Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729)
gp: K = bnfinit(y^36 + 6*y^34 + 27*y^32 + 111*y^30 + 441*y^28 + 1728*y^26 + 6732*y^24 + 12906*y^22 + 22032*y^20 + 36234*y^18 + 57834*y^16 + 86994*y^14 + 110160*y^12 + 51516*y^10 + 24057*y^8 + 11178*y^6 + 5103*y^4 + 2187*y^2 + 729, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729)
\( x^{36} + 6 x^{34} + 27 x^{32} + 111 x^{30} + 441 x^{28} + 1728 x^{26} + 6732 x^{24} + 12906 x^{22} + \cdots + 729 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(90067643300370785938616861622694756230952958181429238736879616\)
\(\medspace = 2^{36}\cdot 3^{54}\cdot 7^{30}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.60\) | 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: | $2\cdot 3^{3/2}7^{5/6}\approx 52.596911665775266$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(252=2^{2}\cdot 3^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(131,·)$, $\chi_{252}(11,·)$, $\chi_{252}(13,·)$, $\chi_{252}(143,·)$, $\chi_{252}(145,·)$, $\chi_{252}(23,·)$, $\chi_{252}(25,·)$, $\chi_{252}(155,·)$, $\chi_{252}(157,·)$, $\chi_{252}(37,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(47,·)$, $\chi_{252}(179,·)$, $\chi_{252}(181,·)$, $\chi_{252}(59,·)$, $\chi_{252}(61,·)$, $\chi_{252}(191,·)$, $\chi_{252}(193,·)$, $\chi_{252}(71,·)$, $\chi_{252}(73,·)$, $\chi_{252}(205,·)$, $\chi_{252}(83,·)$, $\chi_{252}(85,·)$, $\chi_{252}(215,·)$, $\chi_{252}(95,·)$, $\chi_{252}(97,·)$, $\chi_{252}(227,·)$, $\chi_{252}(229,·)$, $\chi_{252}(107,·)$, $\chi_{252}(109,·)$, $\chi_{252}(239,·)$, $\chi_{252}(241,·)$, $\chi_{252}(121,·)$, $\chi_{252}(251,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{3}a^{10}$, $\frac{1}{3}a^{11}$, $\frac{1}{9}a^{12}$, $\frac{1}{9}a^{13}$, $\frac{1}{36}a^{14}+\frac{1}{4}$, $\frac{1}{36}a^{15}+\frac{1}{4}a$, $\frac{1}{36}a^{16}+\frac{1}{4}a^{2}$, $\frac{1}{36}a^{17}+\frac{1}{4}a^{3}$, $\frac{1}{108}a^{18}-\frac{1}{4}a^{4}$, $\frac{1}{108}a^{19}-\frac{1}{4}a^{5}$, $\frac{1}{108}a^{20}+\frac{1}{12}a^{6}$, $\frac{1}{108}a^{21}+\frac{1}{12}a^{7}$, $\frac{1}{108}a^{22}+\frac{1}{12}a^{8}$, $\frac{1}{108}a^{23}+\frac{1}{12}a^{9}$, $\frac{1}{324}a^{24}-\frac{1}{12}a^{10}$, $\frac{1}{324}a^{25}-\frac{1}{12}a^{11}$, $\frac{1}{4531788}a^{26}+\frac{3353}{2265894}a^{24}+\frac{901}{251766}a^{22}-\frac{139}{125883}a^{20}+\frac{998}{377649}a^{18}+\frac{1469}{167844}a^{16}-\frac{1189}{503532}a^{14}-\frac{22139}{503532}a^{12}+\frac{3643}{83922}a^{10}-\frac{1777}{27974}a^{8}+\frac{181}{13987}a^{6}-\frac{3083}{13987}a^{4}+\frac{6739}{55948}a^{2}+\frac{27711}{55948}$, $\frac{1}{4531788}a^{27}+\frac{3353}{2265894}a^{25}+\frac{901}{251766}a^{23}-\frac{139}{125883}a^{21}+\frac{998}{377649}a^{19}+\frac{1469}{167844}a^{17}-\frac{1189}{503