Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 13*y^18 - 17*y^17 + 94*y^16 - 151*y^15 + 406*y^14 - 633*y^13 + 3653*y^12 - 772*y^11 + 13426*y^10 + 5719*y^9 + 24764*y^8 + 15388*y^7 + 21892*y^6 + 15269*y^5 + 9379*y^4 + 3157*y^3 + 732*y^2 - 45*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*x + 1)
\( x^{20} - 5 x^{19} + 13 x^{18} - 17 x^{17} + 94 x^{16} - 151 x^{15} + 406 x^{14} - 633 x^{13} + 3653 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2131588214553606414985382080078125\)
\(\medspace = 3^{8}\cdot 5^{15}\cdot 239^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.39\) | 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: | $3^{1/2}5^{3/4}239^{1/2}\approx 89.53381316204927$ | ||
| Ramified primes: |
\(3\), \(5\), \(239\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19}a^{16}-\frac{3}{19}a^{15}+\frac{2}{19}a^{14}-\frac{6}{19}a^{13}-\frac{6}{19}a^{12}+\frac{5}{19}a^{11}+\frac{5}{19}a^{10}-\frac{7}{19}a^{9}+\frac{6}{19}a^{8}-\frac{2}{19}a^{7}-\frac{1}{19}a^{6}+\frac{9}{19}a^{5}+\frac{4}{19}a^{4}+\frac{2}{19}a^{3}-\frac{1}{19}a^{2}-\frac{8}{19}$, $\frac{1}{19}a^{17}-\frac{7}{19}a^{15}-\frac{5}{19}a^{13}+\frac{6}{19}a^{12}+\frac{1}{19}a^{11}+\frac{8}{19}a^{10}+\frac{4}{19}a^{9}-\frac{3}{19}a^{8}-\frac{7}{19}a^{7}+\frac{6}{19}a^{6}-\frac{7}{19}a^{5}-\frac{5}{19}a^{4}+\frac{5}{19}a^{3}-\frac{3}{19}a^{2}-\frac{8}{19}a-\frac{5}{19}$, $\frac{1}{19}a^{18}-\frac{2}{19}a^{15}+\frac{9}{19}a^{14}+\frac{2}{19}a^{13}-\frac{3}{19}a^{12}+\frac{5}{19}a^{11}+\frac{1}{19}a^{10}+\frac{5}{19}a^{9}-\frac{3}{19}a^{8}-\frac{8}{19}a^{7}+\frac{5}{19}a^{6}+\frac{1}{19}a^{5}-\frac{5}{19}a^{4}-\frac{8}{19}a^{3}+\frac{4}{19}a^{2}-\frac{5}{19}a+\frac{1}{19}$, $\frac{1}{15\!\cdots\!79}a^{19}-\frac{32\!\cdots\!16}{15\!\cdots\!79}a^{18}+\frac{22\!\cdots\!40}{15\!\cdots\!79}a^{17}+\frac{98\!\cdots\!67}{81\!\cdots\!41}a^{16}-\frac{62\!\cdots\!36}{15\!\cdots\!79}a^{15}-\frac{12\!\cdots\!24}{15\!\cdots\!79}a^{14}-\frac{14\!\cdots\!14}{15\!\cdots\!79}a^{13}+\frac{33\!\cdots\!07}{15\!\cdots\!79}a^{12}-\frac{46\!\cdots\!01}{15\!\cdots\!79}a^{11}-\frac{68\!\cdots\!07}{15\!\cdots\!79}a^{10}-\frac{11\!\cdots\!37}{15\!\cdots\!79}a^{9}-\frac{30\!\cdots\!46}{15\!\cdots\!79}a^{8}-\frac{48\!\cdots\!58}{15\!\cdots\!79}a^{7}+\frac{78\!\cdots\!00}{15\!\cdots\!79}a^{6}-\frac{99\!\cdots\!62}{15\!\cdots\!79}a^{5}-\frac{88\!\cdots\!79}{15\!\cdots\!79}a^{4}+\frac{22\!\cdots\!57}{15\!\cdots\!79}a^{3}+\frac{46\!\cdots\!05}{15\!\cdots\!79}a^{2}-\frac{33\!\cdots\!14}{15\!\cdots\!79}a+\frac{27\!\cdots\!55}{15\!\cdots\!79}$
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
