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Properties

Label 27.3.177...000.6
Degree $27$
Signature $[3, 12]$
Discriminant $1.778\times 10^{83}$
Root discriminant \(1211.52\)
Ramified primes $2,3,5$
Class number not computed
Class group not computed
Galois group $\SO(5,3)$ (as 27T1161)

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

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200)
 
Copy content gp:K = bnfinit(y^27 - 360*y^25 + 54000*y^23 - 9216*y^22 - 4446720*y^21 + 1935360*y^20 + 223891200*y^19 - 147947520*y^18 - 7264733184*y^17 + 5311365120*y^16 + 159244507440*y^15 - 91269365760*y^14 - 2497379088960*y^13 + 545341833216*y^12 + 28698328270080*y^11 + 7709768294400*y^10 - 231976756423680*y^9 - 199955270615040*y^8 + 1167100609904640*y^7 + 1878904774656000*y^6 - 2532714300702720*y^5 - 7902598720389120*y^4 - 3861559365624000*y^3 + 9381033208184832*y^2 + 13476478435983360*y + 6066476888883200, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200)
 

\( x^{27} - 360 x^{25} + 54000 x^{23} - 9216 x^{22} - 4446720 x^{21} + 1935360 x^{20} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $27$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[3, 12]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(177\!\cdots\!000\) \(\medspace = 2^{70}\cdot 3^{60}\cdot 5^{48}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(1211.52\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{30}a^{4}-\frac{2}{15}a^{3}+\frac{1}{5}a^{2}-\frac{2}{15}a+\frac{1}{3}$, $\frac{1}{30}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{5}a+\frac{1}{3}$, $\frac{1}{30}a^{6}+\frac{1}{3}a^{3}-\frac{1}{5}a^{2}+\frac{1}{3}$, $\frac{1}{150}a^{7}+\frac{1}{75}a^{6}-\frac{1}{75}a^{5}-\frac{1}{150}a^{4}-\frac{2}{25}a^{3}+\frac{31}{75}a^{2}+\frac{11}{25}a+\frac{4}{15}$, $\frac{1}{900}a^{8}-\frac{1}{450}a^{7}+\frac{1}{90}a^{6}-\frac{2}{225}a^{5}-\frac{2}{225}a^{4}+\frac{8}{45}a^{3}+\frac{112}{225}a^{2}+\frac{79}{225}a-\frac{13}{45}$, $\frac{1}{900}a^{9}-\frac{1}{75}a^{5}-\frac{2}{5}a^{3}-\frac{1}{15}a^{2}-\frac{9}{25}a+\frac{22}{45}$, $\frac{1}{4500}a^{10}-\frac{1}{2250}a^{9}+\frac{1}{4500}a^{8}+\frac{1}{450}a^{7}-\frac{17}{1125}a^{6}-\frac{19}{2250}a^{5}-\frac{1}{225}a^{4}+\frac{109}{1125}a^{3}+\frac{97}{1125}a^{2}-\frac{49}{125}a+\frac{14}{75}$, $\frac{1}{22500}a^{11}-\frac{1}{11250}a^{10}+\frac{1}{22500}a^{9}-\frac{1}{2250}a^{8}+\frac{8}{5625}a^{7}-\frac{59}{11250}a^{6}-\frac{14}{1125}a^{5}+\frac{43}{11250}a^{4}+\frac{317}{5625}a^{3}-\frac{2801}{5625}a^{2}+\frac{329}{1125}a+\frac{91}{225}$, $\frac{1}{135000}a^{12}+\frac{1}{11250}a^{10}+\frac{31}{67500}a^{9}-\frac{2}{5625}a^{8}+\frac{4}{5625}a^{7}+\frac{97}{11250}a^{6}-\frac{56}{5625}a^{5}+\frac{19}{2250}a^{4}+\frac{1834}{16875}a^{3}-\frac{2642}{5625}a^{2}-\frac{37}{225}a-\frac{196}{675}$, $\frac{1}{540000}a^{13}-\frac{1}{45000}a^{11}-\frac{1}{13500}a^{10}-\frac{1}{7500}a^{9}-\frac{11}{22500}a^{8}+\frac{29}{22500}a^{7}-\frac{47}{5625}a^{6}+\frac{2}{1125}a^{5}-\frac{131}{16875}a^{4}+\frac{2}{1125}a^{3}-\frac{758}{1875}a^{2}-\frac{6503}{13500}a+\frac{34}{225}$, $\frac{1}{2700