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Properties

Label 40.0.40516673261...3361.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 11^{36}\cdot 19^{20}$
Root discriminant $65.34$
Ramified primes $3, 11, 19$
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![95367431640625, -19073486328125, 19073486328125, -10681152343750, 2136230468750, -2136230468750, 433349609375, -86669921875, 86669921875, 36914062500, -7382812500, -9373046875, -4250000000, -1830937500, 356421875, 180687500, 264127500, 35065125, 13763000, -14934780, -6778669, -2986956, 550520, 280521, 422604, 57820, 22811, -23436, -10880, -4799, -756, 756, 355, -71, 71, -70, 14, -14, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 5*x^38 - 14*x^37 + 14*x^36 - 70*x^35 + 71*x^34 - 71*x^33 + 355*x^32 + 756*x^31 - 756*x^30 - 4799*x^29 - 10880*x^28 - 23436*x^27 + 22811*x^26 + 57820*x^25 + 422604*x^24 + 280521*x^23 + 550520*x^22 - 2986956*x^21 - 6778669*x^20 - 14934780*x^19 + 13763000*x^18 + 35065125*x^17 + 264127500*x^16 + 180687500*x^15 + 356421875*x^14 - 1830937500*x^13 - 4250000000*x^12 - 9373046875*x^11 - 7382812500*x^10 + 36914062500*x^9 + 86669921875*x^8 - 86669921875*x^7 + 433349609375*x^6 - 2136230468750*x^5 + 2136230468750*x^4 - 10681152343750*x^3 + 19073486328125*x^2 - 19073486328125*x + 95367431640625)
 
gp: K = bnfinit(x^40 - x^39 + 5*x^38 - 14*x^37 + 14*x^36 - 70*x^35 + 71*x^34 - 71*x^33 + 355*x^32 + 756*x^31 - 756*x^30 - 4799*x^29 - 10880*x^28 - 23436*x^27 + 22811*x^26 + 57820*x^25 + 422604*x^24 + 280521*x^23 + 550520*x^22 - 2986956*x^21 - 6778669*x^20 - 14934780*x^19 + 13763000*x^18 + 35065125*x^17 + 264127500*x^16 + 180687500*x^15 + 356421875*x^14 - 1830937500*x^13 - 4250000000*x^12 - 9373046875*x^11 - 7382812500*x^10 + 36914062500*x^9 + 86669921875*x^8 - 86669921875*x^7 + 433349609375*x^6 - 2136230468750*x^5 + 2136230468750*x^4 - 10681152343750*x^3 + 19073486328125*x^2 - 19073486328125*x + 95367431640625, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 5 x^{38} - 14 x^{37} + 14 x^{36} - 70 x^{35} + 71 x^{34} - 71 x^{33} + 355 x^{32} + 756 x^{31} - 756 x^{30} - 4799 x^{29} - 10880 x^{28} - 23436 x^{27} + 22811 x^{26} + 57820 x^{25} + 422604 x^{24} + 280521 x^{23} + 550520 x^{22} - 2986956 x^{21} - 6778669 x^{20} - 14934780 x^{19} + 13763000 x^{18} + 35065125 x^{17} + 264127500 x^{16} + 180687500 x^{15} + 356421875 x^{14} - 1830937500 x^{13} - 4250000000 x^{12} - 9373046875 x^{11} - 7382812500 x^{10} + 36914062500 x^{9} + 86669921875 x^{8} - 86669921875 x^{7} + 433349609375 x^{6} - 2136230468750 x^{5} + 2136230468750 x^{4} - 10681152343750 x^{3} + 19073486328125 x^{2} - 19073486328125 x + 95367431640625 \)

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:  \(4051667326120348106672526475600319761775876344932176849950811746241033361=3^{20}\cdot 11^{36}\cdot 19^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.34$
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:  $3, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(627=3\cdot 11\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{627}(512,·)$, $\chi_{627}(1,·)$, $\chi_{627}(514,·)$, $\chi_{627}(134,·)$, $\chi_{627}(265,·)$, $\chi_{627}(398,·)$, $\chi_{627}(400,·)$, $\chi_{627}(20,·)$, $\chi_{627}(533,·)$, $\chi_{627}(151,·)$, $\chi_{627}(284,·)$, $\chi_{627}(419,·)$, $\chi_{627}(37,·)$, $\chi_{627}(170,·)$, $\chi_{627}(172,·)$, $\chi_{627}(305,·)$, $\chi_{627}(436,·)$, $\chi_{627}(56,·)$, $\chi_{627}(569,·)$, $\chi_{627}(58,·)$, $\chi_{627}(571,·)$, $\chi_{627}(191,·)$, $\chi_{627}(322,·)$, $\chi_{627}(455,·)$, $\chi_{627}(457,·)$, $\chi_{627}(590,·)$, $\chi_{627}(208,·)$, $\chi_{627}(343,·)$, $\chi_{627}(476,·)$, $\chi_{627}(94,·)$, $\chi_{627}(607,·)$, $\chi_{627}(227,·)$, $\chi_{627}(229,·)$, $\chi_{627}(362,·