532}a^{15}-\frac{22139}{503532}a^{13}+\frac{3643}{83922}a^{11}-\frac{1777}{27974}a^{9}+\frac{181}{13987}a^{7}-\frac{3083}{13987}a^{5}+\frac{6739}{55948}a^{3}+\frac{27711}{55948}a$, $\frac{1}{18127152}a^{28}+\frac{10333}{1007064}a^{14}+\frac{99225}{223792}$, $\frac{1}{18127152}a^{29}+\frac{10333}{1007064}a^{15}+\frac{99225}{223792}a$, $\frac{1}{54381456}a^{30}+\frac{12769}{1007064}a^{16}-\frac{22873}{223792}a^{2}$, $\frac{1}{54381456}a^{31}+\frac{12769}{1007064}a^{17}-\frac{22873}{223792}a^{3}$, $\frac{1}{54381456}a^{32}+\frac{10333}{3021192}a^{18}+\frac{33075}{223792}a^{4}$, $\frac{1}{54381456}a^{33}+\frac{10333}{3021192}a^{19}+\frac{33075}{223792}a^{5}$, $\frac{1}{54381456}a^{34}+\frac{10333}{3021192}a^{20}+\frac{33075}{223792}a^{6}$, $\frac{1}{54381456}a^{35}+\frac{10333}{3021192}a^{21}+\frac{33075}{223792}a^{7}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{14}\times C_{182}$, which has order $5096$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{19}{2014128} a^{32} + \frac{377893}{3021192} a^{18} - \frac{1022295}{223792} a^{4} \)
(order $14$)
| 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: |
$\frac{33727}{54381456}a^{34}+\frac{8723}{2265894}a^{32}+\frac{8723}{503532}a^{30}+\frac{322751}{4531788}a^{28}+\frac{142477}{503532}a^{26}+\frac{139568}{125883}a^{24}+\frac{1631201}{377649}a^{22}+\frac{8625211}{1007064}a^{20}+\frac{593164}{41961}a^{18}+\frac{5853133}{251766}a^{16}+\frac{1038037}{27974}a^{14}+\frac{28087745}{503532}a^{12}+\frac{2965820}{41961}a^{10}+\frac{462319}{13987}a^{8}+\frac{3808451}{671376}a^{6}+\frac{200629}{27974}a^{4}+\frac{183183}{55948}a^{2}+\frac{22559}{55948}$, $\frac{19}{2014128}a^{32}+\frac{187}{54381456}a^{30}-\frac{377893}{3021192}a^{18}-\frac{45935}{1007064}a^{16}+\frac{1022295}{223792}a^{4}+\frac{478329}{223792}a^{2}$, $\frac{1117}{54381456}a^{32}-\frac{822547}{3021192}a^{18}+\frac{1921327}{223792}a^{4}$, $\frac{139}{18127152}a^{28}-\frac{102283}{1007064}a^{14}+\frac{644495}{223792}$, $\frac{4457}{18127152}a^{34}+\frac{7537}{6042384}a^{32}+\frac{140489}{27190728}a^{30}+\frac{186053}{9063576}a^{28}+\frac{30376}{377649}a^{26}+\frac{710039}{2265894}a^{24}+\frac{1839661}{1510596}a^{22}+\frac{4436597}{3021192}a^{20}+\frac{202721}{111896}a^{18}+\frac{451843}{167844}a^{16}+\frac{679663}{167844}a^{14}+\frac{645490}{125883}a^{12}+\frac{201241}{83922}a^{10}-\frac{1104681}{55948}a^{8}-\frac{9692737}{671376}a^{6}-\frac{969137}{223792}a^{4}+\frac{11391}{111896}a^{2}+\frac{3797}{111896}$, $\frac{33727}{54381456}a^{34}+\frac{69613}{18127152}a^{32}+\frac{941293}{54381456}a^{30}+\frac{322751}{4531788}a^{28}+\frac{142477}{503532}a^{26}+\frac{139568}{125883}a^{24}+\frac{1631201}{377649}a^{22}+\frac{8625211}{1007064}a^{20}+\frac{43085701}{3021192}a^{18}+\frac{