LMFDB - Number field 32.0.788112603159847969611382715115371034050560000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536)

gp: K = bnfinit(y^32 - 12*y^28 + 86*y^24 - 427*y^20 + 1809*y^16 - 6832*y^12 + 22016*y^8 - 49152*y^4 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536);

oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536)

$ x^{32} - 12x^{28} + 86x^{24} - 427x^{20} + 1809x^{16} - 6832x^{12} + 22016x^{8} - 49152x^{4} + 65536 $ x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$788112603159847969611382715115371034050560000000000000000$ 788112603159847969611382715115371034050560000000000000000 $\medspace = 2^{64}\cdot 5^{16}\cdot 101^{8}\cdot 401^{4}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$59.98$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{2}5^{1/2}101^{1/2}401^{1/2}\approx 1800.0222220850496$
Ramified primes:	$2$, $5$, $101$, $401$ 2, 5, 101, 401	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{10}+\frac{1}{4}a^{6}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{2}a^{15}-\frac{1}{4}a^{11}-\frac{3}{8}a^{7}+\frac{1}{8}a^{3}$, $\frac{1}{16}a^{20}+\frac{1}{4}a^{16}+\frac{3}{8}a^{12}+\frac{5}{16}a^{8}+\frac{1}{16}a^{4}$, $\frac{1}{32}a^{21}+\frac{1}{8}a^{17}-\frac{5}{16}a^{13}-\frac{11}{32}a^{9}+\frac{1}{32}a^{5}$, $\frac{1}{64}a^{22}+\frac{1}{16}a^{18}+\frac{11}{32}a^{14}-\frac{11}{64}a^{10}-\frac{31}{64}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{128}a^{23}+\frac{1}{32}a^{19}+\frac{11}{64}a^{15}+\frac{53}{128}a^{11}-\frac{31}{128}a^{7}-\frac{1}{4}a^{3}$, $\frac{1}{29440}a^{24}-\frac{3}{1472}a^{20}-\frac{625}{2944}a^{16}-\frac{1227}{29440}a^{12}-\frac{2451}{5888}a^{8}-\frac{27}{184}a^{4}+\frac{16}{115}$, $\frac{1}{58880}a^{25}-\frac{3}{2944}a^{21}-\frac{625}{5888}a^{17}+\frac{28213}{58880}a^{13}+\frac{3437}{11776}a^{9}-\frac{27}{368}a^{5}+\frac{8}{115}a$, $\frac{1}{117760}a^{26}-\frac{3}{5888}a^{22}-\frac{625}{11776}a^{18}+\frac{28213}{117760}a^{14}-\frac{8339}{23552}a^{10}-\frac{27}{736}a^{6}+\frac{4}{115}a^{2}$, $\frac{1}{235520}a^{27}-\frac{3}{11776}a^{23}-\frac{625}{23552}a^{19}-\frac{89547}{235520}a^{15}-\frac{8339}{47104}a^{11}+\frac{709}{1472}a^{7}-\frac{111}{230}a^{3}$, $\frac{1}{25907200}a^{28}-\frac{79}{6476800}a^{24}-\frac{57969}{2590720}a^{20}+\frac{9518133}{25907200}a^{16}+\frac{9222177}{25907200}a^{12}-\frac{33411}{161920}a^{8}+\frac{2686}{6325}a^{4}-\frac{2556}{6325}$, $\frac{1}{51814400}a^{29}-\frac{79}{12953600}a^{25}-\frac{57969}{5181440}a^{21}+\frac{9518133}{51814400}a^{17}-\frac{16685023}{51814400}a^{13}-\frac{33411}{323840}a^{9}+\frac{1343}{6325}a^{5}-\frac{1278}{6325}a$, $\frac{1}{103628800}a^{30}-\frac{79}{25907200}a^{26}-\frac{57969}{10362880}a^{22}+\frac{9518133}{103628800}a^{18}-\frac{16685023}{103628800}a^{14}-\frac{33411}{647680}a^{10}-\frac{2491}{6325}a^{6}-\frac{639}{6325}a^{2}$, $\frac{1}{207257600}a^{31}-\frac{79}{51814400}a^{27}-\frac{57969}{20725760}a^{23}+\frac{9518133}{207257600}a^{19}+\frac{86943777}{207257600}a^{15}+\frac{614269}{1295360}a^{11}-\frac{2491}{12650}a^{7}-\frac{639}{12650}a^{3}$ 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, 1/2*a^17 - 1/2*a^5 - 1/2*a, 1/4*a^18 - 1/2*a^10 + 1/4*a^6 + 1/4*a^2, 1/8*a^19 - 1/2*a^15 - 1/4*a^11 - 3/8*a^7 + 1/8*a^3, 1/16*a^20 + 1/4*a^16 + 3/8*a^12 + 5/16*a^8 + 1/16*a^4, 1/32*a^21 + 1/8*a^17 - 5/16*a^13 - 11/32*a^9 + 1/32*a^5, 1/64*a^22 + 1/16*a^18 + 11/32*a^14 - 11/64*a^10 - 31/64*a^6 - 1/2*a^2, 1/128*a^23 + 1/32*a^19 + 11/64*a^15 + 53/128*a^11 - 31/128*a^7 - 1/4*a^3, 1/29440*a^24 - 3/1472*a^20 - 625/2944*a^16 - 1227/29440*a^12 - 2451/5888*a^8 - 27/184*a^4 + 16/115, 1/58880*a^25 - 3/2944*a^21 - 625/5888*a^17 + 28213/58880*a^13 + 3437/11776*a^9 - 27/368*a^5 + 8/115*a, 1/117760*a^26 - 3/5888*a^22 - 625/11776*a^18 + 28213/117760*a^14 - 8339/23552*a^10 - 27/736*a^6 + 4/115*a^2, 1/235520*a^27 - 3/11776*a^23 - 625/23552*a^19 - 89547/235520*a^15 - 8339/47104*a^11 + 709/1472*a^7 - 111/230*a^3, 1/25907200*a^28 - 79/6476800*a^24 - 57969/2590720*a^20 + 9518133/25907200*a^16 + 9222177/25907200*a^12 - 33411/161920*a^8 + 2686/6325*a^4 - 2556/6325, 1/51814400*a^29 - 79/12953600*a^25 - 57969/5181440*a^21 + 9518133/51814400*a^17 - 16685023/51814400*a^13 - 33411/323840*a^9 + 1343/6325*a^5 - 1278/6325*a, 1/103628800*a^30 - 79/25907200*a^26 - 57969/10362880*a^22 + 9518133/103628800*a^18 - 16685023/103628800*a^14 - 33411/647680*a^10 - 2491/6325*a^6 - 639/6325*a^2, 1/207257600*a^31 - 79/51814400*a^27 - 57969/20725760*a^23 + 9518133/207257600*a^19 + 86943777/207257600*a^15 + 614269/1295360*a^11 - 2491/12650*a^7 - 639/12650*a^3

