\!31}a^{3}+\frac{72\!\cdots\!07}{17\!\cdots\!31}a^{2}-\frac{35\!\cdots\!01}{17\!\cdots\!31}a+\frac{71\!\cdots\!52}{17\!\cdots\!31}$, $\frac{16\!\cdots\!32}{17\!\cdots\!31}a^{20}-\frac{82\!\cdots\!25}{13\!\cdots\!87}a^{19}+\frac{29\!\cdots\!95}{17\!\cdots\!31}a^{18}-\frac{54\!\cdots\!59}{17\!\cdots\!31}a^{17}+\frac{99\!\cdots\!80}{17\!\cdots\!31}a^{16}-\frac{15\!\cdots\!71}{17\!\cdots\!31}a^{15}+\frac{19\!\cdots\!52}{17\!\cdots\!31}a^{14}-\frac{25\!\cdots\!84}{17\!\cdots\!31}a^{13}+\frac{31\!\cdots\!96}{17\!\cdots\!31}a^{12}-\frac{30\!\cdots\!05}{17\!\cdots\!31}a^{11}+\frac{27\!\cdots\!92}{17\!\cdots\!31}a^{10}-\frac{22\!\cdots\!35}{17\!\cdots\!31}a^{9}+\frac{11\!\cdots\!76}{17\!\cdots\!31}a^{8}-\frac{17\!\cdots\!97}{17\!\cdots\!31}a^{7}-\frac{96\!\cdots\!89}{17\!\cdots\!31}a^{6}+\frac{12\!\cdots\!41}{13\!\cdots\!87}a^{5}-\frac{10\!\cdots\!72}{17\!\cdots\!31}a^{4}-\frac{34\!\cdots\!13}{17\!\cdots\!31}a^{3}+\frac{80\!\cdots\!59}{17\!\cdots\!31}a^{2}-\frac{33\!\cdots\!74}{13\!\cdots\!87}a+\frac{51\!\cdots\!85}{17\!\cdots\!31}$, $\frac{421730305802822}{13\!\cdots\!87}a^{20}-\frac{24\!\cdots\!98}{13\!\cdots\!87}a^{19}+\frac{58\!\cdots\!67}{13\!\cdots\!87}a^{18}-\frac{96\!\cdots\!25}{13\!\cdots\!87}a^{17}+\frac{17\!\cdots\!89}{13\!\cdots\!87}a^{16}-\frac{26\!\cdots\!19}{13\!\cdots\!87}a^{15}+\frac{28\!\cdots\!36}{13\!\cdots\!87}a^{14}-\frac{38\!\cdots\!34}{13\!\cdots\!87}a^{13}+\frac{47\!\cdots\!53}{13\!\cdots\!87}a^{12}-\frac{36\!\cdots\!28}{13\!\cdots\!87}a^{11}+\frac{33\!\cdots\!44}{13\!\cdots\!87}a^{10}-\frac{24\!\cdots\!44}{13\!\cdots\!87}a^{9}+\frac{13\!\cdots\!13}{13\!\cdots\!87}a^{8}+\frac{47\!\cdots\!38}{13\!\cdots\!87}a^{7}-\frac{22\!\cdots\!56}{13\!\cdots\!87}a^{6}+\frac{26\!\cdots\!18}{13\!\cdots\!87}a^{5}-\frac{31\!\cdots\!91}{13\!\cdots\!87}a^{4}-\frac{19\!\cdots\!41}{13\!\cdots\!87}a^{3}+\frac{12\!\cdots\!16}{13\!\cdots\!87}a^{2}+\frac{10\!\cdots\!36}{13\!\cdots\!87}a-\frac{22\!\cdots\!64}{13\!\cdots\!87}$, $\frac{59\!\cdots\!12}{17\!\cdots\!31}a^{20}-\frac{37\!\cdots\!66}{17\!\cdots\!31}a^{19}+\frac{97\!\cdots\!89}{17\!\cdots\!31}a^{18}-\frac{13\!\cdots\!78}{13\!\cdots\!87}a^{17}+\frac{33\!\cdots\!67}{17\!\cdots\!31}a^{16}-\frac{51\!\cdots\!55}{17\!\cdots\!31}a^{15}+\frac{64\!\cdots\!58}{17\!\cdots\!31}a^{14}-\frac{84\!\cdots\!27}{17\!\cdots\!31}a^{13}+\frac{10\!\cdots\!75}{17\!\cdots\!31}a^{12}-\frac{98\!\cdots\!21}{17\!\cdots\!31}a^{11}+\frac{92\!\cdots\!49}{17\!\cdots\!31}a^{10}-\frac{71\!\cdots\!48}{17\!\cdots\!31}a^{9}+\frac{35\!\cdots\!41}{17\!\cdots\!31}a^{8}-\frac{85\!\cdots\!81}{17\!\cdots\!31}a^{7}-\frac{25\!\cdots\!67}{13\!\cdots\!87}a^{6}+\frac{37\!\cdots\!60}{13\!\cdots\!87}a^{5}-\frac{30\!\cdots\!02}{17\!