)$, $\chi_{627}(493,·)$, $\chi_{627}(113,·)$, $\chi_{627}(626,·)$, $\chi_{627}(115,·)$, $\chi_{627}(248,·)$, $\chi_{627}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5} a^{21} - \frac{1}{5} a^{20} + \frac{1}{5} a^{18} - \frac{1}{5} a^{17} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{63275} a^{22} - \frac{1}{25} a^{21} + \frac{2}{5} a^{20} + \frac{6}{25} a^{19} - \frac{11}{25} a^{18} + \frac{2}{5} a^{17} - \frac{9}{25} a^{16} + \frac{4}{25} a^{15} + \frac{2}{5} a^{14} + \frac{1}{25} a^{13} - \frac{6}{25} a^{12} + \frac{9631}{63275} a^{11} - \frac{2}{5} a^{10} - \frac{6}{25} a^{9} + \frac{11}{25} a^{8} - \frac{2}{5} a^{7} + \frac{9}{25} a^{6} - \frac{4}{25} a^{5} - \frac{2}{5} a^{4} - \frac{1}{25} a^{3} + \frac{6}{25} a^{2} - \frac{2}{5} a - \frac{807}{2531}$, $\frac{1}{316375} a^{23} - \frac{1}{316375} a^{22} + \frac{1}{25} a^{21} + \frac{31}{125} a^{20} - \frac{56}{125} a^{19} + \frac{11}{25} a^{18} - \frac{59}{125} a^{17} + \frac{34}{125} a^{16} - \frac{4}{25} a^{15} - \frac{49}{125} a^{14} + \frac{24}{125} a^{13} - \frac{3024}{316375} a^{12} - \frac{16606}{63275} a^{11} + \frac{19}{125} a^{10} + \frac{6}{125} a^{9} - \frac{11}{25} a^{8} - \frac{16}{125} a^{7} + \frac{41}{125} a^{6} + \frac{4}{25} a^{5} - \frac{26}{125} a^{4} + \frac{51}{125} a^{3} - \frac{6}{25} a^{2} - \frac{5869}{12655} a - \frac{851}{2531}$, $\frac{1}{1581875} a^{24} - \frac{1}{1581875} a^{23} + \frac{1}{316375} a^{22} + \frac{56}{625} a^{21} - \frac{56}{625} a^{20} + \frac{56}{125} a^{19} - \frac{284}{625} a^{18} + \frac{284}{625} a^{17} - \frac{34}{125} a^{16} + \frac{101}{625} a^{15} - \frac{101}{625} a^{14} + \frac{566451}{1581875} a^{13} - \frac{3951}{316375} a^{12} + \frac{336689}{1581875} a^{11} + \frac{6}{625} a^{10} - \frac{6}{125} a^{9} + \frac{209}{625} a^{8} - \frac{209}{625} a^{7} - \frac{41}{125} a^{6} - \frac{301}{625} a^{5} + \frac{301}{625} a^{4} - \frac{51}{125} a^{3} - \frac{336}{2531} a^{2} + \frac{336}{2531} a + \frac{851}{2531}$, $\frac{1}{7909375} a^{25} - \frac{1}{7909375} a^{24} + \frac{1}{1581875} a^{23} - \frac{14}{7909375} a^{22} - \frac{181}{3125} a^{21} - \frac{69}{625} a^{20} - \frac{784}{3125} a^{19} - \frac{1091}{3125} a^{18} + \frac{216}{625} a^{17} + \frac{851}{3125} a^{16} + \frac{399}{3125} a^{15} - \frac{1015424}{7909375} a^{14} - \frac{573426}{1581875} a^{13} + \frac{3184064}{7909375} a^{12} - \frac{3184689}{7909375} a^{11} + \frac{119}{625} a^{10} + \frac{709}{3125} a^{9} + \frac{1166}{3125} a^{8} - \frac{291}{625} a^{7} + \frac{199}{3125} a^{6} - \frac{1449}{3125} a^{5} - \frac{176}{625} a^{4} - \frac{29521}{63275} a^{3} + \frac{4211}{63275} a^{2} + \frac{851}{12655} a - \frac{1084}{2531}$, $\frac{1}{39546875} a^{26} - \frac{1}{39546875} a^{25} + \frac{1}{7909375} a^{24} - \frac{14}{39546875} a^{23} + \frac{14}{39546875} a^{22} + \frac{181}{3125} a^{21} + \frac{5466}{15625} a^{20} + \frac{784}{15625} a^{19} + \frac{1091}{3125} a^{18} - \frac{5399}{15625} a^{17} - \frac{3976}{15625} a^{16} - \frac{5761049}{39546875} a^{15} + \frac{1008449}{7909375} a^{14} + \frac{7929689}{39546875} a^{13} + \frac{15797811}{39546875} a^{12} + \frac{3219064}{7909375} a^{11} - \frac{2416}{15625} a^{10} - \frac{3834}{15625} a^{9} - \frac{1166}{3125} a^{8} + \frac{3324}{15625} a^{7} + \frac{6051}{15625} a^{6} + \frac{1449}{3125} a^{5} - \frac{156071}{316375} a^{4} + \frac{29521}{316375} a^{3} - \frac{4211}{63275} a^{2} + \frac{3978}{12655} a + \frac{723}{2531}$, $\frac{1}{197734375} a^{27} - \frac{1}{197734375} a^{26} + \frac{1}{39546875} a^{25} - \frac{14}{197734375} a^{24} + \frac{14}{197734375} a^{23} - \frac{14}{39546875} a^{22} - \frac{784}{78125} a^{21} + \frac{784}{78125} a^{20} - \frac{784}{15625} a^{19} - \frac{11649}{78125} a^{18} + \frac{11649}{78125} a^{17} + \frac{49604576}{197734375} a^{16} + \frac{13663449}{39546875} a^{15} - \frac{71164061}{197734375} a^{14} - \frac{47477189}{197734375} a^{13} - \frac{7854061}{39546875} a^{12} + \frac{79125729}{197734375} a^{11} + \frac{27416}{78125} a^{10} + \frac{3834}{15625} a^{9} + \frac{9574}{78125} a^{8} - \frac{9574}{78125} a^{7} - \frac{6051}{15625} a^{6} - \frac{345896}{1581875} a^{5} + \frac{345896}{1581875} a^{4} - \frac{29521}{316375} a^{3} - \frac{3254}{12655} a^{2} + \frac{3254}{12655} a - \frac{723}{2531}$, $\frac{1}{988671875} a^{28} - \frac{1}{988671875} a^{27} + \frac{1}{197734375} a^{26} - \frac{14}{988671875} a^{25} + \frac{14}{988671875} a^{24} - \frac{14}{197734375} a^{23} + \frac{71}{988671875} a^{22} - \frac{30466}{390625} a^{21} - \frac{16409}{78125} a^{20} + \frac{19601}{390625} a^{19} - \frac{175851}{390625} a^{18} + \frac{445073326}{988671875} a^{17} - \frac{49611551}{197734375} a^{16} + \frac{47476564}{988671875} a^{15} - \frac{47477189}{988671875} a^{14} + \frac{47511564}{197734375} a^{13} - \frac{395436771}{988671875} a^{12} + \frac{396139896}{988671875} a^{11} - \frac{11791}{78125} a^{10} - \frac{99801}{390625} a^{9} - \frac{134574}{390625} a^{8} + \frac{25199}{78125} a^{7} + \frac{2185104}{7909375} a^{6} + \frac{978646}{7909375} a^{5} - \frac{29521}{1581875} a^{4} + \frac{4339}{63275} a^{3} + \frac{20971}{63275} a^{2} - \frac{723}{12655} a - \frac{1249}{2531}$, $\frac{1}{4943359375} a^{29} - \frac{1}{4943359375} a^{28} + \frac{1}{988671875} a^{27} - \frac{14}{4943359375} a^{26} + \frac{14}{4943359375} a^{25} - \frac{14}{988671875} a^{24} + \frac{71}{4943359375} a^{23} - \frac{71}{4943359375} a^{22} + \frac{30466}{390625} a^{21} + \frac{410226}{1953125} a^{20} + \frac{761649}{1953125} a^{19} + \frac{49604576}{4943359375} a^{18} - \frac{445080301}{988671875} a^{17} + \frac{1431617189}{4943359375} a^{16} + \frac{545725936}{4943359375} a^{15} + \frac{47511564}{988671875} a^{14} + \frac{988703854}{4943359375} a^{13} + \frac{989343021}{4943359375} a^{12} - \frac{395749271}{988671875} a^{11} + \frac{290824}{1953125} a^{10} + \frac{490426}{1953125} a^{9} + \frac{134574}{390625} a^{8} - \frac{13633646}{39546875} a^{7} - \frac{10094479}{39546875} a^{6} - \frac{978646}{7909375} a^{5} + \frac{67614}{316375} a^{4} + \frac{58936}{316375} a^{3} - \frac{20971}{63275} a^{2} + \frac{1282}{12655} a + \frac{756}{2531}$, $\frac{1}{98867187500} a^{30} + \frac{1}{24716796875} a^{29} + \frac{11}{98867187500} a^{27} - \frac{14}{24716796875} a^{26} - \frac{279}{98867187500} a^{24} + \frac{71}{24716796875} a^{23} + \frac{1}{39062500} a^{21} + \frac{3144601}{9765625} a^{20} - \frac{1779787301}{24716796875} a^{19} - \frac{279}{790937500} a^{18} + \frac{995461}{9765625} a^{17} + \frac{5139136564}{24716796875} a^{16} + \frac{11}{6327500} a^{15} + \frac{3144671}{9765625} a^{14} + \frac{2224719479}{24716796875} a^{13} + \frac{1}{50620} a^{12} - \frac{2442394}{9765625} a^{11} - \frac{2443551}{9765625} a^{10} + \frac{1}{4} a^{9} - \frac{248265359}{988671875} a^{8} + \frac{49641354}{197734375} a^{7} - \frac{1}{4} a^{6} + \frac{2054481}{7909375} a^{5} - \frac{375311}{1581875} a^{4} + \frac{1}{4} a^{3} - \frac{15679}{63275} a^{2} - \frac{756}{12655} a - \frac{1}{4}$, $\frac{1}{3061313212695312500} a^{31} + \frac{3143491}{765328303173828125} a^{30} + \frac{919497}{153065660634765625} a^{29} + \frac{31905911}{3061313212695312500} a^{28} - \frac{5303999}{765328303173828125} a^{27} - \frac{12872958}{153065660634765625} a^{26} - \frac{446682879}{3061313212695312500} a^{25} - \frac{318680389}{765328303173828125} a^{24} + \frac{65284287}{153065660634765625} a^{23} + \frac{2265321431}{3061313212695312500} a^{22} - \frac{30202335304909}{302381787109375} a^{21} + \frac{153299751813098859}{765328303173828125} a^{20} + \frac{82655070398202037}{612262642539062500} a^{19} - \frac{207181838254426559}{765328303173828125} a^{18} + \frac{234354591250403174}{765328303173828125} a^{17} - \frac{43897445894037393}{612262642539062500} a^{16} + \frac{210556716871306701}{765328303173828125} a^{15} - \frac{248912462101986186}{765328303173828125} a^{14} + \frac{78882723366268877}{612262642539062500} a^{13} - \frac{9913349609817689}{765328303173828125} a^{12} + \frac{211612871037566104}{765328303173828125} a^{11} + \frac{48373197572737}{241905429687500} a^{10} - \frac{1731968928513}{30613132126953125} a^{9} - \frac{245110605052164}{1224525285078125} a^{8} + \frac{980384367768877}{4898101140312500} a^{7} + \frac{66287854742}{244905057015625} a^{6} - \frac{1933701179774}{9796202280625} a^{5} + \frac{7911101787107}{39184809122500} a^{4} + \frac{6431511697}{1959240456125} a^{3} - \frac{16889938308}{78369618245} a^{2} + \frac{48278901133}{313478472980} a - \frac{1250632148}{15673923649}$, $\frac{1}{15306566063476562500} a^{32} - \frac{1}{15306566063476562500} a^{31} + \frac{4916541}{3061313212695312500} a^{30} + \frac{222186411}{15306566063476562500} a^{29} - \frac{123855611}{15306566063476562500} a^{28} + \frac{177937551}{3061313212695312500} a^{27} - \frac{3110609879}{15306566063476562500} a^{26} + \frac{1733978679}{15306566063476562500} a^{25} - \frac{3105693339}{3061313212695312500} a^{24} + \frac{15775236931}{15306566063476562500} a^{23} - \frac{8793750131}{15306566063476562500} a^{22} - \frac{726715075501897349}{15306566063476562500} a^{21} - \frac{1342375322645649091}{3061313212695312500} a^{20} + \frac{6052407986241625139}{15306566063476562500} a^{19} + \frac{3201252595609669061}{15306566063476562500} a^{18} - \frac{759438202867464001}{3061313212695312500} a^{17} + \frac{1071416161508381929}{15306566063476562500} a^{16} + \frac{102149908772114271}{15306566063476562500} a^{15} + \frac{872883834522507389}{3061313212695312500} a^{14} - \frac{6222521367492364381}{15306566063476562500} a^{13} - \frac{3615955523627607419}{15306566063476562500} a^{12} + \frac{53944932344459179}{3061313212695312500} a^{11} + \frac{306083076934492753}{612262642539062500} a^{10} - \frac{61220885588520409}{122452528507812500} a^{9} + \frac{12235557894947621}{24490505701562500} a^{8} - \frac{2447204109767277}{4898101140312500} a^{7} + \frac{489604255806181}{979620228062500} a^{6} - \frac{97452765881089}{195924045612500} a^{5} - \frac{19413253444507}{39184809122500} a^{4} + \frac{3898507710371}{7836961824500} a^{3} - \frac{763173311399}{1567392364900} a^{2} + \frac{121902628303}{313478472980} a - \frac{27463982887}{62695694596}$, $\frac{1}{76532830317382812500} a^{33} - \frac{1}{76532830317382812500} a^{32} + \frac{1}{15306566063476562500} a^{31} - \frac{36214639}{76532830317382812500} a^{30} + \frac{1461495889}{76532830317382812500} a^{29} - \frac{321270889}{15306566063476562500} a^{28} + \frac{7633411071}{76532830317382812500} a^{27} - \frac{20460942321}{76532830317382812500} a^{26} + \frac{4497792321}{15306566063476562500} a^{25} - \frac{102340925119}{76532830317382812500} a^{24} + \frac{103766206369}{76532830317382812500} a^{23} + \frac{210520470318951}{76532830317382812500} a^{22} - \frac{1180540528059981601}{15306566063476562500} a^{21} - \frac{9245780772306407561}{76532830317382812500} a^{20} - \frac{27928843833650257439}{76532830317382812500} a^{19} - \frac{1897063201535423811}{15306566063476562500} a^{18} + \frac{15033580224723014729}{76532830317382812500} a^{17} + \frac{11682572950948735271}{76532830317382812500} a^{16} + \frac{5754224130751445979}{15306566063476562500} a^{15} + \frac{26858509583584988919}{76532830317382812500} a^{14} + \frac{36637754245538948581}{76532830317382812500} a^{13} + \frac{3508101726642107669}{15306566063476562500} a^{12} + \frac{74515950126602181}{612262642539062500} a^{11} - \frac{91902864613358181}{612262642539062500} a^{10} + \frac{13483729933586131}{122452528507812500} a^{9} + \frac{438293089233331}{4898101140312500} a^{8} + \frac{1716764525076129}{4898101140312500} a^{7} + \frac{205186269374821}{979620228062500} a^{6} - \frac{302930765579}{39184809122500} a^{5} - \frac{5642037616661}{39184809122500} a^{4} + \frac{2377649377831}{7836961824500} a^{3} - \frac{120766326709}{1567392364900} a^{2} + \frac{63887877617}{313478472980} a + \frac{12090905153}{31347847298}$, $\frac{1}{382664151586914062500} a^{34} - \frac{1}{382664151586914062500} a^{33} + \frac{1}{76532830317382812500} a^{32} - \frac{7}{191332075793457031250} a^{31} + \frac{71675639}{382664151586914062500} a^{30} + \frac{5440561111}{76532830317382812500} a^{29} - \frac{13458051527}{191332075793457031250} a^{28} + \frac{135368947429}{382664151586914062500} a^{27} - \frac{76167855679}{76532830317382812500} a^{26} + \frac{188412722253}{191332075793457031250} a^{25} - \frac{1904124718881}{382664151586914062500} a^{24} + \frac{212565920678951}{382664151586914062500} a^{23} - \frac{21254556481713}{38266415158691406250} a^{22} - \frac{9221581597600922811}{382664151586914062500} a^{21} - \frac{74388847946885955939}{382664151586914062500} a^{20} + \frac{2726039593129629907}{38266415158691406250} a^{19} + \frac{156391473882912619479}{382664151586914062500} a^{18} + \frac{172469737929811364271}{382664151586914062500} a^{17} - \frac{11946953290397785823}{38266415158691406250} a^{16} + \frac{111209520100080866169}{382664151586914062500} a^{15} - \frac{3925066221067732419}{382664151586914062500} a^{14} + \frac{17834472717822294397}{38266415158691406250} a^{13} - \frac{114080821301165821}{3061313212695312500} a^{12} - \frac{957223216692601409}{3061313212695312500} a^{11} - \frac{1737962068392413}{61226264253906250} a^{10} + \frac{3475426112918213}{122452528507812500} a^{9} - \frac{3475929512082623}{24490505701562500} a^{8} - \frac{51995226441783}{489810114031250} a^{7} + \frac{104009513790583}{979620228062500} a^{6} + \frac{91949916514107}{195924045612500} a^{5} + \frac{134757347369}{3918480912250} a^{4} - \frac{267665325391}{7836961824500} a^{3} + \frac{267729433181}{1567392364900} a^{2} - \frac{15625619861}{62695694596} a + \frac{7633001619}{31347847298}$, $\frac{1}{1913320757934570312500} a^{35} - \frac{1}{1913320757934570312500} a^{34} + \frac{1}{382664151586914062500} a^{33} - \frac{7}{956660378967285156250} a^{32} + \frac{7}{956660378967285156250} a^{31} + \frac{399078743}{191332075793457031250} a^{30} + \frac{36530285973}{956660378967285156250} a^{29} - \frac{28548710973}{956660378967285156250} a^{28} + \frac{32938577223}{191332075793457031250} a^{27} - \frac{511424002747}{956660378967285156250} a^{26} + \frac{399681952747}{956660378967285156250} a^{25} + \frac{102762136117913}{956660378967285156250} a^{24} - \frac{20544722089213}{191332075793457031250} a^{23} + \frac{524559345222657}{956660378967285156250} a^{22} + \frac{82246386837233584843}{956660378967285156250} a^{21} - \frac{77228008016025057343}{191332075793457031250} a^{20} + \frac{197893576028916370677}{956660378967285156250} a^{19} - \frac{138810249110001539427}{956660378967285156250} a^{18} - \frac{30347059254366264323}{191332075793457031250} a^{17} - \frac{349415989015190927853}{956660378967285156250} a^{16} + \frac{272488543591389852853}{956660378967285156250} a^{15} - \frac{62240185488962787853}{191332075793457031250} a^{14} + \frac{2140274714578081077}{7653283031738281250} a^{13} + \frac{965253476140776823}{7653283031738281250} a^{12} + \frac{762517784075508147}{1530656606347656250} a^{11} - \frac{74881950798275789}{306131321269531250} a^{10} + \frac{13602148920020423}{61226264253906250} a^{9} - \frac{5787582348908759}{12245252850781250} a^{8} + \frac{1157400902104351}{2449050570156250} a^{7} - \frac{175731623209807}{489810114031250} a^{6} + \frac{5709828940441}{97962022806250} a^{5} - \frac{791004119481}{19592404561250} a^{4} + \frac{197962284217}{783696182450} a^{3} + \frac{88482425847}{1567392364900} a^{2} - \frac{94462816099}{313478472980} a + \frac{17145167227}{62695694596}$, $\frac{1}{9566603789672851562500} a^{36} - \frac{1}{9566603789672851562500} a^{35} + \frac{1}{1913320757934570312500} a^{34} - \frac{7}{4783301894836425781250} a^{33} + \frac{7}{4783301894836425781250} a^{32} - \frac{7}{956660378967285156250} a^{31} + \frac{46178750071}{9566603789672851562500} a^{30} + \frac{112076914027}{4783301894836425781250} a^{29} - \frac{3943882777}{956660378967285156250} a^{28} + \frac{705160391381}{9566603789672851562500} a^{27} - \frac{1569076797253}{4783301894836425781250} a^{26} + \frac{105593332536663}{4783301894836425781250} a^{25} - \frac{45255822142801}{1913320757934570312500} a^{24} + \frac{534543764597657}{4783301894836425781250} a^{23} - \frac{1475841728777657}{4783301894836425781250} a^{22} - \frac{137885478969374785311}{1913320757934570312500} a^{21} - \frac{1265475066530839085573}{4783301894836425781250} a^{20} - \frac{710974654106249039427}{4783301894836425781250} a^{19} - \frac{927956775746750405521}{1913320757934570312500} a^{18} + \frac{71305406672722022147}{4783301894836425781250} a^{17} + \frac{1924367295479867352853}{4783301894836425781250} a^{16} - \frac{386052395326771158831}{1913320757934570312500} a^{15} - \frac{5122522946033025573}{38266415158691406250} a^{14} - \frac{2198196357773546927}{38266415158691406250} a^{13} + \frac{7241614638903660979}{15306566063476562500} a^{12} + \frac{3279316573922687}{306131321269531250} a^{11} + \frac{136005821950448433}{306131321269531250} a^{10} - \frac{45833690141478259}{122452528507812500} a^{9} - \frac{2771365370228011}{12245252850781250} a^{8} - \frac{663671886680647}{2449050570156250} a^{7} + \frac{127715966447551}{979620228062500} a^{6} + \frac{26911113868777}{97962022806250} a^{5} + \frac{1017942577929}{3918480912250} a^{4} + \frac{1255879603583}{3918480912250} a^{3} + \frac{134397975449}{1567392364900} a^{2} + \frac{9285048243}{313478472980} a + \frac{13564823733}{62695694596}$, $\frac{1}{47833018948364257812500} a^{37} - \frac{1}{47833018948364257812500} a^{36} + \frac{1}{9566603789672851562500} a^{35} - \frac{7}{23916509474182128906250} a^{34} + \frac{7}{23916509474182128906250} a^{33} - \frac{7}{4783301894836425781250} a^{32} + \frac{71}{47833018948364257812500} a^{31} + \frac{241265468679}{47833018948364257812500} a^{30} - \frac{289121351527}{4783301894836425781250} a^{29} + \frac{3856275391381}{47833018948364257812500} a^{28} - \frac{16627456797631}{47833018948364257812500} a^{27} + \frac{125555755349163}{23916509474182128906250} a^{26} - \frac{52924475392801}{9566603789672851562500} a^{25} + \frac{1255798818960939}{47833018948364257812500} a^{24} - \frac{1577079730183907}{23916509474182128906250} a^{23} + \frac{644535770714689}{9566603789672851562500} a^{22} - \frac{3012494056828514186771}{47833018948364257812500} a^{21} + \frac{986526965091139866823}{23916509474182128906250} a^{20} + \frac{2662027358454229531979}{9566603789672851562500} a^{19} - \frac{268839105238231502581}{47833018948364257812500} a^{18} + \frac{11645215073697881259103}{23916509474182128906250} a^{17} + \frac{651247612166270653669}{9566603789672851562500} a^{16} + \frac{134717117169227563229}{382664151586914062500} a^{15} - \frac{71519290448490166927}{191332075793457031250} a^{14} + \frac{20575433023507727979}{76532830317382812500} a^{13} - \frac{692351430266385061}{3061313212695312500} a^{12} + \frac{565209896116598623}{1530656606347656250} a^{11} + \frac{107430120458844237}{612262642539062500} a^{10} + \frac{10394416385389621}{24490505701562500} a^{9} + \frac{3395787961642569}{12245252850781250} a^{8} - \frac{2084013360094913}{4898101140312500} a^{7} + \frac{25467864625411}{979620228062500} a^{6} + \frac{26927307012701}{97962022806250} a^{5} - \frac{5969368304579}{19592404561250} a^{4} - \frac{174309127331}{1959240456125} a^{3} - \frac{241472606933}{1567392364900} a^{2} + \frac{28194484225}{62695694596} a - \frac{261038767}{62695694596}$, $\frac{1}{239165094741821289062500} a^{38} - \frac{1}{239165094741821289062500} a^{37} + \frac{1}{47833018948364257812500} a^{36} - \frac{7}{119582547370910644531250} a^{35} + \frac{7}{119582547370910644531250} a^{34} - \frac{7}{23916509474182128906250} a^{33} + \frac{71}{239165094741821289062500} a^{32} - \frac{71}{239165094741821289062500} a^{31} - \frac{73716562429}{47833018948364257812500} a^{30} - \frac{8094907421119}{239165094741821289062500} a^{29} + \frac{6620576171119}{239165094741821289062500} a^{28} + \frac{173477229682701}{239165094741821289062500} a^{27} - \frac{19461163517801}{47833018948364257812500} a^{26} + \frac{960484540992189}{239165094741821289062500} a^{25} - \frac{2382608364039689}{239165094741821289062500} a^{24} + \frac{474828974777189}{47833018948364257812500} a^{23} - \frac{14274355595280521}{239165094741821289062500} a^{22} - \frac{2154458561290910656979}{239165094741821289062500} a^{21} + \frac{15531080325445066281979}{47833018948364257812500} a^{20} - \frac{19111748264792732283831}{239165094741821289062500} a^{19} - \frac{19378634534862623966169}{239165094741821289062500} a^{18} + \frac{17369257371096076466169}{47833018948364257812500} a^{17} + \frac{909151968260758681979}{1913320757934570312500} a^{16} + \frac{727461971663954030521}{1913320757934570312500} a^{15} - \frac{103124036347350085521}{382664151586914062500} a^{14} + \frac{5149267482707198389}{15306566063476562500} a^{13} - \frac{2979494864844672189}{15306566063476562500} a^{12} + \frac{1170988035337722849}{3061313212695312500} a^{11} + \frac{82751294948934123}{612262642539062500} a^{10} + \frac{39795324599408541}{122452528507812500} a^{9} + \frac{1129983651779571}{24490505701562500} a^{8} - \frac{225793623056251}{4898101140312500} a^{7} + \frac{8887698690991}{39184809122500} a^{6} - \frac{42340017637757}{97962022806250} a^{5} + \frac{8159600335067}{19592404561250} a^{4} - \frac{19893517957}{156739236490} a^{3} + \frac{444294736621}{1567392364900} a^{2} - \frac{4280298121}{62695694596} a + \frac{26647716453}{62695694596}$, $\frac{1}{1195825473709106445312500} a^{39} - \frac{1}{1195825473709106445312500} a^{38} + \frac{1}{239165094741821289062500} a^{37} - \frac{7}{597912736854553222656250} a^{36} + \frac{7}{597912736854553222656250} a^{35} - \frac{7}{119582547370910644531250} a^{34} + \frac{71}{1195825473709106445312500} a^{33} - \frac{71}{1195825473709106445312500} a^{32} + \frac{71}{239165094741821289062500} a^{31} + \frac{2920544922253}{597912736854553222656250} a^{30} + \frac{80668724608619}{1195825473709106445312500} a^{29} + \frac{153330156245201}{1195825473709106445312500} a^{28} + \frac{14013929295787}{119582547370910644531250} a^{27} - \frac{76189537132811}{1195825473709106445312500} a^{26} - \frac{2146622187477189}{1195825473709106445312500} a^{25} - \frac{269208633197343}{119582547370910644531250} a^{24} - \frac{9016937056218021}{1195825473709106445312500} a^{23} + \frac{10886441094030521}{1195825473709106445312500} a^{22} + \frac{11121982124296402008177}{119582547370910644531250} a^{21} + \frac{138717780597236709903669}{1195825473709106445312500} a^{20} - \frac{216967821117855303653669}{1195825473709106