LMFDB - Number field 15.15.430807787028125000000000000.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200)

gp: K = bnfinit(y^15 - 35*y^13 + 425*y^11 - 76*y^10 - 2350*y^9 + 820*y^8 + 6225*y^7 - 2580*y^6 - 7880*y^5 + 2750*y^4 + 4525*y^3 - 500*y^2 - 1100*y - 200, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200)

$ x^{15} - 35 x^{13} + 425 x^{11} - 76 x^{10} - 2350 x^{9} + 820 x^{8} + 6225 x^{7} - 2580 x^{6} + \cdots - 200 $ x^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$15$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[15, 0]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$430807787028125000000000000$ 430807787028125000000000000 $\medspace = 2^{12}\cdot 5^{17}\cdot 13^{10}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$59.65$	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:	$2\cdot 5^{203/100}13^{2/3}\approx 290.113545079841$
Ramified primes:	$2$, $5$, $13$ 2, 5, 13	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{5}) $
$\card{ \Aut(K/\Q) }$:	$1$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}$, $\frac{1}{5}a^{11}-\frac{1}{5}a^{6}$, $\frac{1}{5}a^{12}-\frac{1}{5}a^{7}$, $\frac{1}{10}a^{13}-\frac{1}{10}a^{11}-\frac{1}{10}a^{9}-\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{10}a^{5}-\frac{1}{2}a$, $\frac{1}{115156273340}a^{14}-\frac{2036013297}{57578136670}a^{13}+\frac{3819253977}{115156273340}a^{12}+\frac{5485357707}{57578136670}a^{11}-\frac{8975645511}{115156273340}a^{10}-\frac{5691941203}{57578136670}a^{9}+\frac{26551994597}{57578136670}a^{8}-\frac{14372765893}{28789068335}a^{7}+\frac{29997959221}{115156273340}a^{6}-\frac{4694555967}{57578136670}a^{5}-\frac{2150380959}{5757813667}a^{4}+\frac{5416477535}{11515627334}a^{3}+\frac{3975439797}{23031254668}a^{2}+\frac{1614369341}{11515627334}a+\frac{1814679710}{5757813667}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, 1/5*a^9 + 1/5*a^8 + 1/5*a^7 + 1/5*a^6 + 1/5*a^5, 1/5*a^10 - 1/5*a^5, 1/5*a^11 - 1/5*a^6, 1/5*a^12 - 1/5*a^7, 1/10*a^13 - 1/10*a^11 - 1/10*a^9 - 1/5*a^8 + 2/5*a^7 - 1/10*a^5 - 1/2*a, 1/115156273340*a^14 - 2036013297/57578136670*a^13 + 3819253977/115156273340*a^12 + 5485357707/57578136670*a^11 - 8975645511/115156273340*a^10 - 5691941203/57578136670*a^9 + 26551994597/57578136670*a^8 - 14372765893/28789068335*a^7 + 29997959221/115156273340*a^6 - 4694555967/57578136670*a^5 - 2150380959/5757813667*a^4 + 5416477535/11515627334*a^3 + 3975439797/23031254668*a^2 + 1614369341/11515627334*a + 1814679710/5757813667

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

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

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

oscar: UK, fUK = unit_group(OK)

Rank:	$14$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	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:	$\frac{5395040479}{115156273340}a^{14}+\frac{876804649}{11515627334}a^{13}-\frac{172347129181}{115156273340}a^{12}-\frac{138623232591}{57578136670}a^{11}+\frac{1768405584211}{115156273340}a^{10}+\frac{1185803529943}{57578136670}a^{9}-\frac{800910091675}{11515627334}a^{8}-\frac{1947393707046}{28789068335}a^{7}+\frac{16846551806947}{115156273340}a^{6}+\frac{5382505290027}{57578136670}a^{5}-\frac{761349302269}{5757813667}a^{4}-\frac{594570217555}{11515627334}a^{3}+\frac{831920196899}{23031254668}a^{2}+\frac{122326475879}{11515627334}a-\frac{5354674624}{5757813667}$, $\frac{183274617}{28789068335}a^{14}-\frac{390813597}{11515627334}a^{13}-\frac{7729506563}{28789068335}a^{12}+\frac{64023530303}{57578136670}a^{11}+\frac{119821382423}{28789068335}a^{10}-\frac{719306971569}{57578136670}a^{9}-\frac{157020384630}{5757813667}a^{8}+\frac{1838163820043}{28789068335}a^{7}+\frac{2271049201556}{28789068335}a^{6}-\frac{8840081636741}{57578136670}a^{5}-\frac{592113349648}{5757813667}a^{4}+\frac{931450315120}{5757813667}a^{3}+\frac{349802358902}{5757813667}a^{2}-\frac{624159190557}{11515627334}a-\frac{97744763498}{5757813667}$, $\frac{4174756833}{57578136670}a^{14}+\frac{237538223}{5757813667}a^{13}-\frac{141024967813}{57578136670}a^{12}-\frac{38227024432}{28789068335}a^{11}+\frac{1606558601433}{57578136670}a^{10}+\frac{237210797043}{28789068335}a^{9}-\frac{4084457108888}{28789068335}a^{8}-\frac{112751173884}{28789068335}a^{7}+\frac{19096561097583}{57578136670}a^{6}-\frac{1425393931307}{28789068335}a^{5}-\frac{1910454070875}{5757813667}a^{4}+\frac{421770106057}{5757813667}a^{3}+\frac{1243754154329}{11515627334}a^{2}-\frac{71328038498}{5757813667}a-\frac{46614821845}{5757813667}$, $\frac{5927032639}{115156273340}a^{14}+\frac{2108922543}{28789068335}a^{13}-\frac{186781937957}{115156273340}a^{12}-\frac{65014097342}{28789068335}a^{11}+\frac{372688488435}{23031254668}a^{10}+\frac{505050440874}{28789068335}a^{9}-\frac{3972171417111}{57578136670}a^{8}-\frac{1326620012182}{28789068335}a^{7}+\frac{14568569184823}{115156273340}a^{6}+\frac{211951407167}{5757813667}a^{5}-\frac{482819038672}{5757813667}a^{4}+\frac{69695051927}{11515627334}a^{3}+\frac{246293995703}{23031254668}a^{2}-\frac{75825127516}{5757813667}a-\frac{28406626694}{5757813667}$, $\frac{3422375939}{23031254668}a^{14}+\frac{14743759719}{57578136670}a^{13}-\frac{540070898889}{115156273340}a^{12}-\frac{92287776585}{11515627334}a^{11}+\frac{5414719673339}{115156273340}a^{10}+\frac{3888586719357}{57578136670}a^{9}-\frac{11920518947577}{57578136670}a^{8}-\frac{6245192915923}{28789068335}a^{7}+\frac{48740843427659}{115156273340}a^{6}+\frac{3371618424253}{11515627334}a^{5}-\frac{2143922621200}{5757813667}a^{4}-\frac{1921680114625}{11515627334}a^{3}+\frac{2164892389179}{23031254668}a^{2}+\frac{524804056477}{11515627334}a+\frac{26854721403}{5757813667}$, $\frac{600275827}{23031254668}a^{14}+\frac{2829841107}{28789068335}a^{13}-\frac{93755801081}{115156273340}a^{12}-\frac{94560877424}{28789068335}a^{11}+\frac{904468364519}{115156273340}a^{10}+\frac{997685597481}{28789068335}a^{9}-\frac{1951756733467}{57578136670}a^{8}-\frac{927494983853}{5757813667}a^{7}+\frac{1708916013867}{23031254668}a^{6}+\frac{2030754578234}{5757813667}a^{5}-\frac{430627542347}{5757813667}a^{4}-\frac{3905113076213}{11515627334}a^{3}+\frac{275751240947}{23031254668}a^{2}+\frac{599082284666}{5757813667}a+\frac{118328295114}{5757813667}$, $\frac{11250180329}{115156273340}a^{14}+\frac{2273451512}{28789068335}a^{13}-\frac{380752161763}{115156273340}a^{12}-\frac{75181654234}{28789068335}a