Properties

Label 47.47.680...841.1
Degree $47$
Signature $[47, 0]$
Discriminant $6.808\times 10^{153}$
Root discriminant \(1875.18\)
Ramified prime $47$
Class number not computed
Class group not computed
Galois group $C_{47}$ (as 47T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449)
 
gp: K = bnfinit(y^47 - 1081*y^45 - 705*y^44 + 534061*y^43 + 683944*y^42 - 160057842*y^41 - 299647654*y^40 + 32561335719*y^39 + 78763416061*y^38 - 4767066709455*y^37 - 13908937919967*y^36 + 519679734845940*y^35 + 1750397799006104*y^34 - 43074182887990763*y^33 - 162478138072805487*y^32 + 2749329116777664352*y^31 + 11358660939804373275*y^30 - 136126237738133225032*y^29 - 605463962245369166572*y^28 + 5246767648352840100798*y^27 + 24765274103500484906219*y^26 - 157566416740809178648611*y^25 - 778707155396426126313230*y^24 + 3684240622296757942862071*y^23 + 18785271879388363208006658*y^22 - 66986754039408931327563467*y^21 - 345795459416186142486923619*y^20 + 946170254872545787018973465*y^19 + 4814104180441102948763936108*y^18 - 10375613255284454207094751556*y^17 - 50032174963571133079560800941*y^16 + 88105844627120217056303008521*y^15 + 380945711116472915778482842595*y^14 - 573465045108814971325177493888*y^13 - 2067158816911927150047646995633*y^12 + 2787145282695598886975483579427*y^11 + 7670341992113409376277564204296*y^10 - 9619799236229244848389374409877*y^9 - 18271828994795512973409030766888*y^8 + 21757031287162404406727090902813*y^7 + 25318069661907634555608128065064*y^6 - 28749901083465227418574423514388*y^5 - 17172272828610619745842195295332*y^4 + 19384902113022958413624018194070*y^3 + 2828656546972432377231017394241*y^2 - 5355885371863393108886638462583*y + 1052824394331287344099620777449, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449)
 

\( x^{47} - 1081 x^{45} - 705 x^{44} + 534061 x^{43} + 683944 x^{42} - 160057842 x^{41} - 299647654 x^{40} + 32561335719 x^{39} + 78763416061 x^{38} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display

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

Invariants

Degree:  $47$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[47, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(680\!\cdots\!841\) \(\medspace = 47^{92}\) 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:  \(1875.18\)
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:  $47^{92/47}\approx 1875.1785026045545$