\cdots\!31}a^{4}-\frac{13\!\cdots\!21}{17\!\cdots\!31}a^{3}+\frac{25\!\cdots\!39}{17\!\cdots\!31}a^{2}-\frac{15\!\cdots\!18}{17\!\cdots\!31}a+\frac{17\!\cdots\!10}{13\!\cdots\!87}$, $\frac{35\!\cdots\!13}{17\!\cdots\!31}a^{20}-\frac{16\!\cdots\!29}{17\!\cdots\!31}a^{19}+\frac{25\!\cdots\!19}{17\!\cdots\!31}a^{18}-\frac{32\!\cdots\!85}{17\!\cdots\!31}a^{17}+\frac{74\!\cdots\!79}{17\!\cdots\!31}a^{16}-\frac{77\!\cdots\!20}{17\!\cdots\!31}a^{15}+\frac{43\!\cdots\!33}{17\!\cdots\!31}a^{14}-\frac{12\!\cdots\!91}{17\!\cdots\!31}a^{13}+\frac{94\!\cdots\!57}{17\!\cdots\!31}a^{12}+\frac{28\!\cdots\!64}{17\!\cdots\!31}a^{11}+\frac{55\!\cdots\!00}{17\!\cdots\!31}a^{10}+\frac{37\!\cdots\!70}{17\!\cdots\!31}a^{9}-\frac{13\!\cdots\!64}{17\!\cdots\!31}a^{8}+\frac{97\!\cdots\!12}{17\!\cdots\!31}a^{7}-\frac{16\!\cdots\!73}{17\!\cdots\!31}a^{6}+\frac{16\!\cdots\!71}{13\!\cdots\!87}a^{5}+\frac{12\!\cdots\!97}{17\!\cdots\!31}a^{4}-\frac{12\!\cdots\!85}{17\!\cdots\!31}a^{3}-\frac{99\!\cdots\!52}{17\!\cdots\!31}a^{2}+\frac{30\!\cdots\!69}{17\!\cdots\!31}a+\frac{11\!\cdots\!33}{17\!\cdots\!31}$, $\frac{591356710811595}{17\!\cdots\!31}a^{20}-\frac{64\!\cdots\!09}{17\!\cdots\!31}a^{19}+\frac{19\!\cdots\!65}{13\!\cdots\!87}a^{18}-\frac{54\!\cdots\!28}{17\!\cdots\!31}a^{17}+\frac{96\!\cdots\!09}{17\!\cdots\!31}a^{16}-\frac{17\!\cdots\!34}{17\!\cdots\!31}a^{15}+\frac{18\!\cdots\!19}{13\!\cdots\!87}a^{14}-\frac{28\!\cdots\!80}{17\!\cdots\!31}a^{13}+\frac{38\!\cdots\!88}{17\!\cdots\!31}a^{12}-\frac{43\!\cdots\!06}{17\!\cdots\!31}a^{11}+\frac{37\!\cdots\!26}{17\!\cdots\!31}a^{10}-\frac{35\!\cdots\!16}{17\!\cdots\!31}a^{9}+\frac{24\!\cdots\!80}{17\!\cdots\!31}a^{8}-\frac{74\!\cdots\!54}{17\!\cdots\!31}a^{7}-\frac{62\!\cdots\!86}{17\!\cdots\!31}a^{6}+\frac{13\!\cdots\!89}{13\!\cdots\!87}a^{5}-\frac{18\!\cdots\!99}{17\!\cdots\!31}a^{4}+\frac{28\!\cdots\!23}{17\!\cdots\!31}a^{3}+\frac{11\!\cdots\!75}{17\!\cdots\!31}a^{2}-\frac{62\!\cdots\!01}{17\!\cdots\!31}a+\frac{16\!\cdots\!81}{17\!\cdots\!31}$, $\frac{87\!\cdots\!16}{17\!\cdots\!31}a^{20}-\frac{57\!\cdots\!48}{17\!\cdots\!31}a^{19}+\frac{15\!\cdots\!50}{17\!\cdots\!31}a^{18}-\frac{29\!\cdots\!12}{17\!\cdots\!31}a^{17}+\frac{53\!\cdots\!98}{17\!\cdots\!31}a^{16}-\frac{85\!\cdots\!40}{17\!\cdots\!31}a^{15}+\frac{10\!\cdots\!33}{17\!\cdots\!31}a^{14}-\frac{13\!\cdots\!26}{17\!\cdots\!31}a^{13}+\frac{16\!\cdots\!52}{17\!\cdots\!31}a^{12}-\frac{16\!\cdots\!88}{17\!\cdots\!31}a^{11}+\frac{14\!\cdots\!54}{17\!\cdots\!31}a^{10}-\frac{92\!\cdots\!29}{13\!\cdots\!87}a^{9}+\frac{58\!\cdots\!08}{17\!\cdots\!31}a^{8}-\frac{68\!\cdots\!89}{13\!\cdots\!87}a^{7}-\frac{50\!\cdots\!42}{17\!\cdots\!31}a^{6}+\frac{68\!