$C_{2}\times C_{58}$, which has order $116$ (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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{462818769889456179651310650579408872}{15512369205008039820354470756707694779} a^{19} - \frac{2326769500830031245644024226778428991}{15512369205008039820354470756707694779} a^{18} + \frac{6085575599032599985406446477638831728}{15512369205008039820354470756707694779} a^{17} - \frac{8053672957771882543603366298796728394}{15512369205008039820354470756707694779} a^{16} + \frac{43756386035050661857706352506913597896}{15512369205008039820354470756707694779} a^{15} - \frac{71071874341284803146102106521707118592}{15512369205008039820354470756707694779} a^{14} + \frac{190193241180812691445636467388065236429}{15512369205008039820354470756707694779} a^{13} - \frac{298259079478417551045537111755855720027}{15512369205008039820354470756707694779} a^{12} + \frac{1699680945567312614727430986045645406878}{15512369205008039820354470756707694779} a^{11} - \frac{403976620459062232554254939005395478672}{15512369205008039820354470756707694779} a^{10} + \frac{6238463018903085141089317736446009784682}{15512369205008039820354470756707694779} a^{9} + \frac{2499055278537939215613511471752905341233}{15512369205008039820354470756707694779} a^{8} + \frac{11448874579390192561509181938382218220782}{15512369205008039820354470756707694779} a^{7} + \frac{6933857183413525225748035473749834037042}{15512369205008039820354470756707694779} a^{6} + \frac{10074706929359956358303595959856066709952}{15512369205008039820354470756707694779} a^{5} + \frac{7034096685754392810557671000997154146244}{15512369205008039820354470756707694779} a^{4} + \frac{4294624621290086941447127430527707227167}{15512369205008039820354470756707694779} a^{3} + \frac{1532784301452226441723579802687424978039}{15512369205008039820354470756707694779} a^{2} + \frac{380233125383883954419361157061624425244}{15512369205008039820354470756707694779} a + \frac{1934096680512204750531796611047313906}{15512369205008039820354470756707694779} \)
(order $10$)
| 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: |
$\frac{25\!\cdots\!72}{15\!\cdots\!79}a^{19}-\frac{12\!\cdots\!44}{15\!\cdots\!79}a^{18}+\frac{16\!\cdots\!82}{81\!\cdots\!41}a^{17}-\frac{38\!\cdots\!06}{15\!\cdots\!79}a^{16}+\frac{23\!\cdots\!04}{15\!\cdots\!79}a^{15}-\frac{35\!\cdots\!06}{15\!\cdots\!79}a^{14}+\frac{98\!\cdots\!92}{15\!\cdots\!79}a^{13}-\frac{14\!\cdots\!14}{15\!\cdots\!79}a^{12}+\frac{91\!\cdots\!64}{15\!\cdots\!79}a^{11}-\frac{64\!\cdots\!62}{15\!\cdots\!79}a^{10}+\frac{34\!\cdots\!98}{15\!\cdots\!79}a^{9}+\frac{19\!\cdots\!22}{15\!\cdots\!79}a^{8}+\frac{65\!\cdots\!60}{15\!\cdots\!79}a^{7}+\frac{47\!\cdots\!87}{15\!\cdots\!79}a^{6}+\frac{61\!\cdots\!98}{15\!\cdots\!79}a^{5}+\frac{45\!\cdots\!72}{15\!\cdots\!79}a^{4}+\frac{28\!\cdots\!94}{15\!\cdots\!79}a^{3}+\frac{10\!\cdots\!12}{15\!\cdots\!79}a^{2}+\frac{22\!\cdots\!11}{15\!\cdots\!79}a+\frac{13\!\cdots\!14}{15\!\cdots\!79}$, $\frac{43\!\cdots\!03}{15\!