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

$C_{3}\times C_{192}$, which has order $576$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Relative class number: $576$ (assuming GRH)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ \frac{643}{51814400} a^{29} - \frac{13177}{12953600} a^{25} + \frac{36173}{5181440} a^{21} - \frac{1780481}{51814400} a^{17} + \frac{6776051}{51814400} a^{13} - \frac{359641}{647680} a^{9} + \frac{188059}{101200} a^{5} - \frac{44603}{12650} a $ (643)/(51814400)a^(29) - (13177)/(12953600)a^(25) + (36173)/(5181440)a^(21) - (1780481)/(51814400)a^(17) + (6776051)/(51814400)a^(13) - (359641)/(647680)a^(9) + (188059)/(101200)a^(5) - (44603)/(12650)a (order $8$)	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:	$\frac{7599}{51814400}a^{29}-\frac{34261}{12953600}a^{25}+\frac{93729}{5181440}a^{21}-\frac{4301733}{51814400}a^{17}+\frac{16863743}{51814400}a^{13}-\frac{809243}{647680}a^{9}+\frac{786799}{202400}a^{5}-\frac{87629}{12650}a$, $\frac{383}{2072576}a^{30}-\frac{643}{51814400}a^{29}-\frac{755}{518144}a^{26}+\frac{13177}{12953600}a^{25}+\frac{7525}{1036288}a^{22}-\frac{36173}{5181440}a^{21}-\frac{58725}{2072576}a^{18}+\frac{1780481}{51814400}a^{17}+\frac{11105}{90112}a^{14}-\frac{6776051}{51814400}a^{13}-\frac{104725}{259072}a^{10}+\frac{359641}{647680}a^{9}+\frac{14645}{16192}a^{6}-\frac{188059}{101200}a^{5}-\frac{215}{1012}a^{2}+\frac{44603}{12650}a-1$, $\frac{7669}{103628800}a^{31}+\frac{383}{2072576}a^{30}-\frac{29121}{25907200}a^{27}-\frac{755}{518144}a^{26}+\frac{70699}{10362880}a^{23}+\frac{7525}{1036288}a^{22}-\frac{3105623}{103628800}a^{19}-\frac{58725}{2072576}a^{18}+\frac{12562973}{103628800}a^{15}+\frac{11105}{90112}a^{14}-\frac{52117}{112640}a^{11}-\frac{104725}{259072}a^{10}+\frac{1020213}{809600}a^{7}+\frac{14645}{16192}a^{6}-\frac{26547}{12650}a^{3}-\frac{215}{1012}a^{2}+1$, $\frac{4073}{20725760}a^{30}-\frac{7959}{5181440}a^{26}+\frac{18087}{2072576}a^{22}-\frac{678211}{20725760}a^{18}+\frac{129439}{901120}a^{14}-\frac{56937}{129536}a^{10}+\frac{25247}{20240}a^{6}-\frac{3081}{5060}a^{2}$, $\frac{12663}{51814400}a^{31}-\frac{931}{404800}a^{27}+\frac{65793}{5181440}a^{23}-\frac{2804421}{51814400}a^{19}+\frac{11064971}{51814400}a^{15}-\frac{2073619}{2590720}a^{11}+\frac{1519477}{809600}a^{7}-\frac{131427}{50600}a^{3}$, $\frac{1531}{5181440}a^{30}+\frac{19793}{51814400}a^{29}-\frac{91}{471040}a^{28}-\frac{1619}{647680}a^{26}-\frac{50927}{12953600}a^{25}+\frac{197}{117760}a^{24}+\frac{6349}{518144}a^{22}+\frac{111423}{5181440}a^{21}-\frac{469}{47104}a^{20}-\frac{255457}{5181440}a^{18}-\frac{4716731}{51814400}a^{17}+\frac{18857}{471040}a^{16}+\frac{44753}{225280}a^{14}+\frac{19546801}{51814400}a^{13}-\frac{70651}{471040}a^{12}-\frac{176229}{259072}a^{10}-\frac{441633}{323840}a^{9}+\frac{1687}{2944}a^{8}+\frac{60287}{40480}a^{6}+\frac{742243}{202400}a^{5}-\frac{2771}{1840}a^{4}-\frac{1919}{2530}a^{2}-\frac{31314}{6325}a+\frac{158}{115}$, $\frac{2641}{207257600}a^{31}+\frac{643}{51814400}a^{29}-\frac{81}{471040}a^{28}-\frac{18119}{51814400}a^{27}-\frac{13177}{12953600}a^{25}-\frac{133}{117760}a^{24}+\frac{41471}{20725760}a^{23}+\frac{36173}{5181440}a^{21}+\frac{545}{47104}a^{20}-\frac{70589}{9011200}a^{19}-\frac{1780481}{51814400}a^{17}-\frac{27173}{471040}a^{16}+\frac{7065297}{207257600}a^{15}+\frac{6776051}{51814400}a^{13}+\frac{101439}{471040}a^{12}-\frac{21577}{161920}a^{11}-\frac{359641}{647680}a^{9}-\frac{5735}{5888}a^{8}+\frac{168833}{404800}a^{7}+\frac{188059}{101200}a^{5}+\frac{3447}{920}a^{4}-\frac{15083}{25300}a^{3}-\frac{44603}{12650}a-\frac{907}{115}$, $\frac{16509}{103628800}a^{30}-\frac{2549}{12953600}a^{28}-\frac{19631}{25907200}a^{26}+\frac{2931}{3238400}a^{24}+\frac{33779}{10362880}a^{22}-\frac{3099}{1295360}a^{20}-\frac{1312703}{103628800}a^{18}+\frac{142983}{12953600}a^{16}+\frac{5705453}{103628800}a^{14}-\frac{598453}{12953600}a^{12}-\frac{178393}{1295360}a^{10}+\frac{5527}{40480}a^{8}+\frac{28459}{404800}a^{6}+\frac{763}{25300}a^{4}+\frac{24791}{25300}a^{2}-\frac{11607}{6325}$, $\frac{139}{4505600}a^{31}-\frac{2051}{20725760}a^{30}+\frac{999}{25907200}a^{29}+\frac{7807}{25907200}a^{27}+\frac{4993}{5181440}a^{26}-\frac{2471}{6476800}a^{25}-\frac{24413}{10362880}a^{23}-\frac{7309}{2072576}a^{22}+\frac{2649}{2590720}a^{21}+\frac{1398801}{103628800}a^{19}+\frac{343617}{20725760}a^{18}+\frac{78467}{25907200}a^{17}-\frac{4298491}{103628800}a^{15}-\frac{49573}{901120}a^{14}+\frac{144623}{25907200}a^{13}+\frac{553177}{2590720}a^{11}+\frac{62355}{259072}a^{10}+\frac{14409}{647680}a^{9}-\frac{77857}{101200}a^{7}-\frac{9793}{40480}a^{6}-\frac{9613}{50600}a^{5}+\frac{36173}{25300}a^{3}+\frac{757}{5060}a^{2}+\frac{27087}{12650}a$, $\frac{12663}{51814400}a^{31}+\frac{16509}{103628800}a^{30}-\frac{3513}{6476800}a^{28}-\frac{931}{404800}a^{27}-\frac{19631}{25907200}a^{26}+\frac{1993}{404800}a^{24}+\frac{65793}{5181440}a^{23}+\frac{33779}{10362880}a^{22}-\frac{17263}{647680}a^{20}-\frac{2804421}{51814400}a^{19}-\frac{1312703}{103628800}a^{18}+\frac{662571}{6476800}a^{16}+\frac{11064971}{51814400}a^{15}+\frac{5705453}{103628800}a^{14}-\frac{2893461}{6476800}a^{12}-\frac{2073619}{2590720}a^{11}-\frac{178393}{1295360}a^{10}+\frac{479409}{323840}a^{8}+\frac{1519477}{809600}a^{7}+\frac{28459}{404800}a^{6}-\frac{369177}{101200}a^{4}-\frac{131427}{50600}a^{3}-\frac{509}{25300}a^{2}+\frac{16982}{6325