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Properties

Label 40.0.235...625.1
Degree $40$
Signature $[0, 20]$
Discriminant $2.352\times 10^{68}$
Root discriminant \(51.20\)
Ramified primes $5,7,11$
Class number not computed
Class group not computed
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401)
 
gp: K = bnfinit(y^40 - y^38 - 8*y^36 + 17*y^34 + 55*y^32 - 208*y^30 - 287*y^28 + 2159*y^26 + 424*y^24 - 19855*y^22 + 16039*y^20 - 178695*y^18 + 34344*y^16 + 1573911*y^14 - 1883007*y^12 - 12282192*y^10 + 29229255*y^8 + 81310473*y^6 - 344373768*y^4 - 387420489*y^2 + 3486784401, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401)
 

\( x^{40} - x^{38} - 8 x^{36} + 17 x^{34} + 55 x^{32} - 208 x^{30} - 287 x^{28} + 2159 x^{26} + \cdots + 3486784401 \) Copy content Toggle raw display

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:   \(235232623772795025272937433621472877556782506233347450351715087890625\) \(\medspace = 5^{20}\cdot 7^{20}\cdot 11^{36}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(51.20\)
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:  $5^{1/2}7^{1/2}11^{9/10}\approx 51.202060545412486$
Ramified primes:   \(5\), \(7\), \(11\) Copy content Toggle raw display
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:  \(385=5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{385}(384,·)$, $\chi_{385}(1,·)$, $\chi_{385}(134,·)$, $\chi_{385}(139,·)$, $\chi_{385}(141,·)$, $\chi_{385}(216,·)$, $\chi_{385}(146,·)$, $\chi_{385}(279,·)$, $\chi_{385}(281,·)$, $\chi_{385}(111,·)$, $\chi_{385}(29,·)$, $\chi_{385}(34,·)$, $\chi_{385}(36,·)$, $\chi_{385}(6,·)$, $\chi_{385}(41,·)$, $\chi_{385}(174,·)$, $\chi_{385}(181,·)$, $\chi_{385}(314,·)$, $\chi_{385}(316,·)$, $\chi_{385}(309,·)$, $\chi_{385}(64,·)$, $\chi_{385}(321,·)$, $\chi_{385}(69,·)$, $\chi_{385}(71,·)$, $\chi_{385}(76,·)$, $\chi_{385}(204,·)$, $\chi_{385}(211,·)$, $\chi_{385}(344,·)$, $\chi_{385}(349,·)$, $\chi_{385}(351,·)$, $\chi_{385}(251,·)$, $\chi_{385}(356,·)$, $\chi_{385}(104,·)$, $\chi_{385}(106,·)$, $\chi_{385}(274,·)$, $\chi_{385}(239,·)$, $\chi_{385}(244,·)$, $\chi_{385}(246,·)$, $\chi_{385}(169,·)$, $\chi_{385}(379,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{524288}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{21}-\frac{1}{6}a^{19}-\frac{1}{2}a^{18}-\frac{1}{3}a^{17}-\frac{1}{2}a^{16}-\frac{1}{6}a^{15}+\frac{1}{6}a^{13}-\frac{1}{2}a^{12}+\frac{1}{3}a^{11}+\frac{1}{6}a^{9}-\frac{1}{2}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{288702}a^{22}-\frac{1}{18}a^{20}-\frac{1}{2}a^{19}-\frac{4}{9}a^{18}-\frac{1}{2}a^{17}-\frac{1}{18}a^{16}+\frac{1}{18}a^{14}-\frac{1}{2}a^{13}+\frac{4}{9}a^{12}+\frac{1}{18}a^{10}-\frac{1}{2}a^{9}+\frac{4}{9}a^{8}-\frac{1}{2}a^{7}+\frac{1}{18}a^{6}-\frac{1}{18}a^{4}-\frac{1}{2}a^{3}-\frac{4}{9}a^{2}-\frac{1}{2}a-\frac{1908}{16039}$, $\frac{1}{866106}a^{23}-\frac{1}{54}a^{21}+\frac{19}{54}a^{19}-