^{11}+\frac{869171976585}{23031254668}a^{10}+\frac{605920805351}{28789068335}a^{9}-\frac{2225977039103}{11515627334}a^{8}-\frac{1655642190281}{28789068335}a^{7}+\frac{52918497689749}{115156273340}a^{6}+\frac{1537145304232}{28789068335}a^{5}-\frac{2681463119016}{5757813667}a^{4}-\frac{196246923783}{11515627334}a^{3}+\frac{3222196566841}{23031254668}a^{2}+\frac{142001151278}{5757813667}a-\frac{11656398400}{5757813667}$, $\frac{9375057301}{115156273340}a^{14}-\frac{6045424631}{57578136670}a^{13}-\frac{67283067995}{23031254668}a^{12}+\frac{199605865569}{57578136670}a^{11}+\frac{4234613800617}{115156273340}a^{10}-\frac{2542680127371}{57578136670}a^{9}-\frac{11684303682357}{57578136670}a^{8}+\frac{7219119159441}{28789068335}a^{7}+\frac{56922386971981}{115156273340}a^{6}-\frac{35357584473469}{57578136670}a^{5}-\frac{2842838602356}{5757813667}a^{4}+\frac{6796649413149}{11515627334}a^{3}+\frac{3721756900489}{23031254668}a^{2}-\frac{1589780384803}{11515627334}a-\frac{214219468915}{5757813667}$, $\frac{889401151}{115156273340}a^{14}+\frac{1202805163}{57578136670}a^{13}-\frac{29028192021}{115156273340}a^{12}-\frac{41189411003}{57578136670}a^{11}+\frac{306510393851}{115156273340}a^{10}+\frac{88968094403}{11515627334}a^{9}-\frac{145519525393}{11515627334}a^{8}-\frac{1068743530117}{28789068335}a^{7}+\frac{3397295459107}{115156273340}a^{6}+\frac{983877971605}{11515627334}a^{5}-\frac{196674187408}{5757813667}a^{4}-\frac{988718310423}{11515627334}a^{3}+\frac{372644962223}{23031254668}a^{2}+\frac{308791853743}{11515627334}a+\frac{9608406227}{5757813667}$, $\frac{8653016084}{28789068335}a^{14}+\frac{12416121909}{28789068335}a^{13}-\frac{286637686648}{28789068335}a^{12}-\frac{412322537613}{28789068335}a^{11}+\frac{627975130442}{5757813667}a^{10}+\frac{3882837459426}{28789068335}a^{9}-\frac{15377629531939}{28789068335}a^{8}-\frac{15249289113677}{28789068335}a^{7}+\frac{35099850520558}{28789068335}a^{6}+\frac{5760269733287}{5757813667}a^{5}-\frac{6827356178447}{5757813667}a^{4}-\frac{5202830590986}{5757813667}a^{3}+\frac{1804629149099}{5757813667}a^{2}+\frac{1847488060429}{5757813667}a+\frac{317868974359}{5757813667}$, $\frac{11659478847}{23031254668}a^{14}+\frac{3499419512}{5757813667}a^{13}-\frac{1943329197293}{115156273340}a^{12}-\frac{579937670601}{28789068335}a^{11}+\frac{21554975029319}{115156273340}a^{10}+\frac{5253282906269}{28789068335}a^{9}-\frac{53448386307417}{57578136670}a^{8}-\frac{19224499745774}{28789068335}a^{7}+\frac{49236227525511}{23031254668}a^{6}+\frac{33501461045093}{28789068335}a^{5}-\frac{12085998741732}{5757813667}a^{4}-\frac{11582570921849}{11515627334}a^{3}+\frac{13634829683075}{23031254668}a^{2}+\frac{2213106263189}{5757813667}a+\frac{314847593800}{5757813667}$, $\frac{3143750623}{115156273340}a^{14}-\frac{59838121}{57578136670}a^{13}-\frac{105302575101}{115156273340}a^{12}+\frac{1008559065}{11515627334}a^{11}+\frac{1188089394739}{115156273340}a^{10}-\frac{233632479337}{57578136670}a^{9}-\frac{2957689323217}{57578136670}a^{8}+\frac{205266905620}{5757813667}a^{7}+\frac{13261435466699}{115156273340}a^{6}-\frac{1138648193367}{11515627334}a^{5}-\frac{632327429106}{5757813667}a^{4}+\frac{1048232147025}{11515627334}a^{3}+\frac{709389307795}{23031254668}a^{2}-\frac{142277397657}{11515627334}a+\frac{9715278933}{5757813667}$, $\frac{1117066101}{115156273340}a^{14}-\frac{1825120881}{57578136670}a^{13}-\frac{52175486527}{115156273340}a^{12}+\frac{51715795321}{57578136670}a^{11}+\frac{890135113961}{115156273340}a^{10}-\frac{85332769221}{11515627334}a^{9}-\frac{3267939451451}{57578136670}a^{8}+\frac{591495160847}{28789068335}a^{7}+\frac{22174024221309}{115156273340}a^{6}+\frac{78380599793}{11515627334