\cdots\!79}a^{19}-\frac{21\!\cdots\!66}{15\!\cdots\!79}a^{18}+\frac{55\!\cdots\!11}{15\!\cdots\!79}a^{17}-\frac{71\!\cdots\!06}{15\!\cdots\!79}a^{16}+\frac{40\!\cdots\!50}{15\!\cdots\!79}a^{15}-\frac{63\!\cdots\!86}{15\!\cdots\!79}a^{14}+\frac{17\!\cdots\!29}{15\!\cdots\!79}a^{13}-\frac{26\!\cdots\!98}{15\!\cdots\!79}a^{12}+\frac{15\!\cdots\!92}{15\!\cdots\!79}a^{11}-\frac{27\!\cdots\!69}{15\!\cdots\!79}a^{10}+\frac{58\!\cdots\!65}{15\!\cdots\!79}a^{9}+\frac{27\!\cdots\!44}{15\!\cdots\!79}a^{8}+\frac{10\!\cdots\!97}{15\!\cdots\!79}a^{7}+\frac{71\!\cdots\!44}{15\!\cdots\!79}a^{6}+\frac{97\!\cdots\!84}{15\!\cdots\!79}a^{5}+\frac{70\!\cdots\!95}{15\!\cdots\!79}a^{4}+\frac{43\!\cdots\!12}{15\!\cdots\!79}a^{3}+\frac{15\!\cdots\!50}{15\!\cdots\!79}a^{2}+\frac{41\!\cdots\!48}{15\!\cdots\!79}a+\frac{19\!\cdots\!01}{15\!\cdots\!79}$, $\frac{33\!\cdots\!88}{15\!\cdots\!79}a^{19}-\frac{16\!\cdots\!64}{15\!\cdots\!79}a^{18}+\frac{41\!\cdots\!60}{15\!\cdots\!79}a^{17}-\frac{53\!\cdots\!94}{15\!\cdots\!79}a^{16}+\frac{30\!\cdots\!84}{15\!\cdots\!79}a^{15}-\frac{47\!\cdots\!57}{15\!\cdots\!79}a^{14}+\frac{13\!\cdots\!94}{15\!\cdots\!79}a^{13}-\frac{20\!\cdots\!75}{15\!\cdots\!79}a^{12}+\frac{11\!\cdots\!64}{15\!\cdots\!79}a^{11}-\frac{17\!\cdots\!86}{15\!\cdots\!79}a^{10}+\frac{44\!\cdots\!13}{15\!\cdots\!79}a^{9}+\frac{21\!\cdots\!79}{15\!\cdots\!79}a^{8}+\frac{84\!\cdots\!64}{15\!\cdots\!79}a^{7}+\frac{55\!\cdots\!34}{15\!\cdots\!79}a^{6}+\frac{77\!\cdots\!04}{15\!\cdots\!79}a^{5}+\frac{54\!\cdots\!42}{15\!\cdots\!79}a^{4}+\frac{35\!\cdots\!12}{15\!\cdots\!79}a^{3}+\frac{12\!\cdots\!24}{15\!\cdots\!79}a^{2}+\frac{33\!\cdots\!26}{15\!\cdots\!79}a-\frac{66\!\cdots\!94}{15\!\cdots\!79}$, $\frac{40\!\cdots\!09}{15\!\cdots\!79}a^{19}-\frac{21\!\cdots\!88}{15\!\cdots\!79}a^{18}+\frac{60\!\cdots\!50}{15\!\cdots\!79}a^{17}-\frac{89\!\cdots\!60}{15\!\cdots\!79}a^{16}+\frac{41\!\cdots\!72}{15\!\cdots\!79}a^{15}-\frac{76\!\cdots\!03}{15\!\cdots\!79}a^{14}+\frac{19\!\cdots\!66}{15\!\cdots\!79}a^{13}-\frac{32\!\cdots\!66}{15\!\cdots\!79}a^{12}+\frac{15\!\cdots\!04}{15\!\cdots\!79}a^{11}-\frac{87\!\cdots\!86}{15\!\cdots\!79}a^{10}+\frac{56\!\cdots\!53}{15\!\cdots\!79}a^{9}+\frac{29\!\cdots\!36}{15\!\cdots\!79}a^{8}+\frac{95\!\cdots\!24}{15\!\cdots\!79}a^{7}+\frac{26\!\cdots\!40}{15\!\cdots\!79}a^{6}+\frac{72\!\cdots\!00}{15\!\cdots\!79}a^{5}+\frac{33\!\cdots\!75}{15\!\cdots\!79}a^{4}+\frac{20\!\cdots\!96}{15\!\cdots\!79}a^{3}+\frac{31\!\cdots\!06}{15\!\cdots\!79}a^{2}+\frac{13\!\cdots\!00}{15\!\cdots\!79}a-\frac{78\!\cdots\!32}{15\!\cdots\!79}$, $\frac{38\!\cdots\!35}{81\!\cdots\!41}a^{19}-\frac{47\!\cdots\!51}{15\!\cdots\!79}a^{18}+\frac{15\!\cdots\!87}{15\!\cdots\!79}a^{17}-\frac{27\!\cdots\!54}{15\!\cdots\!79}a^{16}+\frac{89\!\cdots\!71}{15\!\cdots\!79}a^{15}-\frac{21\!