LMFDB - Number field 16.16.5191562722311896995363773104453715984.1

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

comment:Define the number field

sage:x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168)

gp:K = bnfinit(y^16 - y^15 - 65*y^14 + 64*y^13 + 1711*y^12 - 1622*y^11 - 23327*y^10 + 20556*y^9 + 174638*y^8 - 133920*y^7 - 701148*y^6 + 398120*y^5 + 1375168*y^4 - 298560*y^3 - 1095936*y^2 - 262656*y + 39168, 1)

magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168);

oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168)

$ x^{16} - x^{15} - 65 x^{14} + 64 x^{13} + 1711 x^{12} - 1622 x^{11} - 23327 x^{10} + 20556 x^{9} + \cdots + 39168 $ x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168

comment:Defining polynomial

sage:K.defining_polynomial()

gp:K.pol

magma:DefiningPolynomial(K);

oscar:defining_polynomial(K)

Invariants

Degree:	$16$	comment:Degree over Q sage:K.degree() gp:poldegree(K.pol) magma:Degree(K); oscar:degree(K)
Signature:	$(16, 0)$	comment:Signature sage:K.signature() gp:K.sign magma:Signature(K); oscar:signature(K)
Discriminant:	$5191562722311896995363773104453715984$ 5191562722311896995363773104453715984 $\medspace = 2^{4}\cdot 3^{4}\cdot 7^{2}\cdot 13^{8}\cdot 17^{4}\cdot 23^{4}\cdot 1439^{4}$	comment:Discriminant sage:K.disc() gp:K.disc magma:OK := Integers(K); Discriminant(OK); oscar:OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$197.11$	comment:Root discriminant sage:(K.disc().abs())^(1./K.degree()) gp:abs(K.disc)^(1/poldegree(K.pol)) magma:Abs(Discriminant(OK))^(1/Degree(K)); oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
Galois root discriminant:	$2\cdot 3^{1/2}7^{1/2}13^{1/2}17^{1/2}23^{1/2}1439^{1/2}\approx 24787.349757487184$
Ramified primes:	$2$, $3$, $7$, $13$, $17$, $23$, $1439$ 2, 3, 7, 13, 17, 23, 1439	comment:Ramified primes sage:K.disc().support() gp:factor(abs(K.disc))[,1]~ magma:PrimeDivisors(Discriminant(OK)); oscar:prime_divisors(discriminant(OK))
Discriminant root field:	$\Q$
$\Aut(K/\Q)$:	$C_2$	comment:Automorphisms sage:K.automorphisms() magma:Automorphisms(K); oscar:automorphism_group(K)
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{7}+\frac{1}{4}a^{6}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{8}-\frac{3}{8}a^{7}+\frac{1}{16}a^{6}-\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{9}-\frac{3}{16}a^{8}-\frac{15}{32}a^{7}+\frac{3}{8}a^{6}-\frac{1}{16}a^{5}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{9569856}a^{14}+\frac{47999}{9569856}a^{13}+\frac{191083}{9569856}a^{12}+\frac{111319}{2392464}a^{11}-\frac{1007333}{9569856}a^{10}-\frac{260575}{4784928}a^{9}+\frac{1297525}{9569856}a^{8}+\frac{215197}{797488}a^{7}-\frac{274091}{4784928}a^{6}-\frac{120315}{398744}a^{5}+\frac{153273}{797488}a^{4}-\frac{103589}{1196232}a^{3}+\frac{78349}{598116}a^{2}+\frac{18827}{49843}a-\frac{7990}{49843}$, $\frac{1}{20412254031744}a^{15}-\frac{536771}{20412254031744}a^{14}+\frac{139339101209}{20412254031744}a^{13}+\frac{46771320991}{3402042338624}a^{12}-\frac{415957193091}{6804084677248}a^{11}+\frac{31705685913}{1701021169312}a^{10}-\frac{602800336393}{6804084677248}a^{9}-\frac{835052024891}{10206127015872}a^{8}+\frac{2260298537}{443744652864}a^{7}+\frac{2252535564685}{5103063507936}a^{6}-\frac{725176283933}{1701021169312}a^{5}+\frac{185007056329}{637882938492}a^{4}+\frac{221059588307}{1275765876984}a^{3}-\frac{68148655928}{159470734623}a^{2}+\frac{49105521561}{106313823082}a+\frac{12799746983}{53156911541}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, 1/2*a^8 - 1/2*a^7 - 1/2*a^6 - 1/2*a^4 - 1/2*a^2, 1/2*a^9 - 1/2*a^6 - 1/2*a^5 - 1/2*a^4 - 1/2*a^3 - 1/2*a^2, 1/4*a^10 - 1/4*a^9 - 1/4*a^8 - 1/4*a^6 - 1/2*a^5 + 1/4*a^4 - 1/2*a^2, 1/8*a^11 - 1/8*a^10 - 1/8*a^9 - 1/8*a^7 + 1/4*a^6 + 1/8*a^5 - 1/2*a^4 - 1/4*a^3 - 1/2*a, 1/16*a^12 - 1/16*a^11 - 1/16*a^10 - 1/16*a^8 - 3/8*a^7 + 1/16*a^6 - 1/4*a^5 - 1/8*a^4 + 1/4*a^2 - 1/2*a, 1/32*a^13 - 1/32*a^12 - 1/32*a^11 - 1/32*a^9 - 3/16*a^8 - 15/32*a^7 + 3/8*a^6 - 1/16*a^5 - 3/8*a^3 + 1/4*a^2, 1/9569856*a^14 + 47999/9569856*a^13 + 191083/9569856*a^12 + 111319/2392464*a^11 - 1007333/9569856*a^10 - 260575/4784928*a^9 + 1297525/9569856*a^8 + 215197/797488*a^7 - 274091/4784928*a^6 - 120315/398744*a^5 + 153273/797488*a^4 - 103589/1196232*a^3 + 78349/598116*a^2 + 18827/49843*a - 7990/49843, 1/20412254031744*a^15 - 536771/20412254031744*a^14 + 139339101209/20412254031744*a^13 + 46771320991/3402042338624*a^12 - 415957193091/6804084677248*a^11 + 31705685913/1701021169312*a^10 - 602800336393/6804084677248*a^9 - 835052024891/10206127015872*a^8 + 2260298537/443744652864*a^7 + 2252535564685/5103063507936*a^6 - 725176283933/1701021169312*a^5 + 185007056329/637882938492*a^4 + 221059588307/1275765876984*a^3 - 68148655928/159470734623*a^2 + 49105521561/106313823082*a + 12799746983/53156911541

comment:Integral basis

sage:K.integral_basis()

gp:K.zk

magma:IntegralBasis(K);

oscar:basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		$C_{4}$, which has order $4$ (assuming GRH)	comment:Class group sage:K.class_group().invariants() gp:K.clgp magma:ClassGroup(K); oscar:class_group(K)
Narrow class group:		$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}$, which has order $64$ (assuming GRH)	comment:Narrow class group sage:K.narrow_class_group().invariants() gp:bnfnarrow(K) magma:NarrowClassGroup(K);

Unit group

comment:Unit group

sage:UK = K.unit_group()

magma:UK, fUK := UnitGroup(K);

oscar:UK, fUK = unit_group(OK)

Rank:	$15$	comment:Unit rank sage:UK.rank() gp:K.fu magma:UnitRank(K); oscar:rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	comment:Generator for roots of unity 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:	$\frac{1658681}{38225194816}a^{15}-\frac{49920877}{114675584448}a^{14}-\frac{197250803}{114675584448}a^{13}+\frac{161321369}{7167224028}a^{12}+\frac{1856589611}{114675584448}a^{11}-\frac{12552552373}{28668896112}a^{10}+\frac{16995573473}{114675584448}a^{9}+\frac{225281566861}{57337792224}a^{8}-\frac{85469515}{25968203}a^{7}-\frac{113705321711}{7167224028}a^{6}+\frac{41241020825}{2389074676}a^{5}+\frac{111681964471}{4778149352}a^{4}-\frac{383530035271}{14334448056}a^{3}-\frac{17266528447}{1791806007}a^{2}+\frac{12962640727}{1194537338}a+\frac{287646039}{597268669}$, $\frac{36081}{567811456}a^{15}-\frac{1589}{567811456}a^{14}-\frac{2047169}{567811456}a^{13}+\frac{4293}{35488216}a^{12}+\frac{46151979}{567811456}a^{11}-\frac{1034637}{283905728}a^{10}-\frac{524990323}{567811456}a^{9}+\frac{5526613}{70976432}a^{8}+\frac{1575409909}{283905728}a^{7}-\frac{30149709}{35488216}a^{6}-\frac{2357252353}{141952864}a^{5}+\frac{283620161}{70976432}a^{4}+\frac{718795095}{35488216}a^{3}-\frac{28752096}{4436027}a^{2}-\frac{24050331}{4436027}a+\frac{4043608}{4436027}$, $\frac{23290445}{76450389632}a^{15}+\frac{240657593}{229351168896}a^{14}+\frac{3637542757}{229351168896}a^{13}-\frac{6438039557}{114675584448}a^{12}-\frac{70900379185}{229351168896}a^{11}+\frac{16285663673}{14334448056}a^{10}+\frac{632279069213}{229351168896}a^{9}-\frac{1232281252397}{114675584448}a^{8}-\frac{18127836673}{1661964992}a^{7}+\frac{2715673745557}{57337792224}a^{6}+\frac{298689284531}{19112597408}a^{5}-\frac{202120661821}{2389074676}a^{4}-\frac{144340627175}{14334448056}a^{3}+\frac{363581871055}{7167224028}a^{2}+\frac{5386570337}{1194537338}a-\frac{3940669446}{597268669}$, $\frac{727603841}{6804084677248}a^{15}+\frac{8943039067}{20412254031744}a^{14}+\frac{108704321315}{20412254031744}a^{13}-\frac{28869532739}{1275765876984}a^{12}-\frac{1989590443829}{20412254031744}a^{11}+\frac{4505902810301}{10206127015872}a^{10}+\frac{15847665043333}{20412254031744}a^{9}-\frac{20447411872103}{5103063507936}a^{8}-\frac{333954205149}{147914884288}a^{7}+\frac{21323399153399}{1275765876984}a^{6}-\frac{1116651881401}{1701021169312}a^{5}-\frac{23278010065101}{850510584656}a^{4}+\frac{2073552067241}{318941469246}a^{3}+\frac{2570330130887}{159470734623}a^{2}-\frac{577284791957}{106313823082}a-\frac{233525786758}{53156911541}$, $\frac{454701193}{1701021169312}a^{15}-\frac{282877863}{3402042338624}a^{14}+\frac{54803387637}{3402042338624}a^{13}+\frac{15777935295}{3402042338624}a^{12}-\frac{82144862777}{212627646164}a^{11}-\frac{378758177075}{3402042338624}a^{10}+\frac{7955366021433}{1701021169312}a^{9}+\frac{5033939546307}{3402042338624}a^{8}-\frac{2202481962461}{73957442144}a^{7}-\frac{19091066802003}{1701021169312}a^{6}+\frac{79117583017311}{850510584656}a^{5}+\frac{37515368903675}{850510584656}a^{4}-\frac{23265860828615}{212627646164}a^{3}-\frac{14021203899823}{212627646164}a^{2}+\frac{436607976441}{106313823082}a-\frac{16695463868}{53156911541}$, $\frac{773841331}{850510584656}a^{15}+\frac{33583736939}{10206127015872}a^{14}+\frac{476822685919}{10206127015872}a^{13}-\frac{1813500308401}{10206127015872}a^{12}-\frac{4540631124431}{5103063507936}a^{11}+\frac{37183484730881}{10206127015872}a^{10}+\frac{9688511392867}{1275765876984}a^{9}-\frac{359332647371965}{10206127015872}a^{8}-\frac{2007786751633}{73957442144}a^{7}+\frac{825646520991305}{5103063507936}a^{6}+\frac{24057356473427}{850510584656}a^{5}-\frac{270701524591523}{850510584656}a^{4}-\frac{15342870386459}{637882938492}a^{3}+\frac{146952012501305}{637882938492}a^{2}+\frac{6695262384319}{106313823082}a-\frac{476228070563}{53156911541}$, $\frac{338955541}{1701021169312}a^{15}+\frac{2532305179}{10206127015872}a^{14}+\frac{116292162983}{10206127015872}a^{13}-\frac{114266413595}{10206127015872}a^{12}-\frac{82497609049}{318941469246}a^{11}+\frac{1759655811103}{10206127015872}a^{10}+\frac{15067244341727}{5103063507936}a^{9}-\frac{8745446140391}{10206127015872}a^{8}-\frac{1301148489373}{73957442144}a^{7}-\frac{13466408930597}{5103063507936}a^{6}+\frac{42961232622673}{850510584656}a^{5}+\frac{27944482217371}{850510584656}a^{4}-\frac{8031435400591}{159470734623}a^{3}-\frac{40197607627343}{637882938492}a^{2}-\frac{865360339085}{53156911541}a+\frac{127745813976}{53156911541}$, $\frac{53128229}{295829768576}a^{15}-\frac{523230569}{887489305728}a^{14}-\frac{8412302869}{887489305728}a^{13}+\frac{13762343453}{443744652864}a^{12}+\frac{168293855041}{887489305728}a^{11}-\frac{33800777573}{55468081608}a^{10}-\frac{1585563094805}{887489305728}a^{9}+\frac{2415123446237}{443744652864}a^{8}+\frac{1197175617279}{147914884288}a^{7}-\frac{4653957438739}{221872326432}a^{6}-\frac{1271262939451}{73957442144}a^{5}+\frac{435382794943}{18489360536}a^{4}+\frac{1092118157195}{55468081608}a^{3}+\frac{100677316585}{13867020402}a^{2}-\frac{2066555066}{2311170067}a-\frac{82206514}{2311170067}$, $\frac{37595331}{1701021169312}a^{15}+\frac{3858806661}{3402042338624}a^{14}-\frac{11262106527}{3402042338624}a^{13}-\frac{195679793925}{3402042338624}a^{12}+\frac{52102107041}{425255292328}a^{11}+\frac{3864214464545}{3402042338624}a^{10}-\frac{3306420560483}{1701021169312}a^{9}-\frac{37575625809249}{3402042338624}a^{8}+\frac{1095480768967}{73957442144}a^{7}+\frac{93631754604245}{1701021169312}a^{6}-\frac{42998378071133}{850510584656}a^{5}-\frac{113892106921553}{850510584656}a^{4}+\frac{10117471090353}{212627646164}a^{3}+\frac{29095298625079}{212627646164}a^{2}+\frac{3715942873719}{106313823082}a-\frac{271854792461}{53156911541}$, $\frac{162113873}{3402042338624}a^{15}-\frac{4406067097}{10206127015872}a^{14}-\frac{49491420251}{10206127015872}a^{13}+\frac{60092548961}{2551531753968}a^{12}+\frac{1718611880795}{10206127015872}a^{11}-\frac{1269567453703}{2551531753968}a^{10}-\frac{27889823231947}{10206127015872}a^{9}+\frac{26050771888537}{5103063507936}a^{8}+\frac{823955827057}{36978721072}a^{7}-\frac{66332429174543}{2551531753968}a^{6}-\frac{37114122587661}{425255292328}a^{5}+\frac{12322921859541}{212627646164}a^{4}+\frac{175826641928747}{1275765876984}a^{3}-\frac{17052928668775}{637882938492}a^{2}-\frac{2841668189362}{53156911541}a-\frac{372754224474}{53156911541}$, $\frac{6567443803}{6804084677248}a^{15}-\frac{35593756255}{20412254031744}a^{14}-\frac{1063532882603}{20412254031744}a^{13}+\frac{926302688167}{10206127015872}a^{12}+\frac{21895466300159}{20412254031744}a^{11}-\frac{2228716187395}{1275765876984}a^{10}-\frac{213073206520195}{20412254031744}a^{9}+\