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Properties

Label 32.32.5104189893...4993.1
Degree $32$
Signature $[32, 0]$
Discriminant $257^{31}$
Root discriminant $216.08$
Ramified prime $257$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_{32}$ (as 32T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14868736, -19804416, -560812992, 135294656, 5620092128, 209255344, -22248890532, -1849486552, 45466050353, 3669187609, -55097215992, -3921224431, 42838349856, 2699160105, -22362241012, -1274400611, 8035225328, 419256941, -2009951528, -95534899, 350544862, 14861939, -42347664, -1550069, 3488200, 105347, -190140, -4425, 6488, 103, -124, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - x^31 - 124*x^30 + 103*x^29 + 6488*x^28 - 4425*x^27 - 190140*x^26 + 105347*x^25 + 3488200*x^24 - 1550069*x^23 - 42347664*x^22 + 14861939*x^21 + 350544862*x^20 - 95534899*x^19 - 2009951528*x^18 + 419256941*x^17 + 8035225328*x^16 - 1274400611*x^15 - 22362241012*x^14 + 2699160105*x^13 + 42838349856*x^12 - 3921224431*x^11 - 55097215992*x^10 + 3669187609*x^9 + 45466050353*x^8 - 1849486552*x^7 - 22248890532*x^6 + 209255344*x^5 + 5620092128*x^4 + 135294656*x^3 - 560812992*x^2 - 19804416*x + 14868736)
 
gp: K = bnfinit(x^32 - x^31 - 124*x^30 + 103*x^29 + 6488*x^28 - 4425*x^27 - 190140*x^26 + 105347*x^25 + 3488200*x^24 - 1550069*x^23 - 42347664*x^22 + 14861939*x^21 + 350544862*x^20 - 95534899*x^19 - 2009951528*x^18 + 419256941*x^17 + 8035225328*x^16 - 1274400611*x^15 - 22362241012*x^14 + 2699160105*x^13 + 42838349856*x^12 - 3921224431*x^11 - 55097215992*x^10 + 3669187609*x^9 + 45466050353*x^8 - 1849486552*x^7 - 22248890532*x^6 + 209255344*x^5 + 5620092128*x^4 + 135294656*x^3 - 560812992*x^2 - 19804416*x + 14868736, 1)
 

Normalized defining polynomial

\( x^{32} - x^{31} - 124 x^{30} + 103 x^{29} + 6488 x^{28} - 4425 x^{27} - 190140 x^{26} + 105347 x^{25} + 3488200 x^{24} - 1550069 x^{23} - 42347664 x^{22} + 14861939 x^{21} + 350544862 x^{20} - 95534899 x^{19} - 2009951528 x^{18} + 419256941 x^{17} + 8035225328 x^{16} - 1274400611 x^{15} - 22362241012 x^{14} + 2699160105 x^{13} + 42838349856 x^{12} - 3921224431 x^{11} - 55097215992 x^{10} + 3669187609 x^{9} + 45466050353 x^{8} - 1849486552 x^{7} - 22248890532 x^{6} + 209255344 x^{5} + 5620092128 x^{4} + 135294656 x^{3} - 560812992 x^{2} - 19804416 x + 14868736 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[32, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(510418989351370756385471987562075329801799753598808217293070249498399284993=257^{31}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.08$
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:  $257$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(257\)
Dirichlet character group:    $\lbrace$$\chi_{257}(128,·)$, $\chi_{257}(1,·)$, $\chi_{257}(2,·)$, $\chi_{257}(4,·)$, $\chi_{257}(129,·)$, $\chi_{257}(8,·)$, $\chi_{257}(137,·)$, $\chi_{257}(256,·)$, $\chi_{257}(15,·)$, $\chi_{257}(16,·)$, $\chi_{257}(17,·)$, $\chi_{257}(30,·)$, $\chi_{257}(32,·)$, $\chi_{257}(34,·)$, $\chi_{257}(136,·)$, $\chi_{257}(60,·)$, $\chi_{257}(189,·)$, $\chi_{257}(64,·)$, $\chi_{257}(193,·)$, $\chi_{257}(68,·)$, $\chi_{257}(197,·)$, $\chi_{257}(121,·)$, $\chi_{257}(223,·)$, $\chi_{257}(225,·)$, $\chi_{257}(227,·)$, $\chi_{257}(240,·)$, $\chi_{257}(241,·)$, $\chi_{257}(242,·)$, $\chi_{257}(120,·)$, $\chi_{257}(249,·)$, $\chi_{257}(253,·)$, $\chi_{257}(255,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{9} - \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{32} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{64} a^{19} - \frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} + \frac{3}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} + \frac{7}{64} a^{7} - \frac{3}{64} a^{6} - \frac{3}{64} a^{5} - \frac{7}{32} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} - \frac{1}{32} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{7}{64} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{15}{64} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{21} - \frac{1}{64} a^{18} - \frac{1}{32} a^{15} - \frac{1}{32} a^{12} + \frac{5}{64} a^{9} + \frac{3}{64} a^{6} + \frac{7}{16} a^{3} - \frac{1}{2}$, $\frac{1}{256} a^{22} - \frac{1}{256} a^{20} - \frac{1}{256} a^{18} - \frac{1}{128} a^{17} - \frac{1}{128} a^{15} + \frac{1}{64} a^{14} - \frac{1}{128} a^{13} + \frac{1}{64} a^{12} - \frac{11}{128} a^{11} - \frac{17}{256} a^{10} + \frac{5}{128} a^{9} + \frac{21}{256} a^{8} - \frac{19}{128} a^{7} - \frac{43}{256} a^{6} - \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{256} a^{23} - \frac{1}{256} a^{21} - \frac{1}{256} a^{19} - \frac{1}{128} a^{18} - \frac{1}{128} a^{16} + \frac{1}{64} a^{15} - \frac{1}{128} a^{14} + \frac{1}{64} a^{13} + \frac{5}{128} a^{12} - \frac{17}{256} a^{11} + \frac{5}{128} a^{10} - \frac{11}{256} a^{9} + \frac{13}{128} a^{8} - \frac{43}{256} a^{7} - \frac{3}{16} a^{6} - \frac{3}{32} a^{5} - \frac{3}{16} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{24} - \frac{1}{128} a^{20} - \frac{1}{128} a^{19} - \frac{1}{256} a^{18} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{3}{256} a^{12} - \frac{7}{64} a^{11} + \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{3}{128} a^{8} + \frac{29}{128} a^{7} - \frac{19}{256} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{256} a^{25} - \frac{1}{128} a^{21} - \frac{1}{128} a^{20} - \frac{1}{256} a^{19} - \frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{64} a^{16} - \frac{1}{32} a^{15} - \frac{1}{32} a^{14} + \frac{3}{256} a^{13} + \frac{1}{64} a^{12} + \frac{1}{64} a^{11} - \frac{3}{64} a^{10} + \frac{13}{128} a^{9} - \frac{3}{128} a^{8} - \frac{19}{256} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{512} a^{26} - \frac{1}{512} a^{24} - \frac{1}{512} a^{23} - \frac{1}{512} a^{22} - \frac{1}{512} a^{21} - \frac{1}{512} a^{19} + \frac{3}{256} a^{18} - \frac{1}{256} a^{17} + \frac{3}{256} a^{16} + \frac{3}{256} a^{15} - \frac{15}{512} a^{14} + \frac{3}{256} a^{13} - \frac{21}{512} a^{12} - \frac{21}{512} a^{11} - \frac{53}{512} a^{10} - \frac{37}{512} a^{9} + \frac{7}{256} a^{8} - \frac{5}{512} a^{7} + \frac{5}{32} a^{6} - \frac{5}{64} a^{5} + \frac{7}{32} a^{4} + \frac{7}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{512} a^{27} - \frac{1}{512} a^{25} - \frac{1}{512} a^{24} - \frac{1}{512} a^{23} - \frac{1}{512} a^{22} - \frac{1}{512} a^{20} - \frac{1}{256} a^{19} + \frac{3}{256} a^{18} - \frac{1}{256} a^{17} + \frac{3}{256} a^{16} + \frac{1}{512} a^{15} - \frac{5}{256} a^{14} + \frac{11}{512} a^{13} - \frac{21}{512} a^{12} + \frac{43}{512} a^{11} - \frac{53}{512} a^{10} - \frac{17}{256} a^{9} + \frac{11}{512} a^{8} + \frac{15}{64} a^{7} - \frac{7}{32} a^{6} - \frac{5}{64} a^{5} - \frac{5}{32} a^{4} - \frac{7}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{1024} a^{28} - \frac{1}{1024} a^{27} + \frac{1}{1024} a^{24} + \frac{1}{1024} a^{23} - \frac{1}{256} a^{21} + \frac{3}{1024} a^{20} + \frac{1}{1024} a^{19} - \frac{1}{128} a^{18} + \frac{3}{512} a^{17} - \frac{11}{1024} a^{16} + \frac{27}{1024} a^{15} + \frac{1}{512} a^{14} + \frac{15}{512} a^{13} + \frac{5}{1024} a^{12} - \frac{127}{1024} a^{11} + \frac{29}{512} a^{10} + \frac{45}{512} a^{9} - \frac{53}{1024} a^{8} + \frac{241}{1024} a^{7} - \frac{87}{512} a^{6} + \frac{9}{128} a^{5} - \frac{3}{32} a^{4} + \frac{11}{32} a^{3} - \frac{1}{2} a^{2} + \frac{3}{16} a + \frac{1}{8}$, $\frac{1}{2048} a^{29} - \frac{1}{2048} a^{27} - \frac{1}{1024} a^{26} + \frac{1}{2048} a^{25} - \frac{1}{2048} a^{23} - \frac{1}{1024} a^{22} - \frac{11}{2048} a^{21} + \frac{3}{512} a^{20} + \frac{7}{2048} a^{19} + \frac{7}{1024} a^{18} - \frac{17}{2048} a^{17} - \frac{1}{512} a^{16} + \frac{49}{2048} a^{15} + \frac{19}{1024} a^{14} - \frac{25}{2048} a^{13} + \frac{23}{512} a^{12} - \frac{231}{2048} a^{11} - \frac{93}{1024} a^{10} - \frac{197}{2048} a^{9} - \frac{7}{128} a^{8} + \frac{81}{2048} a^{7} - \frac{245}{1024} a^{6} + \frac{31}{256} a^{5} + \frac{3}{32} a^{4} + \frac{5}{64} a^{3} - \frac{7}{32} a^{2} + \frac{7}{32} a - \frac{1}{16}$, $\frac{1}{1197554403328} a^{30} - \frac{185072633}{1197554403328} a^{29} - \frac{111974931}{1197554403328} a^{28} - \frac{1059980291}{1197554403328} a^{27} - \frac{145193781}{1197554403328} a^{26} + \frac{216134499}{1197554403328} a^{25} - \frac{576052711}{1197554403328} a^{24} + \frac{396161337}{1197554403328} a^{23} - \frac{1189331565}{1197554403328} a^{22} - \frac{2883266641}{1197554403328} a^{21} - \frac{2064912023}{1197554403328} a^{20} + \frac{3617252845}{1197554403328} a^{19} + \frac{9996647065}{1197554403328} a^{18} - \frac{9171363311}{1197554403328} a^{17} - \frac{4418107709}{1197554403328} a^{16} - \frac{20849543157}{1197554403328} a^{15} + \frac{13206967869}{1197554403328} a^{14} - \frac{20634159779}{1197554403328} a^{13} - \frac{70097355697}{1197554403328} a^{12} - \frac{110585233705}{1197554403328} a^{11} - \frac{45736663811}{1197554403328} a^{10} - \frac{68048736887}{1197554403328} a^{9} - \frac{19228760785}{1197554403328} a^{8} + \frac{123830716211}{1197554403328} a^{7} + \frac{88282698191}{598777201664} a^{6} - \frac{4688836673}{149694300416} a^{5} - \frac{2017243385}{37423575104} a^{4} - \frac{16907028361}{37423575104} a^{3} + \frac{868507357}{9355893776} a^{2} - \frac{3195928615}{18711787552} a - \frac{9278193}{38821136}$, $\frac{1}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{31} - \frac{651665442314698146583710426388369057966010770360289426191230520482507643}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{30} - \frac{30677120509394160358491511914420698043838719277334112637221958394923037336007523}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{29} - \frac{394908921225032722185969793817182546942972749487858710126339285329108373005132449}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{28} - \frac{2647282396790715359403185432166086780471973213233092491325689017016994729809318663}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{27} + \frac{1756589940830823022962328655773993430529710151333625186361672173770225346178806447}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{26} - \frac{1193074364484682655651844297203311721095993981386655876888926027591344194187772825}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{25} - \frac{8171937855196143326994574008722643673251719004855331894699075025719637527435977917}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{24} - \frac{4494015011634507538937171854249467504351301433402597488941743537280090640449198019}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{23} + \frac{2294930077081535615325276364740010857366337369058120305729899930511151609494692211}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{22} - \frac{24214106033503754189942326888476146477377409035708665000881131021601022346048972671}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{21} - \frac{41105019820586441032694089488423405164452521212387077994231354262294386252320430021}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{20} + \frac{1238253557018237957139288617083294673712816649810324208454853542270743460246991861}{472868642254052281574314290376519788444532603300372698696685907502997129112103665152} a^{19} + \frac{10653069296705530830984407138571947910044217830591696967270654726485276732762337