000}a^{14}-\frac{1}{2700000}a^{13}+\frac{1}{135000}a^{11}+\frac{17}{337500}a^{10}+\frac{2}{5625}a^{9}+\frac{13}{56250}a^{8}+\frac{13}{7500}a^{7}-\frac{583}{56250}a^{6}-\frac{371}{33750}a^{5}+\frac{107}{33750}a^{4}-\frac{10796}{28125}a^{3}-\frac{13843}{337500}a^{2}-\frac{2653}{67500}a+\frac{28}{375}$, $\frac{1}{2700000}a^{15}-\frac{1}{2700000}a^{13}+\frac{1}{75000}a^{11}-\frac{13}{337500}a^{10}-\frac{47}{337500}a^{9}+\frac{11}{112500}a^{8}+\frac{289}{112500}a^{7}+\frac{838}{84375}a^{6}+\frac{29}{11250}a^{5}-\frac{1093}{84375}a^{4}-\frac{733}{22500}a^{3}+\frac{4106}{28125}a^{2}-\frac{2533}{67500}a-\frac{1543}{3375}$, $\frac{1}{8100000}a^{16}-\frac{1}{8100000}a^{15}-\frac{1}{1620000}a^{13}-\frac{1}{2025000}a^{12}-\frac{1}{45000}a^{11}+\frac{53}{1012500}a^{10}+\frac{29}{202500}a^{9}-\frac{37}{112500}a^{8}-\frac{113}{40500}a^{7}-\frac{1531}{101250}a^{6}-\frac{187}{28125}a^{5}+\frac{12577}{1012500}a^{4}-\frac{91313}{202500}a^{3}-\frac{22}{625}a^{2}+\frac{18787}{40500}a+\frac{412}{2025}$, $\frac{1}{2025000000}a^{17}+\frac{1}{202500000}a^{16}-\frac{13}{101250000}a^{15}+\frac{1}{12656250}a^{14}+\frac{169}{202500000}a^{13}-\frac{116}{31640625}a^{12}-\frac{97}{25312500}a^{11}+\frac{1001}{25312500}a^{10}-\frac{3829}{25312500}a^{9}-\frac{4553}{25312500}a^{8}+\frac{341861}{126562500}a^{7}-\frac{98833}{12656250}a^{6}-\frac{165811}{50625000}a^{5}-\frac{286427}{25312500}a^{4}+\frac{2955004}{6328125}a^{3}-\frac{2652991}{31640625}a^{2}+\frac{10621681}{25312500}a+\frac{86636}{1265625}$, $\frac{1}{10125000000}a^{18}+\frac{1}{10125000000}a^{17}-\frac{7}{202500000}a^{16}-\frac{1}{20250000}a^{15}+\frac{1}{40500000}a^{14}-\frac{221}{2531250000}a^{13}+\frac{1757}{1265625000}a^{12}-\frac{4577}{253125000}a^{11}+\frac{1531}{63281250}a^{10}+\frac{12127}{31640625}a^{9}+\frac{322871}{632812500}a^{8}+\frac{121273}{316406250}a^{7}+\frac{1331477}{253125000}a^{6}-\frac{608111}{50625000}a^{5}+\frac{416959}{126562500}a^{4}-\frac{13290046}{158203125}a^{3}+\frac{172718081}{632812500}a^{2}+\frac{29560733}{63281250}a+\frac{1019717}{2109375}$, $\frac{1}{50625000000}a^{19}+\frac{1}{25312500000}a^{18}+\frac{1}{50625000000}a^{17}+\frac{53}{1012500000}a^{16}-\frac{1}{10125000}a^{15}-\frac{221}{12656250000}a^{14}+\frac{15461}{25312500000}a^{13}-\frac{2107}{1582031250}a^{12}-\frac{11239}{632812500}a^{11}-\frac{2854}{31640625}a^{10}+\frac{457387}{1054687500}a^{9}+\frac{573917}{3164062500}a^{8}+\frac{15786427}{6328125000}a^{7}-\frac{8985389}{632812500}a^{6}+\frac{10972013}{1265625000}a^{5}-\frac{35058889}{3164062500}a^{4}+\frac{94832011}{1582031250}a^{3}-\frac{9548807}{1582031250}a^{2}+\frac{37929487}{210937500}a+\frac{14989876}{31640625}$, $\frac{1}{759375000000}a^{20}+\frac{1}{189843750000}a^{19}-\frac{1}{37968750000}a^{18}+\frac{59}{253125000000}a^{17}+\frac{17}{316406250}a^{16}-\frac{1547}{10546875000}a^{15}+\frac{1151}{14062500000}a^{14}+\frac{11909}{25312500000}a^{13}+\frac{18893}{31640625000}a^{12}-\frac{115633}{9492187500}a^{11}+\frac{4993861}{47460937500}a^{10}-\frac{10740011}{47460937500}a^{9}-\frac{2510447}{6328125000}a^{8}-\frac{51136831}{15820312500