445312500} a^{19} + \frac{14746847168685879365897}{119582547370910644531250} a^{18} + \frac{208463020670753344479}{9566603789672851562500} a^{17} + \frac{4578892377921301343021}{9566603789672851562500} a^{16} - \frac{451663009783298523073}{956660378967285156250} a^{15} + \frac{14902351667724133689}{76532830317382812500} a^{14} - \frac{1618662462468321189}{76532830317382812500} a^{13} + \frac{1491027031886930907}{7653283031738281250} a^{12} - \frac{275454339455817229}{612262642539062500} a^{11} + \frac{131428917371353729}{612262642539062500} a^{10} + \frac{30570516747395673}{61226264253906250} a^{9} + \frac{2427486373612281}{24490505701562500} a^{8} - \frac{504645961714417}{4898101140312500} a^{7} - \frac{486548068683989}{979620228062500} a^{6} + \frac{10419029044423}{97962022806250} a^{5} - \frac{1094841296561}{19592404561250} a^{4} - \frac{900495979556}{1959240456125} a^{3} + \frac{189739613751}{1567392364900} a^{2} - \frac{21257295515}{62695694596} a - \frac{30196178811}{62695694596}$

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:  \( \frac{1111}{24490505701562500} a^{39} - \frac{1111}{24490505701562500} a^{38} + \frac{1111}{4898101140312500} a^{37} - \frac{7777}{12245252850781250} a^{36} + \frac{7777}{12245252850781250} a^{35} - \frac{7777}{2449050570156250} a^{34} + \frac{78881}{24490505701562500} a^{33} + \frac{1889811}{153065660634765625} a^{32} + \frac{78881}{4898101140312500} a^{31} + \frac{209979}{6122626425390625} a^{30} - \frac{209979}{6122626425390625} a^{29} - \frac{5331689}{24490505701562500} a^{28} - \frac{604384}{1224525285078125} a^{27} - \frac{6509349}{6122626425390625} a^{26} + \frac{25343021}{24490505701562500} a^{25} + \frac{3211901}{1224525285078125} a^{24} + \frac{117378261}{6122626425390625} a^{23} + \frac{311658831}{24490505701562500} a^{22} - \frac{3057504219}{153065660634765625} a^{21} - \frac{829627029}{6122626425390625} a^{20} - \frac{7531101259}{24490505701562500} a^{19} - \frac{829627029}{1224525285078125} a^{18} + \frac{30581386}{48981011403125} a^{17} + \frac{311658831}{195924045612500} a^{16} + \frac{117378261}{9796202280625} a^{15} + \frac{3211901}{391848091225} a^{14} + \frac{25343021}{1567392364900} a^{13} - \frac{6509349}{78369618245} a^{12} - \frac{3021920}{15673923649} a^{11} - \frac{266141636614421}{612262642539062500} a^{10} - \frac{5249475}{15673923649} a^{9} + \frac{26247375}{15673923649} a^{8} + \frac{246503125}{62695694596} a^{7} - \frac{246503125}{62695694596} a^{6} + \frac{1232515625}{62695694596} a^{5} - \frac{3037890625}{31347847298} a^{4} + \frac{3037890625}{31347847298} a^{3} - \frac{15189453125}{31347847298} a^{2} + \frac{54248046875}{62695694596} a - \frac{54248046875}{62695694596} \) (order $66$)
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

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-627}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{57}) \), \(\Q(\sqrt{209}) \), \(\Q(\sqrt{-19}, \sqrt{33})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{33}, \sqrt{57})\), \(\Q(\sqrt{-3}, \sqrt{-19})\), \(\Q(\sqrt{-11}, \sqrt{-19})\), \(\Q(\sqrt{-3}, \sqrt{209})\), \(\Q(\sqrt{-11}, \sqrt{57})\), \(\Q(\zeta_{11})^+\), 8.0.154550410641.1, \(\Q(\zeta_{33})^+\), 10.0.530773810885219.1, 10.0.1418758396496190387.1, 10.0.52089208083.1, \(\Q(\zeta_{11})\), 10.10.128978036045108217.1, 10.10.5838511919737409.1, 20.0.2012875387628441371387683404151209769.2, \(\Q(\zeta_{33})\), 20.20.2012875387628441371387683404151209769.1, 20.0.16635333782053234474278375240919089.2, 20.0.34088221436915805032899514033281.2, 20.0.2012875387628441371387683404151209769.1, 20.0.2012875387628441371387683404151209769.4

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
3Data not computed
11Data not computed
19Data not computed