}$, $\frac{12629}{103628800}a^{31}+\frac{693}{2355200}a^{30}-\frac{383}{1036288}a^{29}+\frac{3439}{25907200}a^{28}-\frac{19461}{25907200}a^{27}-\frac{1957}{588800}a^{26}+\frac{755}{259072}a^{25}+\frac{4419}{6476800}a^{24}+\frac{31659}{10362880}a^{23}+\frac{4683}{235520}a^{22}-\frac{7525}{518144}a^{21}-\frac{19551}{2590720}a^{20}-\frac{1021143}{103628800}a^{19}-\frac{215031}{2355200}a^{18}+\frac{58725}{1036288}a^{17}+\frac{880987}{25907200}a^{16}+\frac{4069293}{103628800}a^{15}+\frac{848141}{2355200}a^{14}-\frac{11105}{45056}a^{13}-\frac{3270497}{25907200}a^{12}-\frac{366701}{2590720}a^{11}-\frac{79597}{58880}a^{10}+\frac{104725}{129536}a^{9}+\frac{151667}{323840}a^{8}+\frac{154383}{809600}a^{7}+\frac{66511}{18400}a^{6}-\frac{14645}{8096}a^{5}-\frac{113243}{50600}a^{4}+\frac{20617}{50600}a^{3}-\frac{12767}{2300}a^{2}-\frac{291}{506}a+\frac{30816}{6325}$, $\frac{5853}{41451520}a^{31}+\frac{5853}{20725760}a^{30}+\frac{1147}{5181440}a^{29}-\frac{14599}{10362880}a^{27}-\frac{14599}{5181440}a^{26}-\frac{2701}{1295360}a^{25}+\frac{31571}{4145152}a^{23}+\frac{31571}{2072576}a^{22}+\frac{5173}{518144}a^{21}-\frac{1409311}{41451520}a^{19}-\frac{1409311}{20725760}a^{18}-\frac{217289}{5181440}a^{17}+\frac{235419}{1802240}a^{15}+\frac{235419}{901120}a^{14}+\frac{33981}{225280}a^{13}-\frac{251039}{518144}a^{11}-\frac{251039}{259072}a^{10}-\frac{4469}{8096}a^{9}+\frac{6359}{5060}a^{7}+\frac{6359}{2530}a^{6}+\frac{47349}{40480}a^{5}-\frac{10353}{5060}a^{3}-\frac{7823}{2530}a^{2}-\frac{233}{2530}a$, $\frac{1}{32384}a^{31}+\frac{11383}{51814400}a^{30}+\frac{20377}{51814400}a^{29}+\frac{107}{1036288}a^{28}+\frac{13}{80960}a^{27}-\frac{30037}{12953600}a^{26}-\frac{50863}{12953600}a^{25}-\frac{1477}{1295360}a^{24}-\frac{111}{32384}a^{23}+\frac{70233}{5181440}a^{22}+\frac{123447}{5181440}a^{21}+\frac{3065}{518144}a^{20}+\frac{581}{32384}a^{19}-\frac{3076861}{51814400}a^{18}-\frac{5351059}{51814400}a^{17}-\frac{22137}{1036288}a^{16}-\frac{10707}{161920}a^{15}+\frac{586097}{2252800}a^{14}+\frac{23015769}{51814400}a^{13}+\frac{431751}{5181440}a^{12}+\frac{6829}{32384}a^{11}-\frac{672381}{647680}a^{10}-\frac{254891}{161920}a^{9}-\frac{6425}{64768}a^{8}-\frac{24269}{32384}a^{7}+\frac{1106241}{404800}a^{6}+\frac{811927}{202400}a^{5}-\frac{609}{2024}a^{4}+\frac{17331}{10120}a^{3}-\frac{98511}{25300}a^{2}-\frac{27466}{6325}a+\frac{3767}{1265}$, $\frac{6139}{207257600}a^{31}-\frac{1249}{20725760}a^{30}+\frac{89}{225280}a^{29}+\frac{1923}{6476800}a^{28}-\frac{43881}{51814400}a^{27}+\frac{1175}{1036288}a^{26}-\frac{5337}{1295360}a^{25}-\frac{3661}{809600}a^{24}+\frac{128789}{20725760}a^{23}-\frac{14735}{2072576}a^{22}+\frac{12145}{518144}a^{21}+\frac{18533}{647