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Properties

Label 40.0.11302165783...0000.6
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}$
Root discriminant $67.04$
Ramified primes $2, 3, 5, 11$
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![62742241, 0, 12356760, 0, -1954634, 0, 14060626, 0, -12025645, 0, -20548220, 0, 22124595, 0, -1432471, 0, -1815737, 0, -963792, 0, 1073159, 0, -673286, 0, 394548, 0, -186766, 0, 71340, 0, -22286, 0, 5612, 0, -1104, 0, 165, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 17*x^38 + 165*x^36 - 1104*x^34 + 5612*x^32 - 22286*x^30 + 71340*x^28 - 186766*x^26 + 394548*x^24 - 673286*x^22 + 1073159*x^20 - 963792*x^18 - 1815737*x^16 - 1432471*x^14 + 22124595*x^12 - 20548220*x^10 - 12025645*x^8 + 14060626*x^6 - 1954634*x^4 + 12356760*x^2 + 62742241)
 
gp: K = bnfinit(x^40 - 17*x^38 + 165*x^36 - 1104*x^34 + 5612*x^32 - 22286*x^30 + 71340*x^28 - 186766*x^26 + 394548*x^24 - 673286*x^22 + 1073159*x^20 - 963792*x^18 - 1815737*x^16 - 1432471*x^14 + 22124595*x^12 - 20548220*x^10 - 12025645*x^8 + 14060626*x^6 - 1954634*x^4 + 12356760*x^2 + 62742241, 1)
 

Normalized defining polynomial

\( x^{40} - 17 x^{38} + 165 x^{36} - 1104 x^{34} + 5612 x^{32} - 22286 x^{30} + 71340 x^{28} - 186766 x^{26} + 394548 x^{24} - 673286 x^{22} + 1073159 x^{20} - 963792 x^{18} - 1815737 x^{16} - 1432471 x^{14} + 22124595 x^{12} - 20548220 x^{10} - 12025645 x^{8} + 14060626 x^{6} - 1954634 x^{4} + 12356760 x^{2} + 62742241 \)

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:  \(11302165783522556415463223790320401501047994110286233600000000000000000000=2^{40}\cdot 3^{20}\cdot 5^{20}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.04$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(660=2^{2}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(391,·)$, $\chi_{660}(521,·)$, $\chi_{660}(371,·)$, $\chi_{660}(271,·)$, $\chi_{660}(401,·)$, $\chi_{660}(149,·)$, $\chi_{660}(151,·)$, $\chi_{660}(409,·)$, $\chi_{660}(431,·)$, $\chi_{660}(29,·)$, $\chi_{660}(491,·)$, $\chi_{660}(419,·)$, $\chi_{660}(421,·)$, $\chi_{660}(301,·)$, $\chi_{660}(559,·)$, $\chi_{660}(179,·)$, $\chi_{660}(181,·)$, $\chi_{660}(641,·)$, $\chi_{660}(569,·)$, $\chi_{660}(571,·)$, $\chi_{660}(581,·)$, $\chi_{660}(199,·)$, $\chi_{660}(329,·)$, $\chi_{660}(589,·)$, $\chi_{660}(211,·)$, $\chi_{660}(469,·)$, $\chi_{660}(599,·)$, $\chi_{660}(349,·)$, $\chi_{660}(611,·)$, $\chi_{660}(361,·)$, $\chi_{660}(619,·)$, $\chi_{660}(109,·)$, $\chi_{660}(59,·)$, $\chi_{660}(221,·)$, $\chi_{660}(499,·)$, $\chi_{660}(629,·)$, $\chi_{660}(119,·)$, $\chi_{660}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{89} a^{21} - \frac{11}{89} a^{19} - \frac{12}{89} a^{17} + \frac{26}{89} a^{15} - \frac{34}{89} a^{13} + \frac{36}{89} a^{11} + \frac{6}{89} a^{9} + \frac{26}{89} a^{7} + \frac{42}{89} a^{5} + \frac{18}{89} a^{3} + \frac{32}{89} a$, $\frac{1}{17711} a^{22} + \frac{3638}{17711} a^{20} - \frac{3394}{17711} a^{18} - \frac{6471}{17711} a^{16} - \frac{301}{17711} a^{14} + \frac{7423}{17711} a^{12} + \frac{7482}{17711} a^{10} - \frac{1843}{17711} a^{8} - \frac{6099}{17711} a^{6} + \frac{730}{17711} a^{4} - \frac{7622}{17711} a^{2} + \frac{29}{199}$, $\frac{1}{17711} a^{23} + \frac{56}{17711} a^{21} + \frac{586}{17711} a^{19} + \frac{1091}{17711} a^{17} - \frac{4878}{17711} a^{15} + \frac{5234}{17711} a^{13} + \frac{2507}{17711} a^{11} - \frac{5624}{17711} a^{9} + \frac{7035}{17711} a^{7} - \frac{8026}{17711} a^{5} - \frac{1254}{17711} a^{3} - \frac{5777}{17711} a$, $\frac{1}{17711} a^{24} - \frac{8321}{17711} a^{20} - \frac{3666}{17711} a^{18} + \frac{3278}{17711} a^{16} + \frac{4379}{17711} a^{14} - \frac{5828}{17711} a^{12} + \frac{448}{17711} a^{10} + \frac{3977}{17711} a^{8} - \frac{2991}{17711} a^{6} - \frac{6712}{17711} a^{4} - \frac{4009}{17711} a^{2} - \frac{32}{199}$, $\frac{1}{17711} a^{25} + \frac{37}{17711} a^{21} - \frac{7049}{17711} a^{19} - \frac{8463}{17711} a^{17} - \frac{8556}{17711} a^{15} - \frac{6624}{17711} a^{13} + \frac{249}{17711} a^{11} + \frac{992}{17711} a^{9} + \frac{1785}{17711} a^{7} + \frac{7815}{17711} a^{5} + \frac{4747}{17711} a^{3} - \frac{1057}{17711} a$, $\frac{1}{17711} a^{26} + \frac{33}{17711} a^{20} - \frac{6862}{17711} a^{18} + \frac{628}{17711} a^{16} + \frac{4513}{17711} a^{14} - \frac{8737}{17711} a^{12} + \frac{7534}{17711} a^{10} - \frac{868}{17711} a^{8} + \frac{3235}{17711} a^{6} - \frac{4552}{17711} a^{4} - \frac{2419}{17711} a^{2} - \frac{78}{199}$, $\frac{1}{17711} a^{27} + \frac{33}{17711} a^{21} - \frac{6862}{17711} a^{19} + \frac{628}{17711} a^{17} + \frac{4513}{17711} a^{15} - \frac{8737}{17711} a^{13} + \frac{7534}{17711} a^{11} - \frac{868}{17711} a^{9} + \frac{3235}{17711} a^{7} - \frac{4552}{17711} a^{5} - \frac{2419}{17711} a^{3} - \frac{78}{199} a$, $\frac{1}{17711} a^{28} - \frac{2939}{17711} a^{20} + \frac{6364}{17711} a^{18} + \frac{5524}{17711} a^{16} + \frac{1196}{17711} a^{14} - \frac{7182}{17711} a^{12} + \frac{180}{17711} a^{10} - \frac{6790}{17711} a^{8} + \frac{1894}{17711} a^{6} - \frac{8798}{17711} a^{4} - \frac{3370}{17711} a^{2} + \frac{38}{199}$, $\frac{1}{17711} a^{29} + \frac{46}{17711} a^{21} - \frac{8760}{17711} a^{19} + \frac{5126}{17711} a^{17} + \frac{7962}{17711} a^{15} - \frac{2406}{17711} a^{13} + \frac{1374}{17711} a^{11} - \frac{6591}{17711} a^{9} + \frac{8660}{17711} a^{7} - \frac{7405}{17711} a^{5} - \frac{2773}{17711} a^{3} - \frac{7364}{17711} a$, $\frac{1}{17711} a^{30} + \frac{1002}{17711} a^{20} + \frac{1851}{17711} a^{18} + \frac{4541}{17711} a^{16} - \frac{6271}{17711} a^{14} - \frac{3575}{17711} a^{12} + \frac{3457}{17711} a^{10} + \frac{4883}{17711} a^{8} + \frac{7484}{17711} a^{6} - \frac{931}{17711} a^{4} + \frac{6739}{17711} a^{2} + \frac{59}{199}$, $\frac{1}{17711} a^{31} + \frac{7}{17711} a^{21} - \frac{4915}{17711} a^{19} - \frac{1230}{17711} a^{17} + \frac{3281}{17711} a^{15} - \frac{5167}{17711} a^{13} + \frac{3059}{17711} a^{11} - \frac{1087}{17711} a^{9} - \frac{675}{17711} a^{7} - \frac{7299}{17711} a^{5} + \frac{6540}{17711} a^{3} + \frac{8833}{17711} a$, $\frac{1}{761573} a^{32} + \frac{14}{761573} a^{30} - \frac{3}{761573} a^{28} - \frac{15}{761573} a^{26} + \frac{17}{761573} a^{24} - \frac{19}{761573} a^{22} - \frac{130055}{761573} a^{20} + \frac{10467}{761573} a^{18} - \frac{177940}{761573} a^{16} + \frac{272245}{761573} a^{14} + \frac{238600}{761573} a^{12} - \frac{111467}{761573} a^{10} - \frac{243882}{761573} a^{8} - \frac{26113}{761573} a^{6} - \frac{151170}{761573} a^{4} - \frac{269448}{761573} a^{2} + \frac{2176}{8557}$, $\frac{1}{761573} a^{33} + \frac{14}{761573} a^{31} - \frac{3}{761573} a^{29} - \frac{15}{761573} a^{27} + \frac{17}{761573} a^{25} - \frac{19}{761573} a^{23} - \frac{1700}{761573} a^{21} + \frac{121708}{761573} a^{19} - \frac{195054}{761573} a^{17} - \frac{198390}{761573} a^{15} - \frac{317605}{761573} a^{13} - \frac{60125}{761573} a^{11} - \frac{235325}{761573} a^{9} + \frac{264825}{761573} a^{7} - \frac{91271}{761573} a^{5} - \frac{243777}{761573} a^{3} - \frac{268414}{761573} a$, $\frac{1}{761573} a^{34} + \frac{16}{761573} a^{30} - \frac{16}{761573} a^{28} + \frac{12}{761573} a^{26} + \frac{1}{761573} a^{24} - \frac{15}{761573} a^{22} - \frac{38317}{761573} a^{20} + \frac{1525}{17711} a^{18} - \frac{109081}{761573} a^{16} + \frac{296224}{761573} a^{14} + \frac{193587}{761573} a^{12} - \frac{250135}{761573} a^{10} - \frac{189408}{761573} a^{8} + \frac{57548}{761573} a^{6} + \frac{48887}{761573} a^{4} - \frac{139618}{761573} a^{2} - \frac{3976}{8557}$, $\frac{1}{761573} a^{35} + \frac{16}{761573} a^{31} - \frac{16}{761573} a^{29} + \frac{12}{761573} a^{27} + \frac{1}{761573} a^{25} - \frac{15}{761573} a^{23} - \frac{4089}{761573} a^{21} - \frac{7231}{17711} a^{19} + \frac{241756}{761573} a^{17} - \frac{336994}{761573} a^{15} - \frac{208592}{761573} a^{13} + \frac{220500}{761573} a^{11} + \frac{15960}{761573} a^{9} + \frac{185903}{761573} a^{7} - \frac{36683}{761573} a^{5} - \frac{285087}{761573} a^{3} - \frac{20141}{761573} a$, $\frac{1}{3485719621} a^{36} + \frac{519}{3485719621} a^{34} - \frac{37}{3485719621} a^{32} - \frac{16534}{3485719621} a^{30} + \frac{15646}{3485719621} a^{28} + \frac{44520}{3485719621} a^{26} + \frac{25274}{3485719621} a^{24} - \frac{84784}{3485719621} a^{22} - \frac{1364643138}{3485719621} a^{20} + \frac{1074922473}{3485719621} a^{18} - \frac{1617596144}{3485719621} a^{16} - \frac{436156653}{3485719621} a^{14} + \frac{676521907}{3485719621} a^{12} + \frac{1256196472}{3485719621} a^{10} + \frac{1024265786}{3485719621} a^{8} + \frac{846795460}{3485719621} a^{6} - \frac{1354855928}{3485719621} a^{4} - \frac{312382051}{3485719621} a^{2} - \frac{18845586}{39165389}$, $\frac{1}{3485719621} a^{37} + \frac{519}{3485719621} a^{35} - \frac{37}{3485719621} a^{33} - \frac{16534}{3485719621} a^{31} + \frac{15646}{3485719621} a^{29} + \frac{44520}{3485719621} a^{27} + \frac{25274}{3485719621} a^{25} - \frac{84784}{3485719621} a^{23} + \frac{6145477}{3485719621} a^{21} - \frac{60873808}{3485719621} a^{19} - \frac{638461419}{3485719621} a^{17} + \frac{347151127}{3485719621} a^{15} - \frac{615935930}{3485719621} a^{13} - \frac{1681207703}{3485719621} a^{11} - \frac{1208161387}{3485719621} a^{9} + \frac{1630103240}{3485719621} a^{7} + \frac{446751966}{3485719621} a^{5} - \frac{38224328}{3485719621} a^{3} + \frac{359343074}{3485719621} a$, $\frac{1}{1772083478061512575685661619268879788125090383} a^{38} + \frac{94515259154376959127911249683665335}{1772083478061512575685661619268879788125090383} a^{36} + \frac{106848331152629075715596514652534644785}{1772083478061512575685661619268879788125090383} a^{34} + \frac{63597970177210652373527126624729216320}{1772083478061512575685661619268879788125090383} a^{32} + \frac{14988082853505257042529689989012658700100}{1772083478061512575685661619268879788125090383} a^{30} - \frac{38068446123525789926456263696363585593037}{1772083478061512575685661619268879788125090383} a^{28} - \frac{17537531159686861849697432119330586035276}{1772083478061512575685661619268879788125090383} a^{26} + \frac{47080018736641812148028827449207557498978}{1772083478061512575685661619268879788125090383} a^{24} - \frac{