\frac{1}{2}a^{18}-\frac{5}{27}a^{17}-\frac{1}{2}a^{16}+\frac{1}{54}a^{15}-\frac{19}{54}a^{13}+\frac{5}{27}a^{11}-\frac{1}{54}a^{9}-\frac{1}{2}a^{8}-\frac{4}{27}a^{7}-\frac{1}{2}a^{6}+\frac{17}{54}a^{5}+\frac{1}{54}a^{3}-\frac{1}{2}a^{2}-\frac{19855}{96234}a-\frac{1}{2}$, $\frac{1}{2598318}a^{24}-\frac{1}{2598318}a^{22}-\frac{4}{81}a^{20}-\frac{32}{81}a^{18}-\frac{13}{81}a^{16}-\frac{23}{81}a^{14}-\frac{1}{2}a^{13}-\frac{22}{81}a^{12}-\frac{1}{2}a^{11}-\frac{14}{81}a^{10}-\frac{31}{81}a^{8}-\frac{5}{81}a^{6}-\frac{40}{81}a^{4}-\frac{19855}{288702}a^{2}-\frac{15615}{32078}$, $\frac{1}{7794954}a^{25}-\frac{1}{7794954}a^{23}-\frac{4}{243}a^{21}-\frac{113}{243}a^{19}-\frac{94}{243}a^{17}-\frac{104}{243}a^{15}-\frac{1}{2}a^{14}-\frac{22}{243}a^{13}-\frac{1}{2}a^{12}-\frac{14}{243}a^{11}-\frac{31}{243}a^{9}-\frac{86}{243}a^{7}-\frac{121}{243}a^{5}-\frac{19855}{866106}a^{3}-\frac{47693}{96234}a$, $\frac{1}{23384862}a^{26}-\frac{1}{23384862}a^{24}-\frac{4}{11692431}a^{22}+\frac{17}{1458}a^{20}-\frac{1}{2}a^{19}-\frac{337}{729}a^{18}-\frac{1}{2}a^{17}+\frac{521}{1458}a^{16}-\frac{1}{2}a^{15}-\frac{287}{1458}a^{14}-\frac{14}{729}a^{12}-\frac{1}{2}a^{11}-\frac{305}{1458}a^{10}-\frac{1}{2}a^{9}-\frac{86}{729}a^{8}-\frac{1}{2}a^{7}+\frac{1}{1458}a^{6}+\frac{639652}{1299159}a^{4}-\frac{1}{2}a^{3}-\frac{143927}{288702}a^{2}-\frac{1}{2}a+\frac{2159}{32078}$, $\frac{1}{70154586}a^{27}-\frac{1}{70154586}a^{25}-\frac{4}{35077293}a^{23}+\frac{17}{4374}a^{21}+\frac{55}{4374}a^{19}-\frac{1}{2}a^{18}-\frac{104}{2187}a^{17}-\frac{287}{4374}a^{15}-\frac{1}{2}a^{14}+\frac{2159}{4374}a^{13}-\frac{1}{2}a^{12}+\frac{212}{2187}a^{11}+\frac{2015}{4374}a^{9}-\frac{1}{2}a^{8}+\frac{365}{2187}a^{7}-\frac{1}{2}a^{6}-\frac{659507}{3897477}a^{5}-\frac{144139}{433053}a^{3}-\frac{1}{2}a^{2}-\frac{6940}{48117}a-\frac{1}{2}$, $\frac{1}{210463758}a^{28}-\frac{1}{210463758}a^{26}-\frac{4}{105231879}a^{24}+\frac{17}{210463758}a^{22}-\frac{2861}{13122}a^{20}-\frac{1}{2}a^{19}+\frac{1354}{6561}a^{18}-\frac{3203}{13122}a^{16}-\frac{1}{2}a^{15}+\frac{5075}{13122}a^{14}-\frac{1}{2}a^{13}-\frac{1246}{6561}a^{12}-\frac{3817}{13122}a^{10}-\frac{1}{2}a^{9}-\frac{3280}{6561}a^{8}-\frac{1}{2}a^{7}+\frac{5836288}{11692431}a^{6}+\frac{212}{1299159}a^{4}-\frac{1}{2}a^{3}-\frac{71096}{144351}a^{2}-\frac{1}{2}a+\frac{7876}{16039}$, $\frac{1}{631391274}a^{29}-\frac{1}{631391274}a^{27}-\frac{4}{315695637}a^{25}+\frac{17}{631391274}a^{23}-\frac{2861}{39366}a^{21}-\frac{3853}{39366}a^{19}-\frac{4882}{19683}a^{17}+\frac{5075}{39366}a^{15}+\frac{4069}{39366}a^{13}-\frac{5189}{19683}a^{11}+\frac{13123}{39366}a^{9}-\frac{1}{2}a^{8}-\frac{23404717}{70154586}a^{7}-\frac{1}{2}a^{6}+\frac{1299371}{3897477}a^{5}-\frac{286543}{866106}a^{3}-\frac{1}{2}a^{2}+\frac{10597}{32078}a-\frac{1}{2}$, $\frac{1}{1894173822}a^{30}-\frac{1}{1894173822}a^{28}-\frac{4}{947086911}a^{26}+\frac{17}{1894173822}a^{24}+\frac{55}{1894173822}a^{22