680}a^{20}-\frac{5857513}{207257600}a^{19}+\frac{576203}{20725760}a^{18}-\frac{542709}{5181440}a^{17}-\frac{887641}{6476800}a^{16}+\frac{18998603}{207257600}a^{15}-\frac{524989}{4145152}a^{14}+\frac{2141631}{5181440}a^{13}+\frac{3543711}{6476800}a^{12}-\frac{926043}{2590720}a^{11}+\frac{143235}{259072}a^{10}-\frac{23245}{16192}a^{9}-\frac{667889}{323840}a^{8}+\frac{420007}{404800}a^{7}-\frac{154159}{80960}a^{6}+\frac{129809}{40480}a^{5}+\frac{265071}{50600}a^{4}-\frac{28271}{12650}a^{3}+\frac{2845}{1012}a^{2}-\frac{4558}{1265}a-\frac{48357}{6325}$, $\frac{7807}{103628800}a^{31}-\frac{33127}{51814400}a^{30}-\frac{7739}{51814400}a^{29}+\frac{6421}{5181440}a^{28}-\frac{1941}{1126400}a^{27}+\frac{4291}{563200}a^{26}+\frac{23981}{12953600}a^{25}-\frac{19211}{1295360}a^{24}+\frac{115937}{10362880}a^{23}-\frac{231017}{5181440}a^{22}-\frac{47669}{5181440}a^{21}+\frac{44987}{518144}a^{20}-\frac{5476069}{103628800}a^{19}+\frac{10104909}{51814400}a^{18}+\frac{1909513}{51814400}a^{17}-\frac{1947687}{5181440}a^{16}+\frac{22253559}{103628800}a^{15}-\frac{40318359}{51814400}a^{14}-\frac{8262203}{51814400}a^{13}+\frac{7712373}{5181440}a^{12}-\frac{2119113}{2590720}a^{11}+\frac{1871269}{647680}a^{10}+\frac{48681}{80960}a^{9}-\frac{180993}{32384}a^{8}+\frac{2082339}{809600}a^{7}-\frac{3294829}{404800}a^{6}-\frac{116707}{101200}a^{5}+\frac{309081}{20240}a^{4}-\frac{260379}{50600}a^{3}+\frac{289029}{25300}a^{2}+\frac{18559}{12650}a-\frac{26234}{1265}$ 7599/51814400a^29 - 34261/12953600a^25 + 93729/5181440a^21 - 4301733/51814400a^17 + 16863743/51814400a^13 - 809243/647680a^9 + 786799/202400a^5 - 87629/12650a, 383/2072576a^30 - 643/51814400a^29 - 755/518144a^26 + 13177/12953600a^25 + 7525/1036288a^22 - 36173/5181440a^21 - 58725/2072576a^18 + 1780481/51814400a^17 + 11105/90112a^14 - 6776051/51814400a^13 - 104725/259072a^10 + 359641/647680a^9 + 14645/16192a^6 - 188059/101200a^5 - 215/1012a^2 + 44603/12650a - 1, 7669/103628800a^31 + 383/2072576a^30 - 29121/25907200a^27 - 755/518144a^26 + 70699/10362880a^23 + 7525/1036288a^22 - 3105623/103628800a^19 - 58725/2072576a^18 + 12562973/103628800a^15 + 11105/90112a^14 - 52117/112640a^11 - 104725/259072a^10 + 1020213/809600a^7 + 14645/16192a^6 - 26547/12650a^3 - 215/1012a^2 + 1, 4073/20725760a^30 - 7959/5181440a^26 + 18087/2072576a^22 - 678211/20725760a^18 + 129439/901120a^14 - 56937/129536a^10 + 25247/20240a^6 - 3081/5060a^2, 12663/51814400a^31 - 931/404800a^27 + 65793/5181440a^23 - 2804421/51814400a^19 + 11064971/51814400a^15 - 2073619/2590720a^11 + 1519477/809600a^7 - 131427/50600a^3, 1531/5181440a^30 + 19793/51814400a^29 - 91/471040a^28 - 1619/647680a^26 - 50927/12953600a^25 + 197/117760a^24 + 6349/518144a^22 + 111423/5181440a^21 - 469/47104a^20 - 255457/5181440a^18 - 4716731/51814400a^17 + 18857/471040a^16 + 44753/225280a^14 + 19546801/51814400a^13 - 70651/471040a^12 - 176229/259072a^10 - 441633/323840a^9 + 1687/2944a^8 + 60287/40480a^6 + 742243/202400a^5 - 2771/1840a^4 - 1919/2530a^2 - 31314/6325a + 158/115, 2641/207257600a^31 + 643/51814400a^29 - 81/471040a^28 - 18119/51814400a^27 - 13177/12953600a^25 - 133/117760a^24 + 41471/20725760a^23 + 36173/5181440a^21 + 545/47104a^20 - 70589/9011200a^19 - 1780481/51814400a^17 - 27173/471040a^16 + 7065297/207257600a^15 + 6776051/51814400a^13 + 101439/471040a^12 - 21577/161920a^11 - 359641/647680a^9 - 5735/5888a^8 + 168833/404800a^7 + 188059/101200a^5 + 3447/920a^4 - 15083/25300a^3 - 44603/12650a - 907/115, 16509/103628800a^30 - 2549/12953600a^28 - 19631/25907200a^26 + 2931/3238400a^24 + 33779/10362880a^22 - 3099/1295360a^20 - 1312703/103628800a^18 + 142983/12953600a^16 + 5705453/103628800a^14 - 598453/12953600a^12 - 178393/1295360a^10 + 5527/40480a^8 + 28459/404800a^6 + 763/25300a^4 + 24791/25300a^2 - 11607/6325, 139/4505600a^31 - 2051/20725760a^30 + 999/25907200a^29 + 7807/25907200a^27 + 4993/5181440a^26 - 2471/6476800a^25 - 24413/10362880a^23 - 7309/2072576a^22 + 2649/2590720a^21 + 1398801/103628800a^19 + 343617/20725760a^18 + 78467/25907200a^17 - 4298491/103628800a^15 - 49573/901120a^14 + 144623/25907200a^13 + 553177/2590720a^11 + 62355/259072a^10 + 14409/647680a^9 - 77857/101200a^7 - 9793/40480a^6 - 9613/50600a^5 + 36173/25300a^3 + 757/5060a^2 + 27087/12650a, 12663/51814400a^31 + 16509/103628800a^30 - 3513/6476800a^28 - 931/404800a^27 - 19631/25907200a^26 + 1993/404800a^24 + 65793/5181440a^23 + 33779/10362880a^22 - 17263/647680a^20 - 2804421/51814400a^19 - 1312703/103628800a^18 + 662571/6476800a^16 + 11064971/51814400a^15 + 5705453/103628800a^14 - 2893461/6476800a^12 - 2073619/2590720a^11 - 178393/1295360a^10 + 479409/323840a^8 + 1519477/809600a^7 + 28459/404800a^6 - 369177/101200a^4 - 131427/50600a^3 - 509/25300a^2 + 16982/6325, 12629/103628800a^31 + 693/2355200a^30 - 383/1036288a^29 + 3439/25907200a^28 - 19461/25907200a^27 - 1957/588800a^26 + 755/259072a^25 + 4419/6476800a^24 + 31659/10362880a^23 + 4683/235520a^22 - 7525/518144a^21 - 19551/2590720a^20 - 1021143/103628800a^19 - 215031/2355200a^18 + 58725/1036288a^17 + 880987/25907200a^16 + 4069293/103628800a^15 + 848141/2355200a^14 - 11105/45056a^13 - 3270497/25907200a^12 - 366701/2590720a^11 - 79597/58880a^10 + 104725/129536a^9 + 151667/323840a^8 + 154383/809600a^7 + 66511/18400a^6 - 14645/8096a^5 - 113243/50600a^4 + 20617/50600a^3 - 12767/2300a^2 - 291/506a + 30816/6325, 5853/41451520a^31 + 5853/20725760a^30 + 1147/5181440a^29 - 