frac{152792901830635}{10206127015872}a^{8}+\frac{7102827249475}{147914884288}a^{7}-\frac{266066213410499}{5103063507936}a^{6}-\frac{153816756148285}{1701021169312}a^{5}+\frac{1823791001427}{53156911541}a^{4}+\frac{63694914455761}{1275765876984}a^{3}+\frac{36180309471451}{637882938492}a^{2}+\frac{1940593853085}{53156911541}a-\frac{236534852882}{53156911541}$, $\frac{756712145}{6804084677248}a^{15}+\frac{4735602287}{20412254031744}a^{14}+\frac{141394377787}{20412254031744}a^{13}-\frac{9305328857}{637882938492}a^{12}-\frac{3569907714193}{20412254031744}a^{11}+\frac{3901027206187}{10206127015872}a^{10}+\frac{45821864592017}{20412254031744}a^{9}-\frac{3368229783257}{637882938492}a^{8}-\frac{2174578961895}{147914884288}a^{7}+\frac{100589803869131}{2551531753968}a^{6}+\frac{67263976273829}{1701021169312}a^{5}-\frac{120677498154875}{850510584656}a^{4}+\frac{5817247561957}{1275765876984}a^{3}+\frac{102795488301229}{637882938492}a^{2}-\frac{7639150192463}{106313823082}a-\frac{103354287897}{53156911541}$, $\frac{129969146713}{20412254031744}a^{15}-\frac{588147184805}{20412254031744}a^{14}-\frac{6317122517629}{20412254031744}a^{13}+\frac{2525453857383}{1701021169312}a^{12}+\frac{37722411026357}{6804084677248}a^{11}-\frac{99527526318603}{3402042338624}a^{10}-\frac{297184181168501}{6804084677248}a^{9}+\frac{700469594242849}{2551531753968}a^{8}+\frac{59270021935481}{443744652864}a^{7}-\frac{15\cdots 35}{1275765876984}a^{6}-\frac{106950645137297}{1701021169312}a^{5}+\frac{62\cdots 89}{2551531753968}a^{4}+\frac{15728955062500}{159470734623}a^{3}-\frac{11\cdots 61}{637882938492}a^{2}-\frac{27966099757340}{53156911541}a+\frac{3964811638699}{53156911541}$, $\frac{12666075041}{6804084677248}a^{15}-\frac{45582695877}{6804084677248}a^{14}+\frac{673398388027}{6804084677248}a^{13}+\frac{1204747125081}{3402042338624}a^{12}-\frac{13963090228371}{6804084677248}a^{11}-\frac{12486010246397}{1701021169312}a^{10}+\frac{141678682334911}{6804084677248}a^{9}+\frac{257146790696859}{3402042338624}a^{8}-\frac{15488446157615}{147914884288}a^{7}-\frac{687217199417857}{1701021169312}a^{6}+\frac{366305253046041}{1701021169312}a^{5}+\frac{226228748823791}{212627646164}a^{4}+\frac{1997286517901}{53156911541}a^{3}-\frac{236631699940979}{212627646164}a^{2}-\frac{50659855430327}{106313823082}a+\frac{3097303179253}{53156911541}$, $\frac{12110228071}{20412254031744}a^{15}-\frac{205814609}{6804084677248}a^{14}+\frac{210339646451}{6804084677248}a^{13}+\frac{26063374469}{10206127015872}a^{12}-\frac{12376935551753}{20412254031744}a^{11}-\frac{646305662741}{5103063507936}a^{10}+\frac{111883122254077}{20412254031744}a^{9}+\frac{30483696876109}{10206127015872}a^{8}-\frac{9930875598431}{443744652864}a^{7}-\frac{163885227948593}{5103063507936}a^{6}+\frac{64063601524143}{1701021169312}a^{5}+\frac{179123503608241}{1275765876984}a^{4}-\frac{20291223856445}{425255292328}a^{3}-\frac{85722306891065}{637882938492}a^{2}-\frac{3315702740937}{106313823082}a+\frac{254635593288}{53156911541}$ 1658681/38225194816a^15 - 49920877/114675584448a^14 - 197250803/114675584448a^13 + 161321369/7167224028a^12 + 1856589611/114675584448a^11 - 12552552373/28668896112a^10 + 16995573473/114675584448a^9 + 225281566861/57337792224a^8 - 85469515/25968203a^7 - 113705321711/7167224028a^6 + 41241020825/2389074676a^5 + 111681964471/4778149352a^4 - 383530035271/14334448056a^3 - 17266528447/1791806007a^2 + 12962640727/1194537338a + 287646039/597268669, 36081/567811456a^15 - 1589/567811456a^14 - 2047169/567811456a^13 + 4293/35488216a^12 + 46151979/567811456a^11 - 1034637/283905728a^10 - 524990323/567811456a^9 + 5526613/70976432a^8 + 1575409909/283905728a^7 - 30149709/35488216a^6 - 2357252353/141952864a^5 + 283620161/70976432a^4 + 718795095/35488216a^3 - 28752096/4436027a^2 - 24050331/4436027a + 4043608/4436027, -23290445/76450389632a^15 + 240657593/229351168896a^14 + 3637542757/229351168896a^13 - 6438039557/114675584448a^12 - 70900379185/229351168896a^11 + 16285663673/14334448056a^10 + 632279069213/229351168896a^9 - 1232281252397/114675584448a^8 - 18127836673/1661964992a^7 + 2715673745557/57337792224a^6 + 298689284531/19112597408a^5 - 202120661821/2389074676a^4 - 144340627175/14334448056a^3 + 363581871055/7167224028a^2 + 5386570337/1194537338a - 3940669446/597268669, -727603841/6804084677248a^15 + 8943039067/20412254031744a^14 + 108704321315/20412254031744a^13 - 28869532739/1275765876984a^12 - 1989590443829/20412254031744a^11 + 4505902810301/10206127015872a^10 + 15847665043333/20412254031744a^9 - 20447411872103/5103063507936a^8 - 333954205149/147914884288a^7 + 21323399153399/1275765876984a^6 - 1116651881401/1701021169312a^5 - 23278010065101/850510584656a^4 + 2073552067241/318941469246a^3 + 2570330130887/159470734623a^2 - 577284791957/106313823082a - 233525786758/53156911541, -454701193/1701021169312a^15 - 282877863/3402042338624a^14 + 54803387637/3402042338624a^13 + 15777935295/3402042338624a^12 - 82144862777/212627646164a^11 - 378758177075/3402042338624a^10 + 7955366021433/1701021169312a^9 + 5033939546307/3402042338624a^8 - 2202481962461/73957442144a^7 - 19091066802003/1701021169312a^6 + 79117583017311/850510584656a^5 + 37515368903675/850510584656a^4 - 23265860828615/212627646164a^3 - 14021203899823/212627646164a^2 + 436607976441/106313823082a - 16695463868/53156911541, -773841331/850510584656a^15 + 33583736939/10206127015872a^14 + 476822685919/10206127015872a^13 - 1813500308401/10206127015872a^12 - 4540631124431/5103063507936a^11 + 37183484730881/10206127015872a^10 + 9688511392867/1275765876984a^9 - 359332647371965/10206127015872a^8 - 2007786751633/73957442144a^7 + 825646520991305/5103063507936a^6 + 24057356473427/850510584656a^5 - 270701524591523/850510584656a^4 - 15342870386459/637882938492a^3 + 146952012501305/637882938492a^2 + 6695262384319/106313823082a - 476228070563/53156911541, -338955541/1701021169312a^15 + 2532305179/10206127015872a^14 + 116292162983/10206127015872a^13 - 114266413595/10206127015872a^12 - 82497609049/318941469246a^11 + 1759655811103/10206127015872a^10 + 15067244341727/5103063507936a^9 - 8745446140391/10206127015872a^8 - 1301148489373/73957442144a^7 - 13466408930597/5103063507936a^6 + 42961232622673/850510584656a^5 + 27944482217371/850510584656a^4 - 8031435400591/159470734623a^3 - 40197607627343/637882938492a^2 - 865360339085/53156911541a + 127745813976/53156911541, 53128229/295829768576a^15 - 523230569/887489305728a^14 - 8412302869/887489305728a^13 + 13762343453/443744652864a^12 + 168293855041/887489305728a^11 - 33800777573/55468081608a^10 - 1585563094805/887489305728a^9 + 2415123446237/443744652864a^8 + 1197175617279/147914884288a^7 - 4653957438739/221872326432a^6 - 1271262939451/73957442144a^5 + 435382794943/18489360536a^4 + 1092118157195/55468081608a^3 + 100677316585/13867020402a^2 - 2066555066/2311170067a - 82206514/2311170067, 37595331/1701021169312a^15 + 3858806661/3402042338624a^14 - 11262106527/3402042338624a^13 - 195679793925/3402042338624a^12 + 52102107041/425255292328a^11 + 3864214464545/3402042338624a^10 - 3306420560483/1701021169312a^9 - 37575625809249/3402042338624a^8 + 1095480768967/73957442144a^7 + 93631754604245/1701021169312a^6 - 42998378071133/850510584656a^5 - 113892106921553/850510584656a^4 + 10117471090353/212627646164a^3 + 29095298625079/212627646164a^2 + 3715942873719/106313823082a - 271854792461/53156911541, 162113873/3402042338624a^15 - 4406067097/10206127015872a^14 - 49491420251/10206127015872a^13 + 60092548961/2551531753968a^12 + 1718611880795/10206127015872a^11 - 1269567453703/2551531753968a^10 - 27889823231947/10206127015872a^9 + 26050771888537/5103063507936a^8 + 823955827057/36978721072a^7 - 66332429174543/2551531753968a^6 - 37114122587661/425255292328a^5 + 12322921859541/212627646164a^4 + 175826641928747/1275765876984a^3 - 17052928668775/637882938492a^2 - 2841668189362/53156911541a - 372754224474/53156911541, 6567443803/6804084677248a^15 - 35593756255/20412254031744a^14 - 1063532882603/20412254031744a^13 + 926302688167/10206127015872a^12 + 21895466300159/20412254031744a^11 - 2228716187395/1275765876984a^10 - 213073206520195/20412254031744a^9 + 152792901830635/10206127015872a^8 + 7102827249475/147914884288a^7 - 266066213410499/5103063507936a^6 - 153816756148285/1701021169312a^5 + 1823791001427/53156911541a^4 + 63694914455761/1275765876984a^3 + 36180309471451/637882938492a^2 + 1940593853085/53156911541a - 236534852882/53156911541, -756712145/6804084677248a^15 + 4735602287/20412254031744a^14 + 141394377787/20412254031744a^13 - 9305328857/637882938492a^12 - 3569907714193/20412254031744a^11 + 3901027206187/10206127015872a^10 + 45821864592017/20412254031744a^9 - 3368229783257/637882938492a^8 - 2174578961895/147914884288a^7 + 100589803869131/2551531753968a^6 + 67263976273829/1701021169312a^5 - 120677498154875/850510584656a^4 + 5817247561957/1275765876984a^3 + 102795488301229/637882938492a^2 - 7639150192463/106313823082a - 103354287897/53156911541, 129969146713/20412254031744a^15 - 588147184805/20412254031744a^14 - 6317122517629/20412254031744a^13 + 2525453857383/1701021169312a^12 + 37722411026357/6804084677248a^11 - 99527526318603/3402042338624a^10 - 297184181168501/6804084677248a^9 + 700469594242849/2551531753968a^8 + 59270021935481/443744652864a^7 - 1585537615096535/1275765876984a^6 - 106950645137297/1701021169312a^5 + 6286806681195589/2551531753968a^4 + 15728955062500/159470734623a^3 - 1177849017394061/637882938492a^2 - 27966099757340/53156911541a + 3964811638699/53156911541, -12666075041/6804084677248a^15 - 45582695877/6804084677248a^14 + 673398388027/6804084677248a^13 + 1204747125081/3402042338624a^12 - 13963090228371/6804084677248a^11 - 12486010246397/1701021169312a^10 + 141678682334911/6804084677248a^9 + 257146790696859/3402042338624a^8 - 15488446157615/147914884288a^7 - 687217199417857/1701021169312a^6 + 366305253046041/1701021169312a^5 + 226228748823791/212627646164a^4 + 1997286517901/53156911541a^3 - 236631699940979/212627646164a^2 - 50659855430327/106313823082a + 3097303179253/53156911541, -12110228071/20412254031744a^15 - 205814609/6804084677248a^14 + 210339646451/6804084677248a^13 + 26063374469/10206127015872a^12 - 12376935551753/20412254031744a^11 - 646305662741/5103063507936a^10 + 111883122254077/20412254031744a^9 + 30483696876109/10206127015872a^8 - 9930875598431/443744652864a^7 - 163885227948593/5103063507936a^6 + 64063601524143/1701021169312a^5 + 179123503608241/1275765876984a^4 - 20291223856445/425255292328a^3 - 85722306891065/637882938492a^2 - 3315702740937/106313823082*a + 254635593288/53156911541 (assuming GRH)	comment:Fundamental units 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:	$ 5897811755427.919 $ (assuming GRH)	comment:Regulator sage:K.regulator() gp:K.reg magma:Regulator(K); oscar:regulator(K)
Unit signature rank:	$ 12 $ (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 5897811755427.919 \cdot 4}{2\cdot\sqrt{5191562722311896995363773104453715984}}\cr\approx \mathstrut & 0.339274938443352 \end{aligned}\] (assuming GRH)