\cdots\!54}{15\!\cdots\!79}a^{14}+\frac{47\!\cdots\!77}{15\!\cdots\!79}a^{13}-\frac{92\!\cdots\!87}{15\!\cdots\!79}a^{12}+\frac{34\!\cdots\!24}{15\!\cdots\!79}a^{11}-\frac{46\!\cdots\!97}{15\!\cdots\!79}a^{10}+\frac{11\!\cdots\!17}{15\!\cdots\!79}a^{9}-\frac{10\!\cdots\!80}{15\!\cdots\!79}a^{8}+\frac{13\!\cdots\!22}{15\!\cdots\!79}a^{7}-\frac{13\!\cdots\!24}{15\!\cdots\!79}a^{6}+\frac{21\!\cdots\!91}{15\!\cdots\!79}a^{5}-\frac{74\!\cdots\!49}{15\!\cdots\!79}a^{4}-\frac{74\!\cdots\!02}{15\!\cdots\!79}a^{3}-\frac{34\!\cdots\!11}{15\!\cdots\!79}a^{2}-\frac{17\!\cdots\!16}{15\!\cdots\!79}a-\frac{22\!\cdots\!97}{81\!\cdots\!41}$, $\frac{91\!\cdots\!48}{15\!\cdots\!79}a^{19}-\frac{39\!\cdots\!44}{15\!\cdots\!79}a^{18}+\frac{87\!\cdots\!22}{15\!\cdots\!79}a^{17}-\frac{69\!\cdots\!93}{15\!\cdots\!79}a^{16}+\frac{74\!\cdots\!45}{15\!\cdots\!79}a^{15}-\frac{79\!\cdots\!82}{15\!\cdots\!79}a^{14}+\frac{26\!\cdots\!94}{15\!\cdots\!79}a^{13}-\frac{31\!\cdots\!42}{15\!\cdots\!79}a^{12}+\frac{29\!\cdots\!23}{15\!\cdots\!79}a^{11}+\frac{15\!\cdots\!72}{15\!\cdots\!79}a^{10}+\frac{11\!\cdots\!32}{15\!\cdots\!79}a^{9}+\frac{13\!\cdots\!22}{15\!\cdots\!79}a^{8}+\frac{25\!\cdots\!58}{15\!\cdots\!79}a^{7}+\frac{15\!\cdots\!79}{81\!\cdots\!41}a^{6}+\frac{28\!\cdots\!15}{15\!\cdots\!79}a^{5}+\frac{25\!\cdots\!42}{15\!\cdots\!79}a^{4}+\frac{17\!\cdots\!57}{15\!\cdots\!79}a^{3}+\frac{69\!\cdots\!54}{15\!\cdots\!79}a^{2}+\frac{22\!\cdots\!48}{15\!\cdots\!79}a+\frac{86\!\cdots\!10}{15\!\cdots\!79}$, $\frac{96\!\cdots\!61}{15\!\cdots\!79}a^{19}-\frac{51\!\cdots\!09}{15\!\cdots\!79}a^{18}+\frac{14\!\cdots\!07}{15\!\cdots\!79}a^{17}-\frac{20\!\cdots\!46}{15\!\cdots\!79}a^{16}+\frac{96\!\cdots\!88}{15\!\cdots\!79}a^{15}-\frac{17\!\cdots\!18}{15\!\cdots\!79}a^{14}+\frac{44\!\cdots\!01}{15\!\cdots\!79}a^{13}-\frac{75\!\cdots\!59}{15\!\cdots\!79}a^{12}+\frac{37\!\cdots\!46}{15\!\cdots\!79}a^{11}-\frac{10\!\cdots\!89}{81\!\cdots\!41}a^{10}+\frac{13\!\cdots\!41}{15\!\cdots\!79}a^{9}+\frac{10\!\cdots\!24}{15\!\cdots\!79}a^{8}+\frac{22\!\cdots\!00}{15\!\cdots\!79}a^{7}+\frac{70\!\cdots\!08}{15\!\cdots\!79}a^{6}+\frac{17\!\cdots\!46}{15\!\cdots\!79}a^{5}+\frac{83\!\cdots\!65}{15\!\cdots\!79}a^{4}+\frac{50\!\cdots\!99}{15\!\cdots\!79}a^{3}+\frac{79\!\cdots\!00}{15\!\cdots\!79}a^{2}-\frac{51\!\cdots\!31}{15\!\cdots\!79}a-\frac{15\!\cdots\!55}{15\!\cdots\!79}$, $\frac{26\!\cdots\!10}{15\!\cdots\!79}a^{19}-\frac{13\!\cdots\!30}{15\!\cdots\!79}a^{18}+\frac{37\!\cdots\!80}{15\!\cdots\!79}a^{17}-\frac{54\!\cdots\!94}{15\!\cdots\!79}a^{16}+\frac{13\!\cdots\!20}{81\!\cdots\!41}a^{15}-\frac{46\!\cdots\!74}{15\!\cdots\!79}a^{14}+\frac{11\!\cdots\!01}{15\!\cdots\!79}a^{13}-\frac{19\!\cdots\!56}{15\!\cdots\!79}a^{12}+\frac{99\!\cdots\!66}{15\!\cdots\!79}a^{11}-\frac{48\!\cdots\!80}{15\!\cdots\!79}a^{10}+\frac{35\!\cdots\!95}{15\!\cdots\!79}a^{9}+\frac{47\!\cdots\!74}{15\!\cdots\!79}a^{8}+\frac{58\!\cdots\!18}{15\!\cdots\!79}a^{7}+\frac{22\!\cdots\!31}{15\!\cdots\!79}a^{6}+\frac{44\!\cdots\!04}{15\!\cdots\!79}a^{5}+\frac{25\!\cdots\!13}{15\!\cdots\!79}a^{4}+\frac{13\!\cdots\!05}{15\!\cdots\!79}a^{3}+\frac{28\!\cdots\!20}{15\!\cdots\!79}a^{2}+\frac{95\!\cdots\!82}{15\!\cdots\!79}a-\frac{56\!\cdots\!45}{15\!\cdots\!79}$, $\frac{76\!\cdots\!51}{15\!\cdots\!79}a^{19}-\frac{38\!\cdots\!23}{15\!\cdots\!79}a^{18}+\frac{10\!\cdots\!29}{15\!\cdots\!79}a^{17}-\frac{13\!\cdots\!47}{15\!\cdots\!79}a^{16}+\frac{71\!\cdots\!69}{15\!\cdots\!79}a^{15}-\frac{11\!\cdots\!37}{15\!\cdots\!79}a^{14}+\frac{30\!\cdots\!46}{15\!\cdots\!79}a^{13}-\frac{48\!\cdots\!53}{15\!\cdots\!79}a^{12}+\frac{27\!\cdots\!79}{15\!\cdots\!79}a^{11}-\frac{73\!\cdots\!51}{15\!\cdots\!79}a^{10}+\frac{98\!\cdots\!54}{15\!\cdots\!79}a^{9}+\frac{38\!\cdots\!98}{15\!\cdots\!79}a^{8}+\frac{17\!\cdots\!78}{15\!\cdots\!79}a^{7}+\frac{10\!\cdots\!82}{15\!\cdots\!79}a^{6}+\frac{13\!\cdots\!70}{15\!\cdots\!79}a^{5}+\frac{97\!\cdots\!03}{15\!\cdots\!79}a^{4}+\frac{47\!\cdots\!69}{15\!\cdots\!79}a^{3}+\frac{10\!\cdots\!34}{15\!\cdots\!79}a^{2}+\frac{18\!\cdots\!07}{15\!\cdots\!79}a-\frac{37\!\cdots\!28}{15\!\cdots\!79}$
| 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: | \( 10123076.9997 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 10123076.9997 \cdot 116}{10\cdot\sqrt{2131588214553606414985382080078125}}\cr\approx \mathstrut & 0.243903233471 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*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^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*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^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*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^20 - 5*x^19 + 13*x^18 - 17*x^17 + 94*x^16 - 151*x^15 + 406*x^14 - 633*x^13 + 3653*x^12 - 772*x^11 + 13426*x^10 + 5719*x^9 + 24764*x^8 + 15388*x^7 + 21892*x^6 + 15269*x^5 + 9379*x^4 + 3157*x^3 + 732*x^2 - 45*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
$C_4\times D_5$ (as 20T6):
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 40 |
| The 16 conjugacy class representatives for $C_4\times D_5$ |
| Character table for $C_4\times D_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.12852225.1, 10.10.825898437253125.1 |
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 40 |
| Degree 20 sibling: | deg 20 |
| 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 | $20$ | R | R | $20$ | ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | $20$ | $20$ | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | $20$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
|
\(239\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |