Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 1)
gp: K = bnfinit(y^40 + 57*y^36 + 1280*y^32 + 14374*y^28 + 85046*y^24 + 259698*y^20 + 379157*y^16 + 222625*y^12 + 41990*y^8 + 820*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 1)
\( x^{40} + 57 x^{36} + 1280 x^{32} + 14374 x^{28} + 85046 x^{24} + 259698 x^{20} + 379157 x^{16} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(130305099804548492884220428175380349368393046678311823693003457545895936\)
\(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(59.96\) | 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: | $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$ | ||
| Ramified primes: |
\(2\), \(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(264=2^{3}\cdot 3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(5,·)$, $\chi_{264}(263,·)$, $\chi_{264}(139,·)$, $\chi_{264}(13,·)$, $\chi_{264}(17,·)$, $\chi_{264}(19,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(31,·)$, $\chi_{264}(161,·)$, $\chi_{264}(167,·)$, $\chi_{264}(41,·)$, $\chi_{264}(95,·)$, $\chi_{264}(43,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(53,·)$, $\chi_{264}(59,·)$, $\chi_{264}(61,·)$, $\chi_{264}(65,·)$, $\chi_{264}(199,·)$, $\chi_{264}(203,·)$, $\chi_{264}(205,·)$, $\chi_{264}(211,·)$, $\chi_{264}(85,·)$, $\chi_{264}(215,·)$, $\chi_{264}(221,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(103,·)$, $\chi_{264}(233,·)$, $\chi_{264}(109,·)$, $\chi_{264}(239,·)$, $\chi_{264}(245,·)$, $\chi_{264}(169,·)$, $\chi_{264}(251,·)$, $\chi_{264}(125,·)$, $\chi_{264}(247,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{14\!\cdots\!27}a^{36}-\frac{26\!\cdots\!86}{14\!\cdots\!27}a^{32}-\frac{63\!\cdots\!78}{14\!\cdots\!27}a^{28}-\frac{34\!\cdots\!54}{14\!\cdots\!27}a^{24}+\frac{29\!\cdots\!76}{14\!\cdots\!27}a^{20}+\frac{63\!\cdots\!07}{14\!\cdots\!27}a^{16}-\frac{61\!\cdots\!38}{14\!\cdots\!27}a^{12}+\frac{44\!\cdots\!72}{14\!\cdots\!27}a^{8}+\frac{25\!\cdots\!90}{14\!\cdots\!27}a^{4}-\frac{20\!\cdots\!84}{14\!\cdots\!27}$, $\frac{1}{14\!\cdots\!27}a^{37}-\frac{26\!\cdots\!86}{14\!\cdots\!27}a^{33}-\frac{63\!\cdots\!78}{14\!\cdots\!27}a^{29}-\frac{34\!\cdots\!54}{14\!\cdots\!27}a^{25}+\frac{29\!\cdots\!76}{14\!\cdots\!27}a^{21}+\frac{63\!\cdots\!07}{14\!\cdots\!27}a^{17}-\frac{61\!\cdots\!38}{14\!\cdots\!27}a^{13}+\frac{44\!\cdots\!72}{14\!\cdots\!27}a^{9}+\frac{25\!\cdots\!90}{14\!\cdots\!27}a^{5}-\frac{20\!\cdots\!84}{14\!\cdots\!27}a$, $\frac{1}{14\!\cdots\!27}a^{38}-\frac{26\!\cdots\!86}{14\!\cdots\!27}a^{34}-\frac{63\!\cdots\!78}{14\!\cdots\!27}a^{30}-\frac{34\!\cdots\!54}{14\!\cdots\!27}a^{26}+\frac{29\!\cdots\!76}{14\!\cdots\!27}a^{22}+\frac{63\!\cdots\!07}{14\!\cdots\!27}a^{18}-\frac{61\!\cdots\!38}{14\!\cdots\!27}a^{14}+\frac{44\!\cdots\!72}{14\!\cdots\!27}a^{10}+\frac{25\!\cdots\!90}{14\!\cdots\!27}a^{6}-\frac{20\!\cdots\!84}{14\!\cdots\!27}a^{2}$, $\frac{1}{14\!\cdots\!27}a^{39}-\frac{26\!\cdots\!86}{14\!\cdots\!27}a^{35}-\frac{63\!\cdots\!78}{14\!\cdots\!27}a^{31}-\frac{34\!\cdots\!54}{14\!\cdots\!27}a^{27}+\frac{29\!\cdots\!76}{14\!\cdots\!27}a^{23}+\frac{63\!\cdots\!07}{14\!\cdots\!27}a^{19}-\frac{61\!\cdots\!38}{14\!\cdots\!27}a^{15}+\frac{44\!\cdots\!72}{14\!\cdots\!27}a^{11}+\frac{25\!\cdots\!90}{14\!\cdots\!27}a^{7}-\frac{20\!\cdots\!84}{14\!\cdots\!27}a^{3}$
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
$C_{110}\times C_{110}$, which has order $12100$ (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: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{398772571518731004694}{14118201587196095474227} a^{38} + \frac{22731418606447681518718}{14118201587196095474227} a^{34} + \frac{510507728716834005253668}{14118201587196095474227} a^{30} + \frac{5733729459516139143490252}{14118201587196095474227} a^{26} + \frac{33933956932744042548205033}{14118201587196095474227} a^{22} + \frac{103678878142004921547988119}{14118201587196095474227} a^{18} + \frac{151561773478616221575224766}{14118201587196095474227} a^{14} + \frac{89318133696368223029564203}{14118201587196095474227} a^{10} + \frac{17080580897272744456909535}{14118201587196095474227} a^{6} + \frac{412784981084028852042315}{14118201587196095474227} a^{2} \)
(order $4$)
| 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: |
$\frac{58\!\cdots\!67}{14\!\cdots\!27}a^{39}+\frac{33\!\cdots\!18}{14\!\cdots\!27}a^{35}+\frac{74\!\cdots\!42}{14\!\cdots\!27}a^{31}+\frac{84\!\cdots\!32}{14\!\cdots\!27}a^{27}+\frac{49\!\cdots\!31}{14\!\cdots\!27}a^{23}+\frac{15\!\cdots\!00}{14\!\cdots\!27}a^{19}+\frac{22\!\cdots\!85}{14\!\cdots\!27}a^{15}+\frac{12\!\cdots\!38}{14\!\cdots\!27}a^{11}+\frac{24\!\cdots\!61}{14\!\cdots\!27}a^{7}+\frac{44\!\cdots\!95}{14\!\cdots\!27}a^{3}$, $\frac{69\!\cdots\!00}{14\!\cdots\!27}a^{37}+\frac{39\!\cdots\!40}{14\!\cdots\!27}a^{33}+\frac{88\!\cdots\!80}{14\!\cdots\!27}a^{29}+\frac{99\!\cdots\!45}{14\!\cdots\!27}a^{25}+\frac{59\!\cdots\!35}{14\!\cdots\!27}a^{21}+\frac{18\!\cdots\!40}{14\!\cdots\!27}a^{17}+\frac{27\!\cdots\!65}{14\!\cdots\!27}a^{13}+\frac{16\!\cdots\!75}{14\!\cdots\!27}a^{9}+\frac{41\!\cdots\!48}{14\!\cdots\!27}a^{5}+\frac{68\!\cdots\!65}{14\!\cdots\!27}a$, $\frac{47\!\cdots\!80}{24\!\cdots\!37}a^{36}+\frac{26\!\cdots\!44}{24\!\cdots\!37}a^{32}+\frac{60\!\cdots\!76}{24\!\cdots\!37}a^{28}+\frac{67\!\cdots\!59}{24\!\cdots\!37}a^{24}+\frac{39\!\cdots\!56}{24\!\cdots\!37}a^{20}+\frac{11\!\cdots\!74}{24\!\cdots\!37}a^{16}+\frac{16\!\cdots\!08}{24\!\cdots\!37}a^{12}+\frac{92\!\cdots\!59}{24\!\cdots\!37}a^{8}+\frac{15\!\cdots\!40}{24\!\cdots\!37}a^{4}+\frac{34\!\cdots\!82}{24\!\cdots\!37}$, $\frac{78\!\cdots\!28}{14\!\cdots\!27}a^{38}+\frac{44\!\cdots\!48}{14\!\cdots\!27}a^{34}+\frac{10\!\cdots\!60}{14\!\cdots\!27}a^{30}+\frac{11\!\cdots\!50}{14\!\cdots\!27}a^{26}+\frac{67\!\cdots\!24}{14\!\cdots\!27}a^{22}+\frac{20\!\cdots\!90}{14\!\cdots\!27}a^{18}+\frac{29\!\cdots\!14}{14\!\cdots\!27}a^{14}+\frac{17\!\cdots\!56}{14\!\cdots\!27}a^{10}+\frac{33\!\cdots\!87}{14\!\cdots\!27}a^{6}+\frac{70\!\cdots\!51}{14\!\cdots\!27}a^{2}$, $\frac{15\!\cdots\!06}{14\!\cdots\!27}a^{38}+\frac{87\!\cdots\!04}{14\!\cdots\!27}a^{34}+\frac{19\!\cdots\!24}{14\!\cdots\!27}a^{30}+\frac{21\!\cdots\!60}{14\!\cdots\!27}a^{26}+\frac{12\!\cdots\!10}{14\!\cdots\!27}a^{22}+\frac{38\!\cdots\!93}{14\!\cdots\!27}a^{18}+\frac{55\!\cdots\!45}{14\!\cdots\!27}a^{14}+\frac{30\!\cdots\!11}{14\!\cdots\!27}a^{10}+\frac{46\!\cdots\!00}{14\!\cdots\!27}a^{6}-\frac{10\!\cdots\!29}{14\!\cdots\!27}a^{2}$, $\frac{26\!\cdots\!92}{14\!\cdots\!27}a^{39}+\frac{15\!\cdots\!01}{14\!\cdots\!27}a^{35}+\frac{34\!\cdots\!78}{14\!\cdots\!27}a^{31}+\frac{38\!\cdots\!02}{14\!\cdots\!27}a^{27}+\frac{22\!\cdots\!32}{14\!\cdots\!27}a^{23}+\frac{69\!\cdots\!74}{14\!\cdots\!27}a^{19}+\frac{10\!\cdots\!92}{14\!\cdots\!27}a^{15}+\frac{59\!\cdots\!04}{14\!\cdots\!27}a^{11}+\frac{11\!\cdots\!68}{14\!\cdots\!27}a^{7}+\frac{16\!\cdots\!12}{14\!\cdots\!27}a^{3}$, $\frac{49\!\cdots\!22}{14\!\cdots\!27}a^{39}+\frac{28\!\cdots\!09}{14\!\cdots\!27}a^{35}+\frac{63\!\cdots\!22}{14\!\cdots\!27}a^{31}+\frac{70\!\cdots\!53}{14\!\cdots\!27}a^{27}+\frac{41\!\cdots\!86}{14\!\cdots\!27}a^{23}+\frac{12\!\cdots\!57}{14\!\cdots\!27}a^{19}+\frac{18\!\cdots\!52}{14\!\cdots\!27}a^{15}+\frac{11\!\cdots\!27}{14\!\cdots\!27}a^{11}+\frac{20\!\cdots\!07}{14\!\cdots\!27}a^{7}+\frac{41\!\cdots\!74}{14\!\cdots\!27}a^{3}$, $a$, $\frac{60\!\cdots\!73}{14\!\cdots\!27}a^{39}+\frac{34\!\cdots\!22}{14\!\cdots\!27}a^{35}+\frac{76\!\cdots\!66}{14\!\cdots\!27}a^{31}+\frac{86\!\cdots\!92}{14\!\cdots\!27}a^{27}+\frac{51\!\cdots\!41}{14\!\cdots\!27}a^{23}+\frac{15\!\cdots\!93}{14\!\cdots\!27}a^{19}+\frac{22\!\cdots\!30}{14\!\cdots\!27}a^{15}+\frac{13\!\cdots\!49}{14\!\cdots\!27}a^{11}+\frac{24\!\cdots\!61}{14\!\cdots\!27}a^{7}+\frac{43\!\cdots\!66}{14\!\cdots\!27}a^{3}$, $\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{26\!\cdots\!30}{14\!\cdots\!27}a^{37}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{14\!\cdots\!25}{14\!\cdots\!27}a^{33}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{33\!\cdots\!50}{14\!\cdots\!27}a^{29}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{37\!\cdots\!57}{14\!\cdots\!27}a^{25}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{22\!\cdots\!30}{14\!\cdots\!27}a^{21}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{67\!\cdots\!25}{14\!\cdots\!27}a^{17}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{98\!\cdots\!95}{14\!\cdots\!27}a^{13}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{57\!\cdots\!75}{14\!\cdots\!27}a^{9}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{10\!\cdots\!30}{14\!\cdots\!27}a^{5}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}+\frac{11\!\cdots\!75}{14\!\cdots\!27}a+1$, $\frac{35\!\cdots\!54}{14\!\cdots\!27}a^{39}-\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{20\!\cdots\!62}{14\!\cdots\!27}a^{35}-\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{45\!\cdots\!89}{14\!\cdots\!27}a^{31}-\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{51\!\cdots\!72}{14\!\cdots\!27}a^{27}-\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{30\!\cdots\!36}{14\!\cdots\!27}a^{23}-\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{92\!\cdots\!88}{14\!\cdots\!27}a^{19}-\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{13\!\cdots\!38}{14\!\cdots\!27}a^{15}-\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{79\!\cdots\!37}{14\!\cdots\!27}a^{11}-\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{15\!\cdots\!19}{14\!\cdots\!27}a^{7}-\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{33\!\cdots\!03}{14\!\cdots\!27}a^{3}-\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}+1$, $\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}-\frac{11\!\cdots\!92}{14\!\cdots\!27}a^{37}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}-\frac{66\!\cdots\!69}{14\!\cdots\!27}a^{33}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}-\frac{14\!\cdots\!47}{14\!\cdots\!27}a^{29}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}-\frac{16\!\cdots\!07}{14\!\cdots\!27}a^{25}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}-\frac{99\!\cdots\!26}{14\!\cdots\!27}a^{21}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}-\frac{30\!\cdots\!07}{14\!\cdots\!27}a^{17}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}-\frac{44\!\cdots\!51}{14\!\cdots\!27}a^{13}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}-\frac{27\!\cdots\!38}{14\!\cdots\!27}a^{9}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}-\frac{54\!\cdots\!75}{14\!\cdots\!27}a^{5}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-\frac{14\!\cdots\!93}{14\!\cdots\!27}a+1$, $\frac{13\!\cdots\!68}{14\!\cdots\!27}a^{39}+\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{78\!\cdots\!47}{14\!\cdots\!27}a^{35}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{17\!\cdots\!33}{14\!\cdots\!27}a^{31}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{19\!\cdots\!81}{14\!\cdots\!27}a^{27}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{11\!\cdots\!50}{14\!\cdots\!27}a^{23}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{35\!\cdots\!69}{14\!\cdots\!27}a^{19}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{52\!\cdots\!14}{14\!\cdots\!27}a^{15}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{30\!\cdots\!90}{14\!\cdots\!27}a^{11}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{57\!\cdots\!88}{14\!\cdots\!27}a^{7}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{85\!\cdots\!71}{14\!\cdots\!27}a^{3}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-1$, $\frac{11\!\cdots\!82}{14\!\cdots\!27}a^{39}+\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{68\!\cdots\!54}{14\!\cdots\!27}a^{35}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{15\!\cdots\!04}{14\!\cdots\!27}a^{31}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{17\!\cdots\!56}{14\!\cdots\!27}a^{27}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{10\!\cdots\!99}{14\!\cdots\!27}a^{23}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{31\!\cdots\!57}{14\!\cdots\!27}a^{19}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{45\!\cdots\!98}{14\!\cdots\!27}a^{15}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{26\!\cdots\!09}{14\!\cdots\!27}a^{11}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{51\!\cdots\!05}{14\!\cdots\!27}a^{7}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{12\!\cdots\!18}{14\!\cdots\!27}a^{3}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-1$, $\frac{58\!\cdots\!67}{14\!\cdots\!27}a^{39}-\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{33\!\cdots\!18}{14\!\cdots\!27}a^{35}-\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{74\!\cdots\!42}{14\!\cdots\!27}a^{31}-\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{84\!\cdots\!32}{14\!\cdots\!27}a^{27}-\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{49\!\cdots\!31}{14\!\cdots\!27}a^{23}-\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{15\!\cdots\!00}{14\!\cdots\!27}a^{19}-\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{22\!\cdots\!85}{14\!\cdots\!27}a^{15}-\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{12\!\cdots\!38}{14\!\cdots\!27}a^{11}-\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{24\!\cdots\!61}{14\!\cdots\!27}a^{7}-\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{44\!\cdots\!95}{14\!\cdots\!27}a^{3}-\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}+1$, $\frac{19\!\cdots\!10}{14\!\cdots\!27}a^{39}+\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{11\!\cdots\!42}{14\!\cdots\!27}a^{35}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{25\!\cdots\!44}{14\!\cdots\!27}a^{31}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{28\!\cdots\!51}{14\!\cdots\!27}a^{27}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{16\!\cdots\!58}{14\!\cdots\!27}a^{23}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{51\!\cdots\!87}{14\!\cdots\!27}a^{19}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{75\!\cdots\!77}{14\!\cdots\!27}a^{15}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{44\!\cdots\!40}{14\!\cdots\!27}a^{11}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{85\!\cdots\!40}{14\!\cdots\!27}a^{7}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{19\!\cdots\!34}{14\!\cdots\!27}a^{3}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-1$, $\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}-\frac{73\!\cdots\!68}{14\!\cdots\!27}a^{37}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}-\frac{41\!\cdots\!45}{14\!\cdots\!27}a^{33}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}-\frac{94\!\cdots\!19}{14\!\cdots\!27}a^{29}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}-\frac{10\!\cdots\!67}{14\!\cdots\!27}a^{25}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}-\frac{62\!\cdots\!42}{14\!\cdots\!27}a^{21}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}-\frac{19\!