2622965}{111896}a^{16}+\frac{1038037}{27974}a^{14}+\frac{28087745}{503532}a^{12}+\frac{2965820}{41961}a^{10}+\frac{462319}{13987}a^{8}+\frac{3808451}{671376}a^{6}+\frac{582737}{223792}a^{4}-\frac{644629}{223792}a^{2}+\frac{22559}{55948}$, $\frac{1165}{54381456}a^{34}-\frac{839}{54381456}a^{32}+\frac{187}{54381456}a^{30}-\frac{95341}{335688}a^{20}+\frac{617981}{3021192}a^{18}-\frac{45935}{1007064}a^{16}+\frac{6554473}{671376}a^{6}-\frac{1566261}{223792}a^{4}+\frac{478329}{223792}a^{2}$, $\frac{1165}{54381456}a^{34}+\frac{1117}{54381456}a^{32}+\frac{187}{54381456}a^{30}-\frac{95341}{335688}a^{20}-\frac{822547}{3021192}a^{18}-\frac{45935}{1007064}a^{16}+\frac{6554473}{671376}a^{6}+\frac{1921327}{223792}a^{4}+\frac{478329}{223792}a^{2}$, $\frac{2747}{27190728}a^{34}+\frac{2747}{6042384}a^{32}+\frac{101639}{54381456}a^{30}+\frac{134603}{18127152}a^{28}+\frac{10988}{377649}a^{26}+\frac{513689}{4531788}a^{24}+\frac{665431}{1510596}a^{22}+\frac{46699}{125883}a^{20}+\frac{1843237}{3021192}a^{18}+\frac{326893}{335688}a^{16}+\frac{491713}{335688}a^{14}+\frac{233495}{125883}a^{12}+\frac{145591}{167844}a^{10}-\frac{1148249}{167844}a^{8}+\frac{63181}{335688}a^{6}+\frac{19229}{223792}a^{4}+\frac{8241}{223792}a^{2}-a+\frac{2747}{223792}$, $\frac{8723}{13595364}a^{35}-\frac{1165}{54381456}a^{34}+\frac{8723}{2265894}a^{33}+\frac{8723}{503532}a^{31}+\frac{322751}{4531788}a^{29}+\frac{142477}{503532}a^{27}+\frac{139568}{125883}a^{25}+\frac{1631201}{377649}a^{23}+\frac{2084797}{251766}a^{21}+\frac{95341}{335688}a^{20}+\frac{593164}{41961}a^{19}+\frac{5853133}{251766}a^{17}+\frac{1038037}{27974}a^{15}+\frac{28087745}{503532}a^{13}+\frac{2965820}{41961}a^{11}+\frac{462319}{13987}a^{9}+\frac{863577}{55948}a^{7}-\frac{6554473}{671376}a^{6}+\frac{200629}{27974}a^{5}+\frac{183183}{55948}a^{3}+\frac{78507}{55948}a$, $\frac{1141}{27190728}a^{35}+\frac{8723}{13595364}a^{34}+\frac{8723}{2265894}a^{32}+\frac{941897}{54381456}a^{30}+\frac{322751}{4531788}a^{28}+\frac{142477}{503532}a^{26}+\frac{139568}{125883}a^{24}+\frac{1631201}{377649}a^{22}-\frac{210077}{377649}a^{21}+\frac{2084797}{251766}a^{20}+\frac{593164}{41961}a^{18}+\frac{7819489}{335688}a^{16}+\frac{1038037}{27974}a^{14}+\frac{28087745}{503532}a^{12}+\frac{2965820}{41961}a^{10}+\frac{462319}{13987}a^{8}+\frac{6159227}{335688}a^{7}+\frac{863577}{55948}a^{6}+\frac{200629}{27974}a^{4}+\frac{254403}{223792}a^{2}+\frac{78507}{55948}$, $\frac{1805}{3398841}a^{35}+\frac{33727}{54381456}a^{34}+\frac{1805}{755298}a^{33}+\frac{69613}{18127152}a^{32}+\frac{66785}{6797682}a^{31}+\frac{941897}{54381456}a^{30}+\frac{88445}{2265894}a^{29}+\frac{322751}{4531788}a^{28}+\frac{57760}{377649}a^{27}+\frac{142477}{503532}a^{26}+\frac{675070}{1132947}a^{25}+\frac{139568}{125883}a^{24}+\frac{64779}{27974}a^{23}+\frac{1631201}{377649}a^{22}+\frac{245480}{125883}a^{21}+\frac{8625211}{1007064}a^{20}+\frac{1211155}{377649}a^{19}+\frac{43085701}{3021192}a^{18}+\frac{214795}{41961}a^{17}+\frac{7819489}{335688}a^{16}+\frac{323095}{41961}a^{15}+\frac{1038037}{27974}a^{14}+\frac{1227400}{125883}a^{13}+\frac{28087745}{503532}a^{12}+\frac{191330}{41961}a^{11}+\frac{2965820}{41961}a^{10}-\frac{3112109}{83922}a^{9}+\frac{462319}{13987}a^{8}+\frac{41515}{41961}a^{7}+\frac{3808451}{671376}a^{6}+\frac{12635}{27974}a^{5}+\frac{582737}{223792}a^{4}+\frac{5415}{27974}a^{3}+\frac{254403}{223792}a^{2}+\frac{1805}{27974}a+\frac{22559}{55948}$, $\frac{3797}{13595364}a^{35}+\frac{3797}{3021192}a^{33}+\frac{19}{2014128}a^{32}+\frac{140489}{27190728}a^{31}+\frac{186053}{9063576}a^{29}+\frac{30376}{377649}a^{27}+\frac{710039}{2265894}a^{25}+\frac{1839661}{1510596}a^{23}+\frac{129098}{125883}a^{21}+\frac{2547787}{1510596}a^{19}-\frac{377893}{3021192}a^{18}+\frac{451843}{167844}a^{17}+\frac{679663}{167844}a^{15}+\frac{645490}{125883}a^{13}+\frac{201241}{83922}a^{11}-\frac{1104681}{55948}a^{9}+\frac{87331}{167844}a^{7}+\frac{26579}{111896}a^{5}+\frac{1022295}{223792}a^{4}+\frac{11391}{111896}a^{3}+\frac{3797}{111896}a+1$, $\frac{3797}{13595364}a^{34}-\frac{19}{2014128}a^{33}+\frac{3797}{3021192}a^{32}+\frac{31199}{6042384}a^{30}+\frac{186053}{9063576}a^{28}+\frac{30376}{377649}a^{26}+\frac{710039}{2265894}a^{24}+\frac{1839661}{1510596}a^{22}+\frac{129098}{125883}a^{20}+\frac{377893}{3021192}a^{19}+\frac{2547787}{1510596}a^{18}+\frac{2756993}{1007064}a^{16}+\frac{679663}{167844}a^{14}+\frac{645490}{125883}a^{12}+\frac{201241}{83922}a^{10}-\frac{1104681}{55948}a^{8}+\frac{87331}{167844}a^{6}-\frac{1022295}{223792}a^{5}+\frac{26579}{111896}a^{4}-\frac{455547}{223792}a^{2}+\frac{3797}{111896}$, $\frac{30745}{18127152}a^{35}+\frac{522665}{54381456}a^{33}+\frac{19}{2014128}a^{32}+\frac{768625}{18127152}a^{31}+\frac{522665}{3021192}a^{29}+\frac{3105245}{4531788}a^{27}+\frac{3038002}{1132947}a^{25}+\frac{584155}{55948}a^{23}+\frac{54633865}{3021192}a^{21}+\frac{153725}{5112}a^{19}-\frac{377893}{3021192}a^{18}+\frac{16448575}{335688}a^{17}+\frac{39076895}{503532}a^{15}+\frac{6425705}{55948}a^{13}+\frac{5770475}{41961}a^{11}+\frac{4212065}{167844}a^{9}+\frac{2613325}{223792}a^{7}+\frac{1199055}{223792}a^{5}+\frac{1022295}{223792}a^{4}+\frac{522665}{223792}a^{3}+\frac{92235}{111896}a$, $\frac{23525}{9063576}a^{35}-\frac{14023}{54381456}a^{34}+\frac{399925}{27190728}a^{33}-\frac{7537}{6042384}a^{32}+\frac{588125}{9063576}a^{31}-\frac{31199}{6042384}a^{30}+\frac{399925}{1510596}a^{29}-\frac{186053}{9063576}a^{28}+\frac{2376025}{2265894}a^{27}-\frac{30376}{377649}a^{26}+\frac{9298267}{2265894}a^{25}-\frac{710039}{2265894}a^{24}+\frac{446975}{27974}a^{23}-\frac{1839661}{1510596}a^{22}+\frac{41803925}{1510596}a^{21}-\frac{1318807}{1007064}a^{20}+\frac{117625}{2556}a^{19}-\frac{202721}{111896}a^{18}+\frac{12585875}{167844}a^{17}-\frac{2756993}{1007064}a^{16}+\frac{29900275}{251766}a^{15}-\frac{679663}{167844}a^{14}+\frac{4916725}{27974}a^{13}-\frac{645490}{125883}a^{12}+\frac{5891277}{27974}a^{11}-\frac{201241}{83922}a^{10}+\frac{3222925}{83922}a^{9}+\frac{1104681}{55948}a^{8}+\frac{1999625}{111896}a^{7}+\frac{2068383}{223792}a^{6}+\frac{917475}{111896}a^{5}+\frac{969137}{223792}a^{4}+\frac{399925}{111896}a^{3}+\frac{455547}{223792}a^{2}+\frac{70575}{55948}a+\frac{108099}{111896}$, $\frac{23525}{9063576}a^{35}+\frac{1165}{54381456}a^{34}+\frac{399925}{27190728}a^{33}+\frac{588125}{9063576}a^{31}+\frac{187}{54381456}a^{30}+\frac{399925}{1510596}a^{29}+\frac{2376025}{2265894}a^{27}+\frac{9298267}{2265894}a^{25}+\frac{446975}{27974}a^{23}+\frac{41803925}{1510596}a^{21}-\frac{95341}{335688}a^{20}+\frac{117625}{2556}a^{19}+\frac{12585875}{167844}a^{17}-\frac{45935}{1007064}a^{16}+\frac{29900275}{251766}a^{15}+\frac{4916725}{27974}a^{13}+\frac{5891277}{27974}a^{11}+\frac{3222925}{83922}a^{9}+\frac{1999625}{111896}a^{7}+\frac{6554473}{671376}a^{6}+\frac{917475}{111896}a^{5}+\frac{399925}{111896}a^{3}+\frac{478329}{223792}a^{2}+\frac{70575}{55948}a$
| 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: | \( 36543757083175.945 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 36543757083175.945 \cdot 5096}{14\cdot\sqrt{90067643300370785938616861622694756230952958181429238736879616}}\cr\approx \mathstrut & 0.326488440805346 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729)
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^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729, 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^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729);
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^36 + 6*x^34 + 27*x^32 + 111*x^30 + 441*x^28 + 1728*x^26 + 6732*x^24 + 12906*x^22 + 22032*x^20 + 36234*x^18 + 57834*x^16 + 86994*x^14 + 110160*x^12 + 51516*x^10 + 24057*x^8 + 11178*x^6 + 5103*x^4 + 2187*x^2 + 729);
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
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)
| An abelian group of order 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ is not computed |
Intermediate fields
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]
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 | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.3.0.1}{3} }^{12}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(3\)
| Deg $36$ | $6$ | $6$ | $54$ | |||
|
\(7\)
| Deg $36$ | $6$ | $6$ | $30$ |