}a^{5}-\frac{1703475954400}{5757813667}a^{4}-\frac{1206361674339}{11515627334}a^{3}+\frac{3610629895701}{23031254668}a^{2}+\frac{1238396753599}{11515627334}a+\frac{102581755149}{5757813667}$, $\frac{130298941}{57578136670}a^{14}-\frac{2672410057}{57578136670}a^{13}-\frac{437546131}{57578136670}a^{12}+\frac{106112333393}{57578136670}a^{11}-\frac{57491953767}{57578136670}a^{10}-\frac{1520214993633}{57578136670}a^{9}+\frac{67754849301}{5757813667}a^{8}+\frac{4682330681137}{28789068335}a^{7}-\frac{544128254361}{11515627334}a^{6}-\frac{5069968937207}{11515627334}a^{5}+\frac{347428745501}{5757813667}a^{4}+\frac{2838576267757}{5757813667}a^{3}+\frac{118961087671}{11515627334}a^{2}-\frac{1951871060065}{11515627334}a-\frac{249273285182}{5757813667}$ 5395040479/115156273340a^14 + 876804649/11515627334a^13 - 172347129181/115156273340a^12 - 138623232591/57578136670a^11 + 1768405584211/115156273340a^10 + 1185803529943/57578136670a^9 - 800910091675/11515627334a^8 - 1947393707046/28789068335a^7 + 16846551806947/115156273340a^6 + 5382505290027/57578136670a^5 - 761349302269/5757813667a^4 - 594570217555/11515627334a^3 + 831920196899/23031254668a^2 + 122326475879/11515627334a - 5354674624/5757813667, 183274617/28789068335a^14 - 390813597/11515627334a^13 - 7729506563/28789068335a^12 + 64023530303/57578136670a^11 + 119821382423/28789068335a^10 - 719306971569/57578136670a^9 - 157020384630/5757813667a^8 + 1838163820043/28789068335a^7 + 2271049201556/28789068335a^6 - 8840081636741/57578136670a^5 - 592113349648/5757813667a^4 + 931450315120/5757813667a^3 + 349802358902/5757813667a^2 - 624159190557/11515627334a - 97744763498/5757813667, 4174756833/57578136670a^14 + 237538223/5757813667a^13 - 141024967813/57578136670a^12 - 38227024432/28789068335a^11 + 1606558601433/57578136670a^10 + 237210797043/28789068335a^9 - 4084457108888/28789068335a^8 - 112751173884/28789068335a^7 + 19096561097583/57578136670a^6 - 1425393931307/28789068335a^5 - 1910454070875/5757813667a^4 + 421770106057/5757813667a^3 + 1243754154329/11515627334a^2 - 71328038498/5757813667a - 46614821845/5757813667, 5927032639/115156273340a^14 + 2108922543/28789068335a^13 - 186781937957/115156273340a^12 - 65014097342/28789068335a^11 + 372688488435/23031254668a^10 + 505050440874/28789068335a^9 - 3972171417111/57578136670a^8 - 1326620012182/28789068335a^7 + 14568569184823/115156273340a^6 + 211951407167/5757813667a^5 - 482819038672/5757813667a^4 + 69695051927/11515627334a^3 + 246293995703/23031254668a^2 - 75825127516/5757813667a - 28406626694/5757813667, 3422375939/23031254668a^14 + 14743759719/57578136670a^13 - 540070898889/115156273340a^12 - 92287776585/11515627334a^11 + 5414719673339/115156273340a^10 + 3888586719357/57578136670a^9 - 11920518947577/57578136670a^8 - 6245192915923/28789068335a^7 + 48740843427659/115156273340a^6 + 3371618424253/11515627334a^5 - 2143922621200/5757813667a^4 - 1921680114625/11515627334a^3 + 2164892389179/23031254668a^2 + 524804056477/11515627334a + 26854721403/5757813667, 600275827/23031254668a^14 + 2829841107/28789068335a^13 - 93755801081/115156273340a^12 - 94560877424/28789068335a^11 + 904468364519/115156273340a^10 + 997685597481/28789068335a^9 - 1951756733467/57578136670a^8 - 927494983853/5757813667a^7 + 1708916013867/23031254668a^6 + 2030754578234/5757813667a^5 - 430627542347/5757813667a^4 - 3905113076213/11515627334a^3 + 275751240947/23031254668a^2 + 599082284666/5757813667a + 118328295114/5757813667, 11250180329/115156273340a^14 + 2273451512/28789068335a^13 - 380752161763/115156273340a^12 - 75181654234/28789068335a^11 + 869171976585/23031254668a^10 + 605920805351/28789068335a^9 - 2225977039103/11515627334a^8 - 1655642190281/28789068335a^7 + 52918497689749/115156273340a^6 + 1537145304232/28789068335a^5 - 2681463119016/5757813667a^4 - 196246923783/11515627334a^3 + 3222196566841/23031254668a^2 + 142001151278/5757813667a - 11656398400/5757813667, 9375057301/115156273340a^14 - 6045424631/57578136670a^13 - 67283067995/23031254668a^12 + 199605865569/57578136670a^11 + 4234613800617/115156273340a^10 - 2542680127371/57578136670a^9 - 11684303682357/57578136670a^8 + 7219119159441/28789068335a^7 + 56922386971981/115156273340a^6 - 35357584473469/57578136670a^5 - 2842838602356/5757813667a^4 + 6796649413149/11515627334a^3 + 3721756900489/23031254668a^2 - 1589780384803/11515627334a - 214219468915/5757813667, 889401151/115156273340a^14 + 1202805163/57578136670a^13 - 29028192021/115156273340a^12 - 41189411003/57578136670a^11 + 306510393851/115156273340a^10 + 88968094403/11515627334a^9 - 145519525393/11515627334a^8 - 1068743530117/28789068335a^7 + 3397295459107/115156273340a^6 + 983877971605/11515627334a^5 - 196674187408/5757813667a^4 - 988718310423/11515627334a^3 + 372644962223/23031254668a^2 + 308791853743/11515627334a + 9608406227/5757813667, 8653016084/28789068335a^14 + 12416121909/28789068335a^13 - 286637686648/28789068335a^12 - 412322537613/28789068335a^11 + 627975130442/5757813667a^10 + 3882837459426/28789068335a^9 - 15377629531939/28789068335a^8 - 15249289113677/28789068335a^7 + 35099850520558/28789068335a^6 + 5760269733287/5757813667a^5 - 6827356178447/5757813667a^4 - 5202830590986/5757813667a^3 + 1804629149099/5757813667a^2 + 1847488060429/5757813667a + 317868974359/5757813667, 11659478847/23031254668a^14 + 3499419512/5757813667a^13 - 1943329197293/115156273340a^12 - 579937670601/28789068335a^11 + 21554975029319/115156273340a^10 + 5253282906269/28789068335a^9 - 53448386307417/57578136670a^8 - 19224499745774/28789068335a^7 + 49236227525511/23031254668a^6 + 33501461045093/28789068335a^5 - 12085998741732/5757813667a^4 - 11582570921849/11515627334a^3 + 13634829683075/23031254668a^2 + 2213106263189/5757813667a + 314847593800/5757813667, 3143750623/115156273340a^14 - 59838121/57578136670a^13 - 105302575101/115156273340a^12 + 1008559065/11515627334a^11 + 1188089394739/115156273340a^10 - 233632479337/57578136670a^9 - 2957689323217/57578136670a^8 + 205266905620/5757813667a^7 + 13261435466699/115156273340a^6 - 1138648193367/11515627334a^5 - 632327429106/5757813667a^4 + 1048232147025/11515627334a^3 + 709389307795/23031254668a^2 - 142277397657/11515627334a + 9715278933/5757813667, 1117066101/115156273340a^14 - 1825120881/57578136670a^13 - 52175486527/115156273340a^12 + 51715795321/57578136670a^11 + 890135113961/115156273340a^10 - 85332769221/11515627334a^9 - 3267939451451/57578136670a^8 + 591495160847/28789068335a^7 + 22174024221309/115156273340a^6 + 78380599793/11515627334a^5 - 1703475954400/5757813667a^4 - 1206361674339/11515627334a^3 + 3610629895701/23031254668a^2 + 1238396753599/11515627334a + 102581755149/5757813667, 130298941/57578136670a^14 - 2672410057/57578136670a^13 - 437546131/57578136670a^12 + 106112333393/57578136670a^11 - 57491953767/57578136670a^10 - 1520214993633/57578136670a^9 + 67754849301/5757813667a^8 + 4682330681137/28789068335a^7 - 544128254361/11515627334a^6 - 5069968937207/11515627334a^5 + 347428745501/5757813667a^4 + 2838576267757/5757813667a^3 + 118961087671/11515627334a^2 - 1951871060065/11515627334a - 249273285182/5757813667 (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:	$ 633057748.364 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{0}\cdot 633057748.364 \cdot 1}{2\cdot\sqrt{430807787028125000000000000}}\cr\approx \mathstrut & 0.499713975657 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200)