}a^{7}+\frac{7377137}{527343750}a^{6}-\frac{151715947}{10546875000}a^{5}+\frac{38446687}{3955078125}a^{4}+\frac{272165051}{1582031250}a^{3}+\frac{1223609977}{47460937500}a^{2}+\frac{4094659967}{9492187500}a+\frac{54949027}{474609375}$, $\frac{1}{16706250000000}a^{21}-\frac{1}{464062500000}a^{19}-\frac{67}{4176562500000}a^{18}-\frac{109}{1392187500000}a^{17}-\frac{1301}{43505859375}a^{16}-\frac{10117}{92812500000}a^{15}-\frac{109379}{696093750000}a^{14}+\frac{4078}{43505859375}a^{13}-\frac{2928923}{1044140625000}a^{12}-\frac{2129309}{174023437500}a^{11}-\frac{639211}{34804687500}a^{10}-\frac{119549867}{2088281250000}a^{9}-\frac{13082317}{43505859375}a^{8}+\frac{27790673}{43505859375}a^{7}-\frac{800538383}{58007812500}a^{6}-\frac{422211257}{34804687500}a^{5}-\frac{560233433}{87011718750}a^{4}-\frac{113614301959}{522070312500}a^{3}-\frac{2832105029}{87011718750}a^{2}+\frac{2457450553}{8701171875}a+\frac{1734115621}{5220703125}$, $\frac{1}{83531250000000}a^{22}+\frac{1}{41765625000000}a^{21}-\frac{1}{2320312500000}a^{20}-\frac{17}{4176562500000}a^{19}-\frac{97}{41765625000000}a^{18}+\frac{73}{556875000000}a^{17}-\frac{50161}{1740234375000}a^{16}-\frac{501643}{6960937500000}a^{15}-\frac{15577}{696093750000}a^{14}+\frac{4233259}{20882812500000}a^{13}-\frac{137233}{52207031250}a^{12}-\frac{642923}{870117187500}a^{11}-\frac{95242187}{10441406250000}a^{10}+\frac{21068671}{208828125000}a^{9}-\frac{102867361}{217529296875}a^{8}-\frac{136789937}{43505859375}a^{7}+\frac{5532192709}{1740234375000}a^{6}+\frac{23953201753}{1740234375000}a^{5}+\frac{3726713897}{261035156250}a^{4}+\frac{69215400203}{237304687500}a^{3}-\frac{17395910453}{435058593750}a^{2}+\frac{164348735267}{522070312500}a-\frac{6527624683}{26103515625}$, $\frac{1}{83531250000000}a^{23}-\frac{1}{1740234375000}a^{20}-\frac{37}{41765625000000}a^{19}-\frac{1891}{41765625000000}a^{18}-\frac{1543}{13921875000000}a^{17}+\frac{30223}{1392187500000}a^{16}+\frac{40069}{1740234375000}a^{15}+\frac{2341639}{20882812500000}a^{14}-\frac{576749}{773437500000}a^{13}-\frac{2531153}{870117187500}a^{12}-\frac{8550697}{696093750000}a^{11}+\frac{134815091}{2610351562500}a^{10}+\frac{322920319}{1305175781250}a^{9}-\frac{158978311}{435058593750}a^{8}-\frac{3831651041}{1740234375000}a^{7}+\frac{5778136723}{348046875000}a^{6}-\frac{26462906633}{5220703125000}a^{5}-\frac{3833351143}{290039062500}a^{4}+\frac{207647690251}{435058593750}a^{3}-\frac{223401306323}{870117187500}a^{2}+\frac{194690245741}{522070312500}a+\frac{2574527801}{26103515625}$, $\frac{1}{12\cdots 00}a^{24}-\frac{1}{261035156250000}a^{22}-\frac{67}{62\cdots 00}a^{21}+\frac{43}{696093750000000}a^{20}+\frac{4159}{10\cdots 00}a^{19}+\frac{1681}{31\cdots 00}a^{18}+\frac{1027}{17402343750000}a^{17}+\frac{2844277}{116015625000000}a^{16}+\frac{211904093}{15\cdots 