\cdots\!94}{13\!\cdots\!87}a^{5}-\frac{55\!\cdots\!55}{17\!\cdots\!31}a^{4}-\frac{20\!\cdots\!95}{17\!\cdots\!31}a^{3}+\frac{45\!\cdots\!28}{17\!\cdots\!31}a^{2}-\frac{19\!\cdots\!84}{17\!\cdots\!31}a+\frac{32\!\cdots\!58}{17\!\cdots\!31}$, $\frac{25\!\cdots\!59}{17\!\cdots\!31}a^{20}-\frac{11\!\cdots\!19}{17\!\cdots\!31}a^{19}+\frac{13\!\cdots\!12}{13\!\cdots\!87}a^{18}-\frac{19\!\cdots\!16}{17\!\cdots\!31}a^{17}+\frac{44\!\cdots\!70}{17\!\cdots\!31}a^{16}-\frac{42\!\cdots\!83}{17\!\cdots\!31}a^{15}+\frac{370070085502278}{13\!\cdots\!87}a^{14}-\frac{58\!\cdots\!77}{17\!\cdots\!31}a^{13}+\frac{38\!\cdots\!76}{17\!\cdots\!31}a^{12}+\frac{70\!\cdots\!26}{17\!\cdots\!31}a^{11}-\frac{39\!\cdots\!30}{17\!\cdots\!31}a^{10}+\frac{47\!\cdots\!20}{17\!\cdots\!31}a^{9}-\frac{12\!\cdots\!15}{17\!\cdots\!31}a^{8}+\frac{19\!\cdots\!05}{17\!\cdots\!31}a^{7}-\frac{93\!\cdots\!24}{17\!\cdots\!31}a^{6}-\frac{185894154227660}{13\!\cdots\!87}a^{5}+\frac{15\!\cdots\!60}{17\!\cdots\!31}a^{4}-\frac{10\!\cdots\!83}{17\!\cdots\!31}a^{3}-\frac{63\!\cdots\!21}{17\!\cdots\!31}a^{2}+\frac{56\!\cdots\!86}{17\!\cdots\!31}a+\frac{14\!\cdots\!58}{17\!\cdots\!31}$, $\frac{469295449564926}{17\!\cdots\!31}a^{20}-\frac{48\!\cdots\!92}{17\!\cdots\!31}a^{19}+\frac{21\!\cdots\!18}{17\!\cdots\!31}a^{18}-\frac{41\!\cdots\!53}{13\!\cdots\!87}a^{17}+\frac{10\!\cdots\!10}{17\!\cdots\!31}a^{16}-\frac{17\!\cdots\!08}{17\!\cdots\!31}a^{15}+\frac{26\!\cdots\!90}{17\!\cdots\!31}a^{14}-\frac{33\!\cdots\!85}{17\!\cdots\!31}a^{13}+\frac{41\!\cdots\!86}{17\!\cdots\!31}a^{12}-\frac{50\!\cdots\!57}{17\!\cdots\!31}a^{11}+\frac{49\!\cdots\!77}{17\!\cdots\!31}a^{10}-\frac{40\!\cdots\!76}{17\!\cdots\!31}a^{9}+\frac{33\!\cdots\!72}{17\!\cdots\!31}a^{8}-\frac{14\!\cdots\!75}{17\!\cdots\!31}a^{7}-\frac{23\!\cdots\!11}{13\!\cdots\!87}a^{6}+\frac{11\!\cdots\!42}{13\!\cdots\!87}a^{5}-\frac{26\!\cdots\!19}{17\!\cdots\!31}a^{4}+\frac{12\!\cdots\!12}{17\!\cdots\!31}a^{3}+\frac{90\!\cdots\!31}{17\!\cdots\!31}a^{2}-\frac{15\!\cdots\!55}{17\!\cdots\!31}a+\frac{27\!\cdots\!87}{13\!\cdots\!87}$, $\frac{74\!\cdots\!03}{17\!\cdots\!31}a^{20}-\frac{46\!\cdots\!00}{17\!\cdots\!31}a^{19}+\frac{11\!\cdots\!50}{17\!\cdots\!31}a^{18}-\frac{21\!\cdots\!33}{17\!\cdots\!31}a^{17}+\frac{39\!\cdots\!01}{17\!\cdots\!31}a^{16}-\frac{60\!\cdots\!87}{17\!\cdots\!31}a^{15}+\frac{73\!\cdots\!30}{17\!\cdots\!31}a^{14}-\frac{94\!\cdots\!96}{17\!\cdots\!31}a^{13}+\frac{11\!\cdots\!99}{17\!\cdots\!31}a^{12}-\frac{10\!\cdots\!71}{17\!\cdots\!31}a^{11}+\frac{93\!\cdots\!10}{17\!\cdots\!31}a^{10}-\frac{70\!\cdots\!62}{17\!\cdots\!31}a^{9}+\frac{20\!\cdots\!04}{13\!\cdots\!87}a^{8}+\frac{49\!\cdots\!43}{17\!\cdots\!31}a^{7}-\frac{46\!