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^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200, 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^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200);

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^15 - 35*x^13 + 425*x^11 - 76*x^10 - 2350*x^9 + 820*x^8 + 6225*x^7 - 2580*x^6 - 7880*x^5 + 2750*x^4 + 4525*x^3 - 500*x^2 - 1100*x - 200);

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

$C_5^3:C_{12}$ (as 15T38):

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 1500

The 25 conjugacy class representatives for $C_5^3:C_{12}$

Character table for $C_5^3:C_{12}$

Intermediate fields

3.3.169.1

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 15 siblings:	data not computed
Degree 30 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	${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.3.0.1}{3} }$	R	${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$	R	${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.3.0.1}{3} }$	$15$	${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.3.0.1}{3} }$	${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$	${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$	${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.3.0.1}{3} }$	$15$	${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.3.0.1}{3} }$	${\href{/padicField/47.4.0.1}{4} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$	${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$	$15$

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	2.3.0.1	$x^{3} + x + 1$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
$2$ 2	2.12.12.25	$x^{12} + 12 x^{11} + 60 x^{10} + 160 x^{9} + 308 x^{8} + 736 x^{7} + 2272 x^{6} + 5632 x^{5} + 10608 x^{4} + 15040 x^{3} + 12224 x^{2} + 3584 x + 704$	$2$	$6$	$12$	$C_{12}$	$[2]^{6}$
$5$ 5	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.5.9.3	$x^{5} + 100 x^{2} + 75 x + 5$	$5$	$1$	$9$	$F_5$	$[9/4]_{4}$
	5.5.5.2	$x^{5} + 5 x + 5$	$5$	$1$	$5$	$F_5$	$[5/4]_{4}$
$13$ 13	13.3.2.2	$x^{3} + 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$13$ 13	13.12.8.1	$x^{12} + 9 x^{10} + 88 x^{9} + 33 x^{8} + 216 x^{7} - 1299 x^{6} - 78 x^{5} - 1797 x^{4} - 15494 x^{3} + 21687 x^{2} - 41586 x + 201846$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$

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