}+\frac{14476}{59049}a^{20}-\frac{1}{2}a^{19}+\frac{27923}{59049}a^{18}-\frac{1}{2}a^{17}+\frac{18940}{59049}a^{16}+\frac{24998}{59049}a^{14}-\frac{1}{2}a^{13}-\frac{18311}{59049}a^{12}-\frac{1}{2}a^{11}-\frac{29524}{59049}a^{10}-\frac{19855}{210463758}a^{8}-\frac{11692007}{23384862}a^{6}-\frac{648500}{1299159}a^{4}-\frac{287}{288702}a^{2}+\frac{15831}{32078}$, $\frac{1}{5682521466}a^{31}-\frac{1}{5682521466}a^{29}-\frac{4}{2841260733}a^{27}+\frac{17}{5682521466}a^{25}+\frac{55}{5682521466}a^{23}+\frac{14476}{177147}a^{21}+\frac{114895}{354294}a^{19}-\frac{1}{2}a^{18}-\frac{21169}{354294}a^{17}-\frac{1}{2}a^{16}+\frac{24998}{177147}a^{15}+\frac{140525}{354294}a^{13}-\frac{1}{2}a^{12}+\frac{118099}{354294}a^{11}-\frac{1}{2}a^{10}+\frac{52606012}{315695637}a^{9}+\frac{11692643}{35077293}a^{7}-\frac{1}{2}a^{6}+\frac{650659}{3897477}a^{5}-\frac{1}{2}a^{4}-\frac{72319}{433053}a^{3}-\frac{5381}{16039}a-\frac{1}{2}$, $\frac{1}{17047564398}a^{32}-\frac{1}{17047564398}a^{30}-\frac{4}{8523782199}a^{28}+\frac{17}{17047564398}a^{26}+\frac{55}{17047564398}a^{24}-\frac{104}{8523782199}a^{22}-\frac{90175}{531441}a^{20}+\frac{451223}{1062882}a^{18}+\frac{109045}{1062882}a^{16}+\frac{40738}{531441}a^{14}+\frac{1}{1062882}a^{12}-\frac{19855}{1894173822}a^{10}-\frac{1}{2}a^{9}+\frac{212}{105231879}a^{8}+\frac{2159}{23384862}a^{6}-\frac{1}{2}a^{5}-\frac{287}{2598318}a^{4}-\frac{1}{2}a^{3}-\frac{104}{144351}a^{2}+\frac{55}{32078}$, $\frac{1}{51142693194}a^{33}-\frac{1}{51142693194}a^{31}-\frac{4}{25571346597}a^{29}+\frac{17}{51142693194}a^{27}+\frac{55}{51142693194}a^{25}-\frac{104}{25571346597}a^{23}-\frac{90175}{1594323}a^{21}+\frac{451223}{3188646}a^{19}+\frac{1171927}{3188646}a^{17}+\frac{572179}{1594323}a^{15}+\frac{1062883}{3188646}a^{13}-\frac{1894193677}{5682521466}a^{11}-\frac{1}{2}a^{10}+\frac{105232091}{315695637}a^{9}-\frac{23382703}{70154586}a^{7}-\frac{1}{2}a^{6}+\frac{2598031}{7794954}a^{5}-\frac{1}{2}a^{4}-\frac{144455}{433053}a^{3}+\frac{10711}{32078}a$, $\frac{1}{153428079582}a^{34}-\frac{1}{153428079582}a^{32}-\frac{4}{76714039791}a^{30}+\frac{17}{153428079582}a^{28}+\frac{55}{153428079582}a^{26}-\frac{104}{76714039791}a^{24}-\frac{287}{153428079582}a^{22}-\frac{40109}{4782969}a^{20}-\frac{1}{2}a^{19}-\frac{3079601}{9565938}a^{18}-\frac{1}{2}a^{17}+\frac{3801563}{9565938}a^{16}-\frac{2391484}{4782969}a^{14}-\frac{1}{2}a^{13}-\frac{19855}{17047564398}a^{12}-\frac{1}{2}a^{11}-\frac{947086487}{1894173822}a^{10}-\frac{1}{2}a^{9}+\frac{2159}{210463758}a^{8}+\frac{5846072}{11692431}a^{6}-\frac{1}{2}a^{5}+\frac{1298951}{2598318}a^{4}-\frac{1}{2}a^{3}+\frac{55}{288702}a^{2}-\frac{1}{2}a-\frac{8011}{16039}$, $\frac{1}{460284238746}a^{35}-\frac{1}{460284238746}a^{33}-\frac{4}{230142119373}a^{31}+\frac{17}{460284238746}a^{29}+\frac{55}{460284238746}a^{27}-\frac{104}{230142119373}a^{25}-\frac{287}{460284238746}a^{23}-\frac{40109}{14348907}a^{21}+\frac{5634653}{14348