Ramified primes:   \(47\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $47$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2209=47^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{2209}(1,·)$, $\chi_{2209}(1411,·)$, $\chi_{2209}(518,·)$, $\chi_{2209}(1928,·)$, $\chi_{2209}(1035,·)$, $\chi_{2209}(142,·)$, $\chi_{2209}(1552,·)$, $\chi_{2209}(659,·)$, $\chi_{2209}(2069,·)$, $\chi_{2209}(1176,·)$, $\chi_{2209}(283,·)$, $\chi_{2209}(1693,·)$, $\chi_{2209}(800,·)$, $\chi_{2209}(1317,·)$, $\chi_{2209}(424,·)$, $\chi_{2209}(1834,·)$, $\chi_{2209}(941,·)$, $\chi_{2209}(48,·)$, $\chi_{2209}(1458,·)$, $\chi_{2209}(565,·)$, $\chi_{2209}(1975,·)$, $\chi_{2209}(1082,·)$, $\chi_{2209}(189,·)$, $\chi_{2209}(1599,·)$, $\chi_{2209}(706,·)$, $\chi_{2209}(2116,·)$, $\chi_{2209}(1223,·)$, $\chi_{2209}(330,·)$, $\chi_{2209}(1740,·)$, $\chi_{2209}(847,·)$, $\chi_{2209}(1364,·)$, $\chi_{2209}(471,·)$, $\chi_{2209}(1881,·)$, $\chi_{2209}(988,·)$, $\chi_{2209}(95,·)$, $\chi_{2209}(1505,·)$, $\chi_{2209}(612,·)$, $\chi_{2209}(2022,·)$, $\chi_{2209}(1129,·)$, $\chi_{2209}(236,·)$, $\chi_{2209}(1646,·)$, $\chi_{2209}(753,·)$, $\chi_{2209}(2163,·)$, $\chi_{2209}(1270,·)$, $\chi_{2209}(377,·)$, $\chi_{2209}(1787,·)$, $\chi_{2209}(894,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{3763}a^{33}+\frac{1769}{3763}a^{32}+\frac{1597}{3763}a^{31}+\frac{1239}{3763}a^{30}+\frac{323}{3763}a^{29}+\frac{905}{3763}a^{28}-\frac{1681}{3763}a^{27}-\frac{1291}{3763}a^{26}+\frac{1080}{3763}a^{25}+\frac{948}{3763}a^{24}-\frac{291}{3763}a^{23}+\frac{580}{3763}a^{22}+\frac{1261}{3763}a^{21}-\frac{836}{3763}a^{20}-\frac{1312}{3763}a^{19}-\frac{57}{3763}a^{18}-\frac{757}{3763}a^{17}+\frac{1572}{3763}a^{16}-\frac{87}{3763}a^{15}-\frac{336}{3763}a^{14}+\frac{1602}{3763}a^{13}+\frac{1831}{3763}a^{12}+\frac{1548}{3763}a^{11}-\frac{1678}{3763}a^{10}+\frac{1443}{3763}a^{9}-\frac{735}{3763}a^{8}+\frac{1321}{3763}a^{7}-\frac{1773}{3763}a^{6}-\frac{698}{3763}a^{5}+\frac{535}{3763}a^{4}+\frac{124}{3763}a^{3}+\frac{666}{3763}a^{2}-\frac{1128}{3763}a+\frac{19}{71}$, $\frac{1}{3763}a^{34}-\frac{711}{3763}a^{32}-\frac{1604}{3763}a^{31}-\frac{1402}{3763}a^{30}+\frac{1494}{3763}a^{29}+\frac{412}{3763}a^{28}-\frac{372}{3763}a^{27}+\frac{718}{3763}a^{26}-\frac{1731}{3763}a^{25}+\frac{995}{3763}a^{24}-\frac{172}{3763}a^{23}-\frac{1223}{3763}a^{22}-\frac{86}{3763}a^{21}-\frac{1287}{3763}a^{20}-\frac{900}{3763}a^{19}-\frac{1525}{3763}a^{18}+\frac{1077}{3763}a^{17}-\frac{98}{3763}a^{16}-\frac{716}{3763}a^{15}+\frac{1432}{3763}a^{14}+\frac{1432}{3763}a^{13}-\frac{1311}{3763}a^{12}-\frac{626}{3763}a^{11}+\frac{818}{3763}a^{10}+\frac{1675}{3763}a^{9}-\frac{462}{3763}a^{8}-\frac{1799}{3763}a^{7}+\frac{1160}{3763}a^{6}+\frac{1033}{3763}a^{5}-\frac{1778}{3763}a^{4}-\frac{436}{3763}a^{3}-\frac{1463}{3763}a^{2}-\frac{1714}{3763}a-\frac{28}{71}$, $\frac{1}{3763}a^{35}-\frac{687}{3763}a^{32}+\frac{1402}{3763}a^{31}+\frac{1881}{3763}a^{30}+\frac{522}{3763}a^{29}-\frac{390}{3763}a^{28}-\frac{1602}{3763}a^{27}-\frac{1460}{3763}a^{26}+\frac{1223}{3763}a^{25}+\frac{279}{3763}a^{24}-\frac{1159}{3763}a^{23}-\