\cdots\!67}{14\!\cdots\!27}a^{17}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}-\frac{28\!\cdots\!50}{14\!\cdots\!27}a^{13}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}-\frac{17\!\cdots\!68}{14\!\cdots\!27}a^{9}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}-\frac{36\!\cdots\!62}{14\!\cdots\!27}a^{5}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-\frac{13\!\cdots\!39}{14\!\cdots\!27}a+1$, $\frac{27\!\cdots\!38}{14\!\cdots\!27}a^{39}-\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}+\frac{15\!\cdots\!90}{14\!\cdots\!27}a^{35}-\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}+\frac{35\!\cdots\!04}{14\!\cdots\!27}a^{31}-\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}+\frac{39\!\cdots\!01}{14\!\cdots\!27}a^{27}-\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}+\frac{23\!\cdots\!82}{14\!\cdots\!27}a^{23}-\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}+\frac{72\!\cdots\!77}{14\!\cdots\!27}a^{19}-\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}+\frac{10\!\cdots\!91}{14\!\cdots\!27}a^{15}-\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}+\frac{62\!\cdots\!96}{14\!\cdots\!27}a^{11}-\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}+\frac{11\!\cdots\!27}{14\!\cdots\!27}a^{7}-\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}+\frac{26\!\cdots\!85}{14\!\cdots\!27}a^{3}-\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}+1$, $\frac{39\!\cdots\!94}{14\!\cdots\!27}a^{38}-\frac{14\!\cdots\!59}{14\!\cdots\!27}a^{37}+\frac{22\!\cdots\!18}{14\!\cdots\!27}a^{34}-\frac{84\!\cdots\!79}{14\!\cdots\!27}a^{33}+\frac{51\!\cdots\!68}{14\!\cdots\!27}a^{30}-\frac{18\!\cdots\!25}{14\!\cdots\!27}a^{29}+\frac{57\!\cdots\!52}{14\!\cdots\!27}a^{26}-\frac{21\!\cdots\!42}{14\!\cdots\!27}a^{25}+\frac{33\!\cdots\!33}{14\!\cdots\!27}a^{22}-\frac{12\!\cdots\!41}{14\!\cdots\!27}a^{21}+\frac{10\!\cdots\!19}{14\!\cdots\!27}a^{18}-\frac{38\!\cdots\!66}{14\!\cdots\!27}a^{17}+\frac{15\!\cdots\!66}{14\!\cdots\!27}a^{14}-\frac{56\!\cdots\!62}{14\!\cdots\!27}a^{13}+\frac{89\!\cdots\!03}{14\!\cdots\!27}a^{10}-\frac{33\!\cdots\!20}{14\!\cdots\!27}a^{9}+\frac{17\!\cdots\!35}{14\!\cdots\!27}a^{6}-\frac{62\!\cdots\!27}{14\!\cdots\!27}a^{5}+\frac{41\!\cdots\!15}{14\!\cdots\!27}a^{2}-\frac{63\!\cdots\!92}{14\!\cdots\!27}a+1$
| 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: | \( 1535102994471753.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 1535102994471753.0 \cdot 12100}{4\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.118298587302334 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 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^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 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^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 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^40 + 57*x^36 + 1280*x^32 + 14374*x^28 + 85046*x^24 + 259698*x^20 + 379157*x^16 + 222625*x^12 + 41990*x^8 + 820*x^4 + 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
$C_2^2\times C_{10}$ (as 40T7):
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)
| An abelian group of order 40 |
| The 40 conjugacy class representatives for $C_2^2\times C_{10}$ |
| Character table for $C_2^2\times C_{10}$ |
Intermediate fields
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.10.0.1}{10} }^{4}$ | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.5.0.1}{5} }^{8}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $40$ | |||
| Deg $20$ | $4$ | $5$ | $40$ | ||||
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |
| 11.20.18.1 | $x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$ | $10$ | $2$ | $18$ | 20T3 | $[\ ]_{10}^{2}$ |