907}a^{19}-\frac{1}{2}a^{18}-\frac{5273672}{14348907}a^{17}-\frac{1}{2}a^{16}-\frac{2391484}{14348907}a^{15}+\frac{4261881172}{25571346597}a^{13}-\frac{1}{2}a^{12}+\frac{947087123}{2841260733}a^{11}+\frac{52617019}{315695637}a^{9}+\frac{23384575}{70154586}a^{7}+\frac{1298951}{7794954}a^{5}-\frac{72148}{433053}a^{3}-\frac{1}{2}a^{2}-\frac{10687}{32078}a-\frac{1}{2}$, $\frac{1}{1380852716238}a^{36}-\frac{1}{1380852716238}a^{34}-\frac{4}{690426358119}a^{32}+\frac{17}{1380852716238}a^{30}+\frac{55}{1380852716238}a^{28}-\frac{104}{690426358119}a^{26}-\frac{287}{1380852716238}a^{24}+\frac{2159}{1380852716238}a^{22}+\frac{16052275}{86093442}a^{20}+\frac{13858204}{43046721}a^{18}+\frac{1}{86093442}a^{16}-\frac{19855}{153428079582}a^{14}+\frac{212}{8523782199}a^{12}+\frac{2159}{1894173822}a^{10}-\frac{1}{2}a^{9}-\frac{287}{210463758}a^{8}-\frac{1}{2}a^{7}-\frac{104}{11692431}a^{6}+\frac{55}{2598318}a^{4}+\frac{17}{288702}a^{2}-\frac{4}{16039}$, $\frac{1}{4142558148714}a^{37}-\frac{1}{4142558148714}a^{35}-\frac{4}{2071279074357}a^{33}+\frac{17}{4142558148714}a^{31}+\frac{55}{4142558148714}a^{29}-\frac{104}{2071279074357}a^{27}-\frac{287}{4142558148714}a^{25}+\frac{2159}{4142558148714}a^{23}+\frac{16052275}{258280326}a^{21}+\frac{13858204}{129140163}a^{19}+\frac{86093443}{258280326}a^{17}-\frac{153428099437}{460284238746}a^{15}+\frac{8523782411}{25571346597}a^{13}-\frac{1894171663}{5682521466}a^{11}-\frac{1}{2}a^{10}+\frac{210463471}{631391274}a^{9}-\frac{1}{2}a^{8}-\frac{11692535}{35077293}a^{7}+\frac{2598373}{7794954}a^{5}-\frac{288685}{866106}a^{3}+\frac{5345}{16039}a$, $\frac{1}{12427674446142}a^{38}-\frac{1}{12427674446142}a^{36}-\frac{4}{6213837223071}a^{34}+\frac{17}{12427674446142}a^{32}+\frac{55}{12427674446142}a^{30}-\frac{104}{6213837223071}a^{28}-\frac{287}{12427674446142}a^{26}+\frac{2159}{12427674446142}a^{24}+\frac{212}{6213837223071}a^{22}-\frac{72235238}{387420489}a^{20}+\frac{1}{774840978}a^{18}-\frac{19855}{1380852716238}a^{16}+\frac{212}{76714039791}a^{14}+\frac{2159}{17047564398}a^{12}-\frac{287}{1894173822}a^{10}-\frac{1}{2}a^{9}-\frac{104}{105231879}a^{8}+\frac{55}{23384862}a^{6}+\frac{17}{2598318}a^{4}-\frac{4}{144351}a^{2}-\frac{1}{32078}$, $\frac{1}{37283023338426}a^{39}-\frac{1}{37283023338426}a^{37}-\frac{4}{18641511669213}a^{35}+\frac{17}{37283023338426}a^{33}+\frac{55}{37283023338426}a^{31}-\frac{104}{18641511669213}a^{29}-\frac{287}{37283023338426}a^{27}+\frac{2159}{37283023338426}a^{25}+\frac{212}{18641511669213}a^{23}-\frac{72235238}{1162261467}a^{21}+\frac{774840979}{2324522934}a^{19}-\frac{1380852736093}{4142558148714}a^{17}+\frac{76714040003}{230142119373}a^{15}-\frac{17047562239}{51142693194}a^{13}+\frac{1894173535}{5682521466}a^{11}-\frac{1}{2}a^{10}-\frac{105231983}{315695637}a^{9}+\frac{23384917}{70154586}a^{7}-\frac{2598301}{7794954}a^{5}+\frac{144347}{433053}a^{3}-\frac{10693}{32078}a$ Copy content Toggle raw display

