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Properties

Label 40.0.23523262377...0625.1
Degree $40$
Signature $[0, 20]$
Discriminant $5^{20}\cdot 7^{20}\cdot 11^{36}$
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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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3486784401, 0, -387420489, 0, -344373768, 0, 81310473, 0, 29229255, 0, -12282192, 0, -1883007, 0, 1573911, 0, 34344, 0, -178695, 0, 16039, 0, -19855, 0, 424, 0, 2159, 0, -287, 0, -208, 0, 55, 0, 17, 0, -8, 0, -1, 0, 1]);
 
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(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)
 

Normalized defining polynomial

\( x^{40} - x^{38} - 8 x^{36} + 17 x^{34} + 55 x^{32} - 208 x^{30} - 287 x^{28} + 2159 x^{26} + 424 x^{24} - 19855 x^{22} + 16039 x^{20} - 178695 x^{18} + 34344 x^{16} + 1573911 x^{14} - 1883007 x^{12} - 12282192 x^{10} + 29229255 x^{8} + 81310473 x^{6} - 344373768 x^{4} - 387420489 x^{2} + 3486784401 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(235232623772795025272937433621472877556782506233347450351715087890625=5^{20}\cdot 7^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.20$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
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.

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

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

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1}{144351} a^{24} - \frac{162656}{144351} a^{2} \) (order $22$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

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.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{4}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
5Data not computed
7Data not computed
$11$11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$