frac{1636}{3763}a^{22}-\frac{310}{3763}a^{21}-\frac{14}{71}a^{20}-\frac{1133}{3763}a^{19}-\frac{1820}{3763}a^{18}-\frac{216}{3763}a^{17}-\frac{635}{3763}a^{16}-\frac{217}{3763}a^{15}-\frac{395}{3763}a^{14}+\frac{1285}{3763}a^{13}-\frac{783}{3763}a^{12}-\frac{21}{71}a^{11}+\frac{1488}{3763}a^{10}-\frac{1788}{3763}a^{9}-\frac{1327}{3763}a^{8}-\frac{359}{3763}a^{7}+\frac{1035}{3763}a^{6}-\frac{1340}{3763}a^{5}-\frac{114}{3763}a^{4}+\frac{152}{3763}a^{3}+\frac{1437}{3763}a^{2}+\frac{1790}{3763}a+\frac{19}{71}$, $\frac{1}{3763}a^{36}+\frac{1256}{3763}a^{32}+\frac{224}{3763}a^{31}+\frac{1277}{3763}a^{30}-\frac{506}{3763}a^{29}-\frac{762}{3763}a^{28}-\frac{1066}{3763}a^{27}-\frac{1389}{3763}a^{26}+\frac{928}{3763}a^{25}-\frac{882}{3763}a^{24}+\frac{1649}{3763}a^{23}-\frac{728}{3763}a^{22}+\frac{75}{3763}a^{21}+\frac{274}{3763}a^{20}-\frac{44}{3763}a^{19}-\frac{1745}{3763}a^{18}-\frac{1400}{3763}a^{17}-\frac{234}{3763}a^{16}+\frac{44}{3763}a^{15}-\frac{4}{3763}a^{14}+\frac{995}{3763}a^{13}-\frac{58}{3763}a^{12}+\frac{35}{3763}a^{11}+\frac{667}{3763}a^{10}+\frac{345}{3763}a^{9}-\frac{1062}{3763}a^{8}+\frac{1679}{3763}a^{7}-\frac{179}{3763}a^{6}-\frac{1739}{3763}a^{5}-\frac{1077}{3763}a^{4}+\frac{76}{3763}a^{3}+\frac{246}{3763}a^{2}+\frac{1249}{3763}a-\frac{11}{71}$, $\frac{1}{3763}a^{37}-\frac{1470}{3763}a^{32}+\frac{1124}{3763}a^{31}+\frac{1192}{3763}a^{30}-\frac{46}{3763}a^{29}-\frac{1320}{3763}a^{28}-\frac{1096}{3763}a^{27}+\frac{571}{3763}a^{26}+\frac{1081}{3763}a^{25}+\frac{69}{3763}a^{24}-\frac{243}{3763}a^{23}+\frac{1617}{3763}a^{22}+\frac{681}{3763}a^{21}+\frac{95}{3763}a^{20}+\frac{32}{71}a^{19}-\frac{1305}{3763}a^{18}-\frac{1481}{3763}a^{17}+\frac{1187}{3763}a^{16}+\frac{141}{3763}a^{15}+\frac{1555}{3763}a^{14}+\frac{1035}{3763}a^{13}-\frac{508}{3763}a^{12}+\frac{1850}{3763}a^{11}+\frac{633}{3763}a^{10}+\frac{296}{3763}a^{9}-\frac{859}{3763}a^{8}+\frac{128}{3763}a^{7}+\frac{1216}{3763}a^{6}-\frac{1168}{3763}a^{5}+\frac{1693}{3763}a^{4}-\frac{1215}{3763}a^{3}+\frac{139}{3763}a^{2}+\frac{1297}{3763}a-\frac{8}{71}$, $\frac{1}{3763}a^{38}+\frac{1321}{3763}a^{32}+\frac{670}{3763}a^{31}-\frac{8}{3763}a^{30}-\frac{648}{3763}a^{29}+\frac{915}{3763}a^{28}+\frac{1792}{3763}a^{27}-\frac{137}{3763}a^{26}-\frac{317}{3763}a^{25}+\frac{19}{71}a^{24}-\frac{934}{3763}a^{23}-\frac{920}{3763}a^{22}-\frac{1394}{3763}a^{21}-\frac{486}{3763}a^{20}+\frac{474}{3763}a^{19}+\frac{18}{53}a^{18}-\frac{1518}{3763}a^{17}+\frac{499}{3763}a^{16}+\frac{1607}{3763}a^{15}+\frac{68}{3763}a^{14}-\frac{1206}{3763}a^{13}-\frac{888}{3763}a^{12}-\frac{422}{3763}a^{11}-\frac{1599}{3763}a^{10}+\frac{1782}{3763}a^{9}-\frac{341}{3763}a^{8}+\frac{26}{71}a^{7}+\frac{281}{3763}a^{6}-\frac{831}{3763}a^{5}-\frac{1232}{3763}a^{4}+\frac{