00}a^{15}+\frac{3579923}{21752929687500}a^{14}-\frac{97642151}{130517578125000}a^{13}+\frac{383252773}{313242187500000}a^{12}-\frac{74641511}{5438232421875}a^{11}-\frac{7165741643}{65258789062500}a^{10}+\frac{33539645413}{783105468750000}a^{9}+\frac{6666647791}{87011718750000}a^{8}-\frac{369637043}{580078125000}a^{7}+\frac{2870954691653}{391552734375000}a^{6}+\frac{14877564106}{5438232421875}a^{5}+\frac{2120316686077}{130517578125000}a^{4}+\frac{17953710084091}{195776367187500}a^{3}+\frac{577503685039}{1812744140625}a^{2}-\frac{963897993934}{3262939453125}a+\frac{324339789682}{1957763671875}$, $\frac{1}{40\cdots 00}a^{25}+\frac{73}{16\cdots 00}a^{23}+\frac{19}{62\cdots 00}a^{22}+\frac{71}{27\cdots 00}a^{21}+\frac{337}{10\cdots 00}a^{20}+\frac{1603}{284765625000000}a^{19}+\frac{4691}{104414062500000}a^{18}-\frac{523}{87011718750000}a^{17}-\frac{1103068}{48944091796875}a^{16}+\frac{25973137}{174023437500000}a^{15}-\frac{8985247}{522070312500000}a^{14}-\frac{1113405001}{50\cdots 00}a^{13}+\frac{4023533}{21752929687500}a^{12}-\frac{30930516043}{20\cdots 00}a^{11}+\frac{40882406759}{783105468750000}a^{10}+\frac{215860006303}{522070312500000}a^{9}+\frac{923523233}{1740234375000}a^{8}-\frac{397786364323}{195776367187500}a^{7}+\frac{347646908707}{21752929687500}a^{6}-\frac{50508442537}{5932617187500}a^{5}-\frac{658003120528}{48944091796875}a^{4}+\frac{260185150087}{805664062500}a^{3}+\frac{2462776575377}{13051757812500}a^{2}-\frac{26038684356347}{250593750000000}a-\frac{181723486}{474609375}$, $\frac{1}{41\cdots 00}a^{26}+\frac{15\cdots 03}{16\cdots 00}a^{25}-\frac{20\cdots 37}{52\cdots 00}a^{24}-\frac{25\cdots 41}{18\cdots 00}a^{23}-\frac{53\cdots 49}{26\cdots 00}a^{22}+\frac{84\cdots 03}{40\cdots 00}a^{21}+\frac{22\cdots 43}{81\cdots 00}a^{20}+\frac{16\cdots 81}{20\cdots 00}a^{19}-\frac{42\cdots 61}{16\cdots 00}a^{18}-\frac{33\cdots 41}{16\cdots 00}a^{17}+\frac{45\cdots 39}{20\cdots 00}a^{16}-\frac{18\cdots 79}{63\cdots 00}a^{15}-\frac{28\cdots 57}{26\cdots 00}a^{14}+\frac{32\cdots 69}{10\cdots 00}a^{13}-\frac{42\cdots 97}{65\cdots 00}a^{12}+\frac{30\cdots 93}{25\cdots 00}a^{11}+\frac{97\cdots 81}{16\cdots 00}a^{10}-\frac{12\cdots 11}{50\cdots 00}a^{9}+\frac{10\cdots 21}{40\cdots 00}a^{8}+\frac{22\cdots 03}{25\cdots 00}a^{7}+\frac{47\cdots 49}{25\cdots 00}a^{6}+\frac{19\cdots 73}{63\cdots 00}a^{5}+\frac{35\cdots 47}{31\cdots 00}a^{4}+\frac{62\cdots 12}{39\cdots 25}a^{3}-\frac{22\cdots 71}{65\cdots 00}a^{2}-\frac{13\cdots 41}{50\cdots 00}a-\frac{32\cdots 91}{15\cdots 25}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $14$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{3}\cdot(2\pi)^{12}\cdot R \cdot h}{2\cdot\sqrt{177801404718372423264819198352687104000000000000000000000000000000000000000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^27 - 360*x^25 + 54000*x^23 - 9216*x^22 - 4446720*x^21 + 1935360*x^20 + 223891200*x^19 - 147947520*x^18 - 7264733184*x^17 + 5311365120*x^16 + 159244507440*x^15 - 91269365760*x^14 - 2497379088960*x^13 + 545341833216*x^12 + 28698328270080*x^11 + 7709768294400*x^10 - 231976756423680*x^9 - 199955270615040*x^8 + 1167100609904640*x^7 + 1878904774656000*x^6 - 2532714300702720*x^5 - 7902598720389120*x^4 - 3861559365624000*x^3 + 9381033208184832*x^2 + 13476478435983360*x + 6066476888883200); 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