14599/10362880a^27 - 14599/5181440a^26 - 2701/1295360a^25 + 31571/4145152a^23 + 31571/2072576a^22 + 5173/518144a^21 - 1409311/41451520a^19 - 1409311/20725760a^18 - 217289/5181440a^17 + 235419/1802240a^15 + 235419/901120a^14 + 33981/225280a^13 - 251039/518144a^11 - 251039/259072a^10 - 4469/8096a^9 + 6359/5060a^7 + 6359/2530a^6 + 47349/40480a^5 - 10353/5060a^3 - 7823/2530a^2 - 233/2530a, 1/32384a^31 + 11383/51814400a^30 + 20377/51814400a^29 + 107/1036288a^28 + 13/80960a^27 - 30037/12953600a^26 - 50863/12953600a^25 - 1477/1295360a^24 - 111/32384a^23 + 70233/5181440a^22 + 123447/5181440a^21 + 3065/518144a^20 + 581/32384a^19 - 3076861/51814400a^18 - 5351059/51814400a^17 - 22137/1036288a^16 - 10707/161920a^15 + 586097/2252800a^14 + 23015769/51814400a^13 + 431751/5181440a^12 + 6829/32384a^11 - 672381/647680a^10 - 254891/161920a^9 - 6425/64768a^8 - 24269/32384a^7 + 1106241/404800a^6 + 811927/202400a^5 - 609/2024a^4 + 17331/10120a^3 - 98511/25300a^2 - 27466/6325a + 3767/1265, 6139/207257600a^31 - 1249/20725760a^30 + 89/225280a^29 + 1923/6476800a^28 - 43881/51814400a^27 + 1175/1036288a^26 - 5337/1295360a^25 - 3661/809600a^24 + 128789/20725760a^23 - 14735/2072576a^22 + 12145/518144a^21 + 18533/647680a^20 - 5857513/207257600a^19 + 576203/20725760a^18 - 542709/5181440a^17 - 887641/6476800a^16 + 18998603/207257600a^15 - 524989/4145152a^14 + 2141631/5181440a^13 + 3543711/6476800a^12 - 926043/2590720a^11 + 143235/259072a^10 - 23245/16192a^9 - 667889/323840a^8 + 420007/404800a^7 - 154159/80960a^6 + 129809/40480a^5 + 265071/50600a^4 - 28271/12650a^3 + 2845/1012a^2 - 4558/1265a - 48357/6325, 7807/103628800a^31 - 33127/51814400a^30 - 7739/51814400a^29 + 6421/5181440a^28 - 1941/1126400a^27 + 4291/563200a^26 + 23981/12953600a^25 - 19211/1295360a^24 + 115937/10362880a^23 - 231017/5181440a^22 - 47669/5181440a^21 + 44987/518144a^20 - 5476069/103628800a^19 + 10104909/51814400a^18 + 1909513/51814400a^17 - 1947687/5181440a^16 + 22253559/103628800a^15 - 40318359/51814400a^14 - 8262203/51814400a^13 + 7712373/5181440a^12 - 2119113/2590720a^11 + 1871269/647680a^10 + 48681/80960a^9 - 180993/32384a^8 + 2082339/809600a^7 - 3294829/404800a^6 - 116707/101200a^5 + 309081/20240a^4 - 260379/50600a^3 + 289029/25300a^2 + 18559/12650a - 26234/1265 (assuming GRH)	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:	$ 18060264289735.246 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 18060264289735.246 \cdot 576}{8\cdot\sqrt{788112603159847969611382715115371034050560000000000000000}}\cr\approx \mathstrut & 0.273300455328539 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 12*x^28 + 86*x^24 - 427*x^20 + 1809*x^16 - 6832*x^12 + 22016*x^8 - 49152*x^4 + 65536);