1795}{3763}a^{3}-\frac{1826}{3763}a^{2}+\frac{899}{3763}a+\frac{27}{71}$, $\frac{1}{252121}a^{39}-\frac{19}{252121}a^{38}+\frac{30}{252121}a^{37}-\frac{20}{252121}a^{36}+\frac{20}{252121}a^{35}-\frac{13}{252121}a^{34}+\frac{29}{252121}a^{33}+\frac{84834}{252121}a^{32}+\frac{30327}{252121}a^{31}-\frac{7440}{252121}a^{30}-\frac{99528}{252121}a^{29}-\frac{40787}{252121}a^{28}+\frac{48068}{252121}a^{27}-\frac{81}{3763}a^{26}+\frac{4605}{252121}a^{25}+\frac{1736}{252121}a^{24}+\frac{79463}{252121}a^{23}+\frac{39234}{252121}a^{22}+\frac{1891}{4757}a^{21}+\frac{80597}{252121}a^{20}-\frac{435}{3763}a^{19}+\frac{4441}{252121}a^{18}+\frac{88329}{252121}a^{17}-\frac{45748}{252121}a^{16}-\frac{37685}{252121}a^{15}+\frac{86817}{252121}a^{14}-\frac{87816}{252121}a^{13}-\frac{36365}{252121}a^{12}-\frac{86471}{252121}a^{11}+\frac{98820}{252121}a^{10}-\frac{1370}{4757}a^{9}-\frac{79834}{252121}a^{8}-\frac{120550}{252121}a^{7}-\frac{108198}{252121}a^{6}-\frac{42288}{252121}a^{5}+\frac{6649}{252121}a^{4}+\frac{56824}{252121}a^{3}-\frac{74189}{252121}a^{2}-\frac{17803}{252121}a+\frac{1152}{4757}$, $\frac{1}{252121}a^{40}+\frac{4}{252121}a^{38}+\frac{14}{252121}a^{37}-\frac{25}{252121}a^{36}+\frac{32}{252121}a^{35}-\frac{17}{252121}a^{34}+\frac{27}{252121}a^{33}+\frac{125226}{252121}a^{32}+\frac{58702}{252121}a^{31}+\frac{26241}{252121}a^{30}+\frac{11315}{252121}a^{29}-\frac{106733}{252121}a^{28}-\frac{38711}{252121}a^{27}-\frac{9934}{252121}a^{26}+\frac{54994}{252121}a^{25}-\frac{16796}{252121}a^{24}-\frac{117393}{252121}a^{23}-\frac{1559}{3551}a^{22}+\frac{23141}{252121}a^{21}+\frac{118581}{252121}a^{20}-\frac{59209}{252121}a^{19}+\frac{85541}{252121}a^{18}+\frac{45809}{252121}a^{17}-\frac{55126}{252121}a^{16}-\frac{108273}{252121}a^{15}+\frac{99834}{252121}a^{14}-\frac{45681}{252121}a^{13}-\frac{42349}{252121}a^{12}+\frac{80085}{252121}a^{11}+\frac{89033}{252121}a^{10}-\frac{106426}{252121}a^{9}-\frac{74688}{252121}a^{8}+\frac{20521}{252121}a^{7}+\frac{11244}{252121}a^{6}-\frac{43743}{252121}a^{5}-\frac{84778}{252121}a^{4}-\frac{119463}{252121}a^{3}+\frac{97593}{252121}a^{2}+\frac{37230}{252121}a+\frac{850}{4757}$, $\frac{1}{252121}a^{41}+\frac{23}{252121}a^{38}-\frac{11}{252121}a^{37}-\frac{22}{252121}a^{36}-\frac{30}{252121}a^{35}+\frac{12}{252121}a^{34}+\frac{21}{252121}a^{33}+\frac{103343}{252121}a^{32}+\frac{94744}{252121}a^{31}+\frac{67205}{252121}a^{30}+\frac{14803}{252121}a^{29}-\frac{68590}{252121}a^{28}+\frac{1511}{3763}a^{27}+\frac{82933}{252121}a^{26}-\frac{6942}{252121}a^{25}+\frac{52275}{252121}a^{24}-\frac{85903}{252121}a^{23}+\frac{21511}{252121}a^{22}-\frac{32200}{252121}a^{21}+\frac{111925}{252121}a^{20}+\frac{121788}{252121}a^{19}+\frac{52299}{252121}a^{18}-\frac{5102}{252121}a^{17}-\frac{41928}{252121}a^{16}-\frac{21245}{252121}a^{15}+\frac{118931}{252121}a^{14}-\frac{75397}{252121}a^{13}-\frac{104095}{252121}a^{12}-\frac{91435}{252121}a^{11}+\frac{32552}{252121}a^{10}+\frac{125101}{252121}a^{9}-\frac{3920}{252121}a^{8}-\frac{65671}{252121}a^{7}-\frac{39148}{252121}a^{6}-\frac{116291}{252121}a^{5}+\frac{56482}{252121}a^{4}-\frac{10979}{252121}a^{3}+\frac{22704}{252121}a^{2}-\frac{42260}{252121}a-\frac{1057}{4757}$, $\frac{1}{17900591}a^{42}-\frac{32}{17900591}a^{41}+\frac{5}{17900591}a^{40}-\frac{13}{17900591}a^{39}+\frac{2101}{17900591}a^{38}-\frac{2221}{17900591}a^{37}-\frac{808}{17900591}a^{36}+\frac{1283}{17900591}a^{35}-\frac{2124}{17900591}a^{34}+\frac{793}{17900591}a^{33}-\frac{153521}{337747}a^{32}+\frac{4784526}{17900591}a^{31}+\frac{2890777}{17900591}a^{30}+\frac{4278038}{17900591}a^{29}-\frac{8910688}{17900591}a^{28}+\frac{4891425}{17900591}a^{27}+\frac{2692613}{17900591}a^{26}+\frac{1261175}{17900591}a^{25}-\frac{1840457}{17900591}a^{24}-\frac{3209558}{17900591}a^{23}+\frac{4817043}{17900591}a^{22}+\frac{7488416}{17900591}a^{21}-\frac{1933587}{17900591}a^{20}+\frac{6033524}{17900591}a^{19}+\frac{5832060}{17900591}a^{18}-\frac{6123719}{17900591}a^{17}-\frac{5123064}{17900591}a^{16}-\frac{8818169}{17900591}a^{15}+\frac{3104791}{17900591}a^{14}+\frac{6481214}{17900591}a^{13}+\frac{6152030}{17900591}a^{12}-\frac{1759030}{17900591}a^{11}-\frac{2963551}{17900591}a^{10}+\frac{392007}{17900591}a^{9}-\frac{2533657}{17900591}a^{8}-\frac{2790851}{17900591}a^{7}+\frac{4379603}{17900591}a^{6}-\frac{7646007}{17900591}a^{5}+\frac{1178620}{17900591}a^{4}+\frac{4447066}{17900591}a^{3}+\frac{1200664}{17900591}a^{2}+\frac{844394}{17900591}a-\frac{153813}{337747}$, $\frac{1}{17900591}a^{43}-\frac{25}{17900591}a^{41}+\frac{5}{17900591}a^{40}-\frac{19}{17900591}a^{39}+\frac{756}{17900591}a^{38}+\frac{2031}{17900591}a^{37}+\frac{703}{17900591}a^{36}-\frac{970}{17900591}a^{35}-\frac{2139}{17900591}a^{34}+\frac{7}{337747}a^{33}-\frac{122259}{267173}a^{32}+\frac{3507844}{17900591}a^{31}+\frac{199614}{17900591}a^{30}+\frac{3639613}{17900591}a^{29}+\frac{6772944}{17900591}a^{28}+\frac{7166246}{17900591}a^{27}+\frac{2245097}{17900591}a^{26}+\frac{6522839}{17900591}a^{25}+\frac{1861564}{17900591}a^{24}-\frac{3172044}{17900591}a^{23}-\frac{2325656}{17900591}a^{22}+\frac{6876997}{17900591}a^{21}+\frac{191514}{17900591}a^{20}-\frac{2284535}{17900591}a^{19}+\frac{2599347}{17900591}a^{18}-\frac{104124}{267173}a^{17}-\frac{8169626}{17900591}a^{16}+\frac{734312}{17900591}a^{15}-\frac{3968891}{17900591}a^{14}+\frac{2761037}{17900591}a^{13}-\frac{7940793}{17900591}a^{12}-\frac{3803641}{17900591}a^{11}-\frac{8325654}{17900591}a^{10}+\frac{3745385}{17900591}a^{9}+\frac{2236878}{17900591}a^{8}+\frac{3610720}{17900591}a^{7}-\frac{5182124}{17900591}a^{6}-\frac{2527343}{17900591}a^{5}-\frac{7589776}{17900591}a^{4}-\frac{6105276}{17900591}a^{3}+\frac{624815}{17900591}a^{2}-\frac{4938349}{17900591}a+\frac{149443}{337747}$, $\frac{1}{58194821341}a^{44}+\frac{306}{58194821341}a^{43}+\frac{979}{58194821341}a^{42}+\frac{65804}{58194821341}a^{41}+\frac{37203}{58194821341}a^{40}-\frac{66816}{58194821341}a^{39}+\frac{4406315}{58194821341}a^{38}-\frac{3734216}{58194821341}a^{37}+\frac{6126048}{58194821341}a^{36}-\frac{6590006}{58194821341}a^{35}-\frac{5713066}{58194821341}a^{34}+\frac{6191305}{58194821341}a^{33}-\frac{4650975295}{58194821341}a^{32}-\frac{16950920777}{58194821341}a^{31}+\frac{26049853216}{58194821341}a^{30}-\frac{7238201177}{58194821341}a^{29}+\frac{16905049964}{58194821341}a^{28}+\frac{28375571180}{58194821341}a^{27}+\frac{16890999478}{58194821341}a^{26}-\frac{27834053897}{58194821341}a^{25}-\frac{28634017869}{58194821341}a^{24}-\frac{13055807950}{58194821341}a^{23}-\frac{20959904954}{58194821341}a^{22}+\frac{11053509283}{58194821341}a^{21}+\frac{3122837419}{58194821341}a^{20}+\frac{28769349778}{58194821341}a^{19}-\frac{16644918760}{58194821341}a^{18}+\frac{25929521013}{58194821341}a^{17}-\frac{26682799204}{58194821341}a^{16}+\frac{7985367926}{58194821341}a^{15}-\frac{16646483413}{58194821341}a^{14}-\frac{26035244004}{58194821341}a^{13}+\frac{6852114667}{58194821341}a^{12}+\frac{16948016843}{58194821341}a^{11}-\frac{12955610389}{58194821341}a^{10}+\frac{21910767660}{58194821341}a^{9}+\frac{24029190317}{58194821341}a^{8}+\frac{2408090626}{58194821341}a^{7}-\frac{4843592981}{58194821341}a^{6}+\frac{10275904904}{58194821341}a^{5}+\frac{24652340149}{58194821341}a^{4}-\frac{11962704383}{58194821341}a^{3}-\frac{16585859502}{58194821341}a^{2}-\frac{6441696227}{58194821341}a-\frac{75721253}{1098015497}$, $\frac{1}{35\!\cdots\!47}a^{45}-\frac{2657193}{35\!\cdots\!47}a^{44}-\frac{6930008733}{35\!\cdots\!47}a^{43}-\frac{204546381}{35\!\cdots\!47}a^{42}+\frac{2673870763}{49\!\cdots\!57}a^{41}+\frac{263571318067}{35\!\cdots\!47}a^{40}-\frac{677130996371}{35\!\cdots\!47}a^{39}-\frac{38489790373294}{35\!\cdots\!47}a^{38}-\frac{5660630542343}{35\!\cdots\!47}a^{37}-\frac{91509342495}{66\!\cdots\!99}a^{36}+\frac{1597176005624}{35\!\cdots\!47}a^{35}+\frac{44445377245561}{35\!\cdots\!47}a^{34}-\frac{35051836607279}{35\!\cdots\!47}a^{33}+\frac{73\!\cdots\!43}{35\!\cdots\!47}a^{32}-\frac{16\!\cdots\!63}{35\!\cdots\!47}a^{31}-\frac{18\!\cdots\!46}{35\!\cdots\!47}a^{30}-\frac{17\!\cdots\!17}{35\!\cdots\!47}a^{29}+\frac{81\!\cdots\!47}{35\!\cdots\!47}a^{28}-\frac{80\!\cdots\!81}{35\!\cdots\!47}a^{27}-\frac{56\!\cdots\!87}{35\!\cdots\!47}a^{26}-\frac{95\!\cdots\!20}{35\!\cdots\!47}a^{25}+\frac{96\!