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

not computed

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:   \( \frac{1}{144351} a^{24} - \frac{162656}{144351} a^{2} \)  (order $22$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401)
 
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 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401, 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 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401);
 
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 - x^38 - 8*x^36 + 17*x^34 + 55*x^32 - 208*x^30 - 287*x^28 + 2159*x^26 + 424*x^24 - 19855*x^22 + 16039*x^20 - 178695*x^18 + 34344*x^16 + 1573911*x^14 - 1883007*x^12 - 12282192*x^10 + 29229255*x^8 + 81310473*x^6 - 344373768*x^4 - 387420489*x^2 + 3486784401);
 
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

\(\Q(\sqrt{77}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{385}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}, \sqrt{77})\), \(\Q(\sqrt{-7}, \sqrt{-11})\), \(\Q(\sqrt{-35}, \sqrt{-55})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-11}, \sqrt{-35})\), \(\Q(\sqrt{-7}, \sqrt{-55})\), \(\Q(\zeta_{11})^+\), 8.0.21970650625.1, 10.10.39630026842637.1, 10.10.669871503125.1, 10.10.123843833883240625.1, \(\Q(\zeta_{11})\), 10.0.3602729712967.1, 10.0.7368586534375.1, 10.0.11258530353021875.4, 20.20.15337295190899698702745251650390625.1, 20.0.1570539027548129147161113769.2, 20.0.15337295190899698702745251650390625.5, 20.0.54296067514572573056640625.1, 20.0.126754505709914865311944228515625.3, 20.0.15337295190899698702745251650390625.6, 20.0.15337295190899698702745251650390625.2

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 ${\href{/padicField/2.10.0.1}{10} }^{4}$ ${\href{/padicField/3.10.0.1}{10} }^{4}$ R R R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display Deg $20$$2$$10$$10$
Deg $20$$2$$10$$10$
\(7\) Copy content Toggle raw display 7.20.10.1$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$$2$$10$$10$20T3$[\ ]_{2}^{10}$
7.20.10.1$x^{20} + 70 x^{18} + 2207 x^{16} + 2 x^{15} + 41168 x^{14} - 68 x^{13} + 501639 x^{12} - 3674 x^{11} + 4175501 x^{10} - 48430 x^{9} + 24202032 x^{8} - 163712 x^{7} + 97377995 x^{6} + 430996 x^{5} + 259701777 x^{4} + 2947158 x^{3} + 412861211 x^{2} + 7541370 x + 287825400$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$