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

$D_4^2:C_2^3$ (as 32T12882):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

A solvable group of order 512

The 80 conjugacy class representatives for $D_4^2:C_2^3$

Character table for $D_4^2:C_2^3$

Intermediate fields

$\Q(\sqrt{5}) $, $\Q(\sqrt{-5}) $, $\Q(\sqrt{-2}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{-1}) $, $\Q(\sqrt{-10}) $, $\Q(\sqrt{10}) $, 4.4.2525.1, 4.0.40400.1, 4.0.161600.4, 4.4.161600.1, $\Q(i, \sqrt{10})$, $\Q(i, \sqrt{5})$, $\Q(\zeta_{8})$, $\Q(\sqrt{-2}, \sqrt{5})$, $\Q(\sqrt{2}, \sqrt{-5})$, $\Q(\sqrt{-2}, \sqrt{-5})$, $\Q(\sqrt{2}, \sqrt{5})$, 8.0.2556625625.1, 8.8.654496160000.1, 8.8.10471938560000.1, 8.0.10471938560000.1, 8.0.40960000.1, 8.0.1632160000.5, 8.0.417832960000.16, 8.0.26114560000.3, 8.0.417832960000.5, 8.0.417832960000.10, 8.8.26114560000.1, 16.0.174584382462361600000000.1, 16.0.428365223454745600000000.1, 16.0.28073343284330207641600000000.2, 16.0.109661497204414873600000000.2, 16.0.28073343284330207641600000000.1, 16.0.109661497204414873600000000.1, 16.16.28073343284330207641600000000.1

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

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

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

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

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

Sibling fields

Degree 32 siblings:	data not computed
Minimal sibling:	not computed

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.8.0.1}{8} }^{4}$	R	${\href{/padicField/7.4.0.1}{4} }^{8}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$	${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{8}$	${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$	${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{8}$	${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$	${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{8}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{8}$	${\href{/padicField/43.2.0.1}{2} }^{16}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.8.0.1}{8} }^{4}$	${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2		Deg $16$	$4$	$4$	$32$
$2$ 2		Deg $16$	$4$	$4$	$32$
$5$ 5	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$101$ 101	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.2.0.1	$x^{2} + 97 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	101.4.2.1	$x^{4} + 16556 x^{3} + 69319047 x^{2} + 6570770114 x + 216554003$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$401$ 401	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{401}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more