\cdots\!73}{35\!\cdots\!47}a^{24}+\frac{54\!\cdots\!48}{35\!\cdots\!47}a^{23}-\frac{43\!\cdots\!74}{35\!\cdots\!47}a^{22}+\frac{25\!\cdots\!52}{66\!\cdots\!99}a^{21}+\frac{68\!\cdots\!51}{35\!\cdots\!47}a^{20}-\frac{12\!\cdots\!98}{35\!\cdots\!47}a^{19}+\frac{33\!\cdots\!73}{35\!\cdots\!47}a^{18}+\frac{52\!\cdots\!66}{35\!\cdots\!47}a^{17}-\frac{10\!\cdots\!09}{35\!\cdots\!47}a^{16}-\frac{13\!\cdots\!94}{35\!\cdots\!47}a^{15}+\frac{17\!\cdots\!66}{35\!\cdots\!47}a^{14}-\frac{16\!\cdots\!80}{35\!\cdots\!47}a^{13}+\frac{99\!\cdots\!30}{35\!\cdots\!47}a^{12}+\frac{10\!\cdots\!19}{35\!\cdots\!47}a^{11}-\frac{10\!\cdots\!14}{35\!\cdots\!47}a^{10}-\frac{40\!\cdots\!04}{35\!\cdots\!47}a^{9}+\frac{38\!\cdots\!62}{35\!\cdots\!47}a^{8}+\frac{14\!\cdots\!19}{35\!\cdots\!47}a^{7}+\frac{10\!\cdots\!99}{35\!\cdots\!47}a^{6}-\frac{14\!\cdots\!75}{35\!\cdots\!47}a^{5}+\frac{14\!\cdots\!37}{35\!\cdots\!47}a^{4}-\frac{73\!\cdots\!63}{35\!\cdots\!47}a^{3}+\frac{17\!\cdots\!14}{35\!\cdots\!47}a^{2}-\frac{14\!\cdots\!00}{66\!\cdots\!99}a-\frac{52818476998726}{125436192361783}$, $\frac{1}{23\!\cdots\!23}a^{46}+\frac{17\!\cdots\!56}{23\!\cdots\!23}a^{45}-\frac{92\!\cdots\!33}{23\!\cdots\!23}a^{44}-\frac{34\!\cdots\!69}{23\!\cdots\!23}a^{43}+\frac{33\!\cdots\!62}{23\!\cdots\!23}a^{42}+\frac{30\!\cdots\!92}{23\!\cdots\!23}a^{41}-\frac{26\!\cdots\!84}{23\!\cdots\!23}a^{40}-\frac{12\!\cdots\!80}{23\!\cdots\!23}a^{39}+\frac{12\!\cdots\!26}{23\!\cdots\!23}a^{38}-\frac{51\!\cdots\!79}{23\!\cdots\!23}a^{37}+\frac{85\!\cdots\!02}{23\!\cdots\!23}a^{36}+\frac{28\!\cdots\!22}{23\!\cdots\!23}a^{35}+\frac{31\!\cdots\!92}{23\!\cdots\!23}a^{34}-\frac{17\!\cdots\!58}{23\!\cdots\!23}a^{33}-\frac{30\!\cdots\!99}{23\!\cdots\!23}a^{32}+\frac{76\!\cdots\!27}{23\!\cdots\!23}a^{31}-\frac{15\!\cdots\!60}{23\!\cdots\!23}a^{30}+\frac{71\!\cdots\!47}{23\!\cdots\!23}a^{29}-\frac{27\!\cdots\!54}{23\!\cdots\!23}a^{28}-\frac{11\!\cdots\!16}{23\!\cdots\!23}a^{27}-\frac{71\!\cdots\!98}{23\!\cdots\!23}a^{26}-\frac{50\!\cdots\!65}{23\!\cdots\!23}a^{25}+\frac{65\!\cdots\!88}{23\!\cdots\!23}a^{24}+\frac{10\!\cdots\!99}{23\!\cdots\!23}a^{23}-\frac{86\!\cdots\!15}{23\!\cdots\!23}a^{22}+\frac{10\!\cdots\!05}{23\!\cdots\!23}a^{21}-\frac{45\!\cdots\!90}{23\!\cdots\!23}a^{20}+\frac{72\!\cdots\!02}{23\!\cdots\!23}a^{19}-\frac{15\!\cdots\!85}{35\!\cdots\!69}a^{18}-\frac{10\!\cdots\!39}{23\!\cdots\!23}a^{17}-\frac{11\!\cdots\!15}{23\!\cdots\!23}a^{16}-\frac{46\!\cdots\!67}{23\!\cdots\!23}a^{15}+\frac{81\!\cdots\!89}{23\!\cdots\!23}a^{14}-\frac{93\!\cdots\!39}{23\!\cdots\!23}a^{13}-\frac{78\!\cdots\!09}{23\!\cdots\!23}a^{12}-\frac{11\!\cdots\!45}{23\!\cdots\!23}a^{11}-\frac{22\!\cdots\!98}{23\!\cdots\!23}a^{10}-\frac{45\!\cdots\!65}{23\!\cdots\!23}a^{9}-\frac{99\!