$\SO(5,3)$ (as 27T1161):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 51840
The 25 conjugacy class representatives for $\SO(5,3)$
Character table for $\SO(5,3)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 R ${\href{/padicField/7.9.0.1}{9} }^{3}$ ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.9.0.1}{9} }^{3}$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }^{4}{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.12.0.1}{12} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.9.0.1}{9} }^{3}$ ${\href{/padicField/47.8.0.1}{8} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$

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

Copy content comment:Frobenius cycle types
 
Copy content 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)]
 
Copy content 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]])
 
Copy content 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))];
 
Copy content 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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.4.22a1.23$x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 19 x^{4} + 16 x^{3} + 18 x^{2} + 20 x + 11$$4$$2$$22$$(C_8:C_2):C_2$$$[2, 2, 3, 4]^{2}$$
2.2.8.48c13.729$x^{16} + 8 x^{15} + 40 x^{14} + 144 x^{13} + 408 x^{12} + 936 x^{11} + 1780 x^{10} + 2832 x^{9} + 3797 x^{8} + 4296 x^{7} + 4096 x^{6} + 3272 x^{5} + 2164 x^{4} + 1168 x^{3} + 500 x^{2} + 168 x + 39$$8$$2$$48$16T41$$[2, 2, 3, 4]^{2}$$
\(3\) Copy content Toggle raw display Deg $27$$27$$1$$60$
\(5\) Copy content Toggle raw display 5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.1.5.9a1.1$x^{5} + 5$$5$$1$$9$$F_5$$$[\frac{9}{4}]_{4}$$
5.1.10.19a2.1$x^{10} + 5$$10$$1$$19$$F_5$$$[\frac{9}{4}]_{4}$$
5.1.10.19a1.4$x^{10} + 75 x^{2} + 10$$10$$1$$19$$F_{5}\times C_2$$$[\frac{9}{4}]_{4}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)