comment:Analytic class number formula

sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168) 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))))

gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168, 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)))]

magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168); 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)));

oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - x^15 - 65*x^14 + 64*x^13 + 1711*x^12 - 1622*x^11 - 23327*x^10 + 20556*x^9 + 174638*x^8 - 133920*x^7 - 701148*x^6 + 398120*x^5 + 1375168*x^4 - 298560*x^3 - 1095936*x^2 - 262656*x + 39168); 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

$C_2^5:(C_2\times S_4)$ (as 16T1298):

comment:Galois group

sage:K.galois_group()

gp:polgalois(K.pol)

magma:G = GaloisGroup(K);

oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)

A solvable group of order 1536

The 62 conjugacy class representatives for $C_2^5:(C_2\times S_4)$

Character table for $C_2^5:(C_2\times S_4)$

Intermediate fields

$\Q(\sqrt{13}) $, 4.4.33097.1, 8.8.31286045252449.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

comment:Intermediate fields

sage:K.subfields()[1:-1]

gp:L = nfsubfields(K); L[2..length(L)]

magma:L := Subfields(K); L[2..#L];

oscar:subfields(K)[2:end-1]

Sibling fields

Degree 16 siblings:	data not computed
Degree 32 siblings:	data not computed
Minimal sibling:	This field is its own minimal sibling

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	${\href{/padicField/5.8.0.1}{8} }^{2}$	R	${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	R	R	${\href{/padicField/19.4.0.1}{4} }^{4}$	R	${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{4}$	${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.4.0.1}{4} }$	${\href{/padicField/43.8.0.1}{8} }^{2}$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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

comment:Frobenius cycle types

sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.2.2.4a1.1	$x^{4} + 2 x^{3} + 5 x^{2} + 4 x + 5$	$2$	$2$	$4$	$C_2^2$	$$[2]^{2}$$
	2.6.1.0a1.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
	2.6.1.0a1.1	$x^{6} + x^{4} + x^{3} + x + 1$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$3$ 3	3.2.2.2a1.1	$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	3.4.1.0a1.1	$x^{4} + 2 x^{3} + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	3.2.2.2a1.1	$x^{4} + 4 x^{3} + 8 x^{2} + 11 x + 4$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
	3.4.1.0a1.1	$x^{4} + 2 x^{3} + 2$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
$7$ 7	7.2.2.2a1.1	$x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$	$2$	$2$	$2$	$C_4$	$$[\ ]_{2}^{2}$$
$7$ 7	7.12.1.0a1.1	$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$	$1$	$12$	$0$	$C_{12}$	$$[\ ]^{12}$$
$13$ 13	13.2.2.2a1.2	$x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
	13.3.2.3a1.2	$x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	13.3.2.3a1.2	$x^{6} + 4 x^{4} + 22 x^{3} + 4 x^{2} + 44 x + 134$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
$17$ 17	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	17.1.2.1a1.1	$x^{2} + 17$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	17.3.2.3a1.1	$x^{6} + 2 x^{4} + 28 x^{3} + x^{2} + 45 x + 196$	$2$	$3$	$3$	$C_6$	$$[\ ]_{2}^{3}$$
	17.6.1.0a1.1	$x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3$	$1$	$6$	$0$	$C_6$	$$[\ ]^{6}$$
$23$ 23	23.1.2.1a1.1	$x^{2} + 23$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	23.1.2.1a1.1	$x^{2} + 23$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
	23.4.1.0a1.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	23.4.1.0a1.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
	23.2.2.2a1.2	$x^{4} + 42 x^{3} + 451 x^{2} + 210 x + 48$	$2$	$2$	$2$	$C_2^2$	$$[\ ]_{2}^{2}$$
$1439$ 1439		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $2$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$1$	$4$	$0$	$C_4$	$$[\ ]^{4}$$
		Deg $4$	$2$	$2$	$2$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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