\cdots\!17}{23\!\cdots\!23}a^{8}+\frac{11\!\cdots\!05}{23\!\cdots\!23}a^{7}-\frac{68\!\cdots\!27}{23\!\cdots\!23}a^{6}-\frac{83\!\cdots\!00}{23\!\cdots\!23}a^{5}-\frac{25\!\cdots\!08}{23\!\cdots\!23}a^{4}+\frac{91\!\cdots\!48}{23\!\cdots\!23}a^{3}+\frac{65\!\cdots\!68}{23\!\cdots\!23}a^{2}-\frac{11\!\cdots\!51}{45\!\cdots\!91}a-\frac{28\!\cdots\!44}{84\!\cdots\!47}$ 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sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
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:  $46$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449)
 
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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449, 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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449);
 
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^47 - 1081*x^45 - 705*x^44 + 534061*x^43 + 683944*x^42 - 160057842*x^41 - 299647654*x^40 + 32561335719*x^39 + 78763416061*x^38 - 4767066709455*x^37 - 13908937919967*x^36 + 519679734845940*x^35 + 1750397799006104*x^34 - 43074182887990763*x^33 - 162478138072805487*x^32 + 2749329116777664352*x^31 + 11358660939804373275*x^30 - 136126237738133225032*x^29 - 605463962245369166572*x^28 + 5246767648352840100798*x^27 + 24765274103500484906219*x^26 - 157566416740809178648611*x^25 - 778707155396426126313230*x^24 + 3684240622296757942862071*x^23 + 18785271879388363208006658*x^22 - 66986754039408931327563467*x^21 - 345795459416186142486923619*x^20 + 946170254872545787018973465*x^19 + 4814104180441102948763936108*x^18 - 10375613255284454207094751556*x^17 - 50032174963571133079560800941*x^16 + 88105844627120217056303008521*x^15 + 380945711116472915778482842595*x^14 - 573465045108814971325177493888*x^13 - 2067158816911927150047646995633*x^12 + 2787145282695598886975483579427*x^11 + 7670341992113409376277564204296*x^10 - 9619799236229244848389374409877*x^9 - 18271828994795512973409030766888*x^8 + 21757031287162404406727090902813*x^7 + 25318069661907634555608128065064*x^6 - 28749901083465227418574423514388*x^5 - 17172272828610619745842195295332*x^4 + 19384902113022958413624018194070*x^3 + 2828656546972432377231017394241*x^2 - 5355885371863393108886638462583*x + 1052824394331287344099620777449);
 
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

$C_{47}$ (as 47T1):

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 cyclic group of order 47
The 47 conjugacy class representatives for $C_{47}$
Character table for $C_{47}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ $47$ R ${\href{/padicField/53.1.0.1}{1} }^{47}$ $47$

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
\(47\) Copy content Toggle raw display Deg $47$$47$$1$$92$