14104690853878554773400428390616618069954}{1772083478061512575685661619268879788125090383} a^{22} - \frac{588407116132499806684298241530352648639224294}{1772083478061512575685661619268879788125090383} a^{20} - \frac{318741983877580590868919849678483408861116541}{1772083478061512575685661619268879788125090383} a^{18} + \frac{806868099488557160836798078601441473077574828}{1772083478061512575685661619268879788125090383} a^{16} + \frac{30349825921566694217415270308391326912062380}{77047107741804894595028766055168686440221321} a^{14} + \frac{668442372038314723520635667774034422662234845}{1772083478061512575685661619268879788125090383} a^{12} + \frac{546575793724070983438718834501095952193098810}{1772083478061512575685661619268879788125090383} a^{10} - \frac{503443552565757143508590576073035215222130417}{1772083478061512575685661619268879788125090383} a^{8} + \frac{286260352201070104200280680706434773773068708}{1772083478061512575685661619268879788125090383} a^{6} + \frac{738458729299239767370349145498227802629408631}{1772083478061512575685661619268879788125090383} a^{4} + \frac{317932082716111526223600420047673047671247494}{1772083478061512575685661619268879788125090383} a^{2} - \frac{88868492948842206924648659088927241778995}{223719666464021282121659085881691678844223}$, $\frac{1}{157715429547474619236023884114930301143133044087} a^{39} + \frac{8737040304930488102823437851985496426}{157715429547474619236023884114930301143133044087} a^{37} - \frac{102443828710548039382820752349448822169652}{157715429547474619236023884114930301143133044087} a^{35} + \frac{95145608220848652710202171207525133056764}{157715429547474619236023884114930301143133044087} a^{33} + \frac{3597415882709121528817125196630995649707077}{157715429547474619236023884114930301143133044087} a^{31} + \frac{723081276577107429119733380499712296776948}{157715429547474619236023884114930301143133044087} a^{29} + \frac{2754599148585013222915655666065888311018290}{157715429547474619236023884114930301143133044087} a^{27} + \frac{479583491824504185478018221137176050794244}{157715429547474619236023884114930301143133044087} a^{25} - \frac{64068459002438744380054011282587819602797}{3667800687150572540372648467789076770770535909} a^{23} + \frac{846607139943193403580465302092857712142634361}{157715429547474619236023884114930301143133044087} a^{21} + \frac{805627099781682318555340971033176683161147821}{6857192589020635618957560178910013093179697569} a^{19} + \frac{16960935040925091956713018876329204834326881021}{157715429547474619236023884114930301143133044087} a^{17} - \frac{53054523494068673906309443595150814447748448657}{157715429547474619236023884114930301143133044087} a^{15} - \frac{61363646253590887736834762156847867722528349926}{157715429547474619236023884114930301143133044087} a^{13} - \frac{27953377985107534730862986315903428407678261021}{157715429547474619236023884114930301143133044087} a^{11} - \frac{171128653844309576387287181036484977501331725}{3667800687150572540372648467789076770770535909} a^{9} + \frac{48889081630374789567314083609848991023105009185}{157715429547474619236023884114930301143133044087} a^{7} - \frac{78656437759782443693925868849546420805375630808}{157715429547474619236023884114930301143133044087} a^{5} - \frac{971390039781412004615305412789873415670096493}{3667800687150572540372648467789076770770535909} a^{3} + \frac{7903607066520047721247036734499455568305032}{19911050315297894108827658643470559417135847} 