769}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{18} + \frac{27489633913653855397555816313965992301644115550107928944119126218864541652350042447}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{17} + \frac{57500778763075611489115642388091321547370685747693701861228732168109961192514682613}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{16} + \frac{81235501271757094980293696207015713859652475195930659836253846623919108095797688887}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{15} - \frac{49224928508660782319121915889028630224909670882241830368082767692774768323844419787}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{14} + \frac{104121417641084800356813046734519817003116434037449130164769973625696913873193305981}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{13} - \frac{141429163600147046299496881901614607612765298502468309035388149246401654219888656855}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{12} + \frac{206582665263376295222619040108562209819448011928443870574958703880632417809503461803}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{11} - \frac{851060157228564442058799265783059226804303684969485436907870866629990063632275150791}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{10} + \frac{39225597848473857226671149564670987299481537807426191890697216784708338714198636911}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{9} - \frac{150464989465188391914123753937848190030253462997218017159497372862870962341827432895}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{8} - \frac{1671758770477011501443621581727635840789261262211120225226618877839966329352175478265}{7565898276064836505189028646024316615112521652805963179146974520047954065793658642432} a^{7} - \frac{286626030499498251500066302189552359825103458676650728017110240237776646919645074301}{3782949138032418252594514323012158307556260826402981589573487260023977032896829321216} a^{6} + \frac{140752967362250430846660160995778296569998446047355953689369257630103771693887713239}{945737284508104563148628580753039576889065206600745397393371815005994258224207330304} a^{5} - \frac{20089411559699595133575901499292481985997072207275992653400951212114723320585097725}{236434321127026140787157145188259894222266301650186349348342953751498564556051832576} a^{4} - \frac{64806516440295384250950749745628969171002319249938007797978941489623471946518153887}{236434321127026140787157145188259894222266301650186349348342953751498564556051832576} a^{3} - \frac{4306725019161642212297525972963147662585087789926138582751038104910050482731833375}{14777145070439133799197321574266243388891643853136646834271434609468660284753239536} a^{2} + \frac{53294571670673792490972784867047771453338847353564632509468717788601291219881284133}{118217160563513070393578572594129947111133150825093174674171476875749282278025916288} a - \frac{121235274987659701984514148874389900708036107879386439181303525267490539033028525}{245263818596500146044768822809398230521023134491894553265915927128110544145281984}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $31$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 39222759909157020000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{32}$ (as 32T33):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{257}) \), 4.4.16974593.1, 8.8.74051159531521793.1, 16.16.1409278576586462959586218741521256193.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{16}$ $32$ $32$ $32$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ $16^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ $32$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ $16^{2}$ $16^{2}$ $32$ $32$ $32$ $32$ $32$ $16^{2}$

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
257Data not computed