\cdots\!42}{17\!\cdots\!31}a^{6}+\frac{48\!\cdots\!26}{13\!\cdots\!87}a^{5}-\frac{35\!\cdots\!23}{17\!\cdots\!31}a^{4}-\frac{26\!\cdots\!12}{17\!\cdots\!31}a^{3}+\frac{30\!\cdots\!85}{17\!\cdots\!31}a^{2}-\frac{11\!\cdots\!32}{17\!\cdots\!31}a-\frac{16\!\cdots\!81}{17\!\cdots\!31}$, $\frac{11\!\cdots\!37}{17\!\cdots\!31}a^{20}-\frac{77\!\cdots\!43}{17\!\cdots\!31}a^{19}+\frac{20\!\cdots\!38}{17\!\cdots\!31}a^{18}-\frac{37\!\cdots\!43}{17\!\cdots\!31}a^{17}+\frac{68\!\cdots\!98}{17\!\cdots\!31}a^{16}-\frac{10\!\cdots\!63}{17\!\cdots\!31}a^{15}+\frac{13\!\cdots\!49}{17\!\cdots\!31}a^{14}-\frac{16\!\cdots\!08}{17\!\cdots\!31}a^{13}+\frac{20\!\cdots\!90}{17\!\cdots\!31}a^{12}-\frac{19\!\cdots\!34}{17\!\cdots\!31}a^{11}+\frac{17\!\cdots\!05}{17\!\cdots\!31}a^{10}-\frac{14\!\cdots\!26}{17\!\cdots\!31}a^{9}+\frac{58\!\cdots\!01}{17\!\cdots\!31}a^{8}+\frac{73\!\cdots\!98}{17\!\cdots\!31}a^{7}-\frac{67\!\cdots\!60}{17\!\cdots\!31}a^{6}+\frac{88\!\cdots\!28}{13\!\cdots\!87}a^{5}-\frac{59\!\cdots\!67}{17\!\cdots\!31}a^{4}-\frac{42\!\cdots\!02}{17\!\cdots\!31}a^{3}+\frac{57\!\cdots\!99}{17\!\cdots\!31}a^{2}-\frac{19\!\cdots\!98}{17\!\cdots\!31}a+\frac{816914385126902}{17\!\cdots\!31}$
| 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: | \( 10096.3106241 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 10096.3106241 \cdot 1}{2\cdot\sqrt{7174552902718171733819392}}\cr\approx \mathstrut & 0.230114554958 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1)
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^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1, 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^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1);
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^21 - 7*x^20 + 21*x^19 - 42*x^18 + 77*x^17 - 126*x^16 + 168*x^15 - 213*x^14 + 266*x^13 - 280*x^12 + 259*x^11 - 217*x^10 + 133*x^9 - 42*x^8 - 53*x^7 + 126*x^6 - 112*x^5 + 7*x^4 + 63*x^3 - 49*x^2 + 14*x - 1);
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 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.1.52706752.1 |
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: | data not computed |
| Degree 7 sibling: | 7.1.52706752.1 |
| Degree 14 sibling: | 14.0.19446011944726528.1 |
| Minimal sibling: | 7.1.52706752.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.18.1 | $x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 62 x^{15} + 111 x^{14} + 98 x^{13} - 189 x^{12} - 189 x^{11} + 259 x^{10} + 1496 x^{9} + 2586 x^{8} + 925 x^{7} + 798 x^{6} - 1092 x^{5} + 1029 x^{4} - 174 x^{3} - 53 x^{2} - 313 x + 131$ | $7$ | $3$ | $18$ | 21T2 | $[\ ]_{7}^{3}$ |
|
\(7\)
| Deg $21$ | $21$ | $1$ | $23$ |