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{22376766566381608239457712740713733559}{1772083478061512575685661619268879788125090383} a^{38} + \frac{8879971323691437349072106933882941589}{41211243675849129667108409750439064840118381} a^{36} - \frac{3726290842919149540399826703760760653159}{1772083478061512575685661619268879788125090383} a^{34} + \frac{25162189736592384075361062596576117344156}{1772083478061512575685661619268879788125090383} a^{32} - \frac{129601743043348837178223448645510775975394}{1772083478061512575685661619268879788125090383} a^{30} + \frac{525215202316720421795855148352686310810966}{1772083478061512575685661619268879788125090383} a^{28} - \frac{1732389341419547774637933079205564845533560}{1772083478061512575685661619268879788125090383} a^{26} + \frac{4744618642107370675293374524526690048904946}{1772083478061512575685661619268879788125090383} a^{24} - \frac{10747697938134295078274603238425615405714488}{1772083478061512575685661619268879788125090383} a^{22} + \frac{20516818374990591872444539320063615014264853}{1772083478061512575685661619268879788125090383} a^{20} - \frac{36811779262523402570374849359274508539177148}{1772083478061512575685661619268879788125090383} a^{18} + \frac{46602819616369449668894639586596148608779439}{1772083478061512575685661619268879788125090383} a^{16} - \frac{301661613092720269919030981347802327570067}{1772083478061512575685661619268879788125090383} a^{14} + \frac{84733117124686794718806090909388232987058883}{1772083478061512575685661619268879788125090383} a^{12} - \frac{492857975863173292770699388438409073834580377}{1772083478061512575685661619268879788125090383} a^{10} + \frac{439832080889393915283820265076478051272380433}{1772083478061512575685661619268879788125090383} a^{8} - \frac{43725747120674590181811910820285118335705736}{1772083478061512575685661619268879788125090383} a^{6} + \frac{43977666862380174425396100095386166927373875}{1772083478061512575685661619268879788125090383} a^{4} - \frac{679671382673061069247524505437921480735268648}{1772083478061512575685661619268879788125090383} a^{2} + \frac{112677771811806937911458297428763433871942}{223719666464021282121659085881691678844223} \) (order $6$)
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

\(\Q(\sqrt{165}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-55}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-5}, \sqrt{-33})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{11}, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-5}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{15}, \sqrt{-33})\), \(\Q(\zeta_{11})^+\), 8.0.189747360000.3, 10.10.1790566527853125.1, 10.0.685948419200000.1, 10.0.586732839846912.1, 10.0.52089208083.1, 10.0.7368586534375.1, 10.10.166685465865600000.1, \(\Q(\zeta_{44})^+\), 20.0.3361869388230684433628866560000000000.1, 20.0.3206128490667995866421572265625.2, 20.20.3361869388230684433628866560000000000.2, 20.0.27784044530832102757263360000000000.2, 20.0.56933553290160450365440000000000.1, 20.0.344255425354822086003595935744.2, 20.0.3361869388230684433628866560000000000.8

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 R R R ${\href{/LocalNumberField/7.5.0.1}{5} }^{8}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{40}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed
11Data not computed