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Properties

Label 30.2.140...000.1
Degree $30$
Signature $(2, 14)$
Discriminant $1.403\times 10^{46}$
Root discriminant \(34.53\)
Ramified primes $2,5,131$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16)
 
Copy content gp:K = bnfinit(y^30 - y^29 + 11*y^28 + 4*y^27 + 44*y^26 - 6*y^25 - 74*y^24 - 446*y^23 - 1543*y^22 + 183*y^21 - 1721*y^20 + 7116*y^19 + 11884*y^18 - 3168*y^17 + 7664*y^16 - 18756*y^15 + 17109*y^14 - 3363*y^13 - 2431*y^12 + 11594*y^11 - 15080*y^10 + 6868*y^9 + 3884*y^8 - 3768*y^7 + 2512*y^6 - 1864*y^5 - 1696*y^4 + 656*y^3 + 448*y^2 + 16*y - 16, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16)
 

\( x^{30} - x^{29} + 11 x^{28} + 4 x^{27} + 44 x^{26} - 6 x^{25} - 74 x^{24} - 446 x^{23} - 1543 x^{22} + \cdots - 16 \) 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:  $30$
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:  $(2, 14)$
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:   \(14026461290181639205847118647072000000000000000\) \(\medspace = 2^{20}\cdot 5^{15}\cdot 131^{14}\) 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:  \(34.53\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}5^{1/2}131^{1/2}\approx 40.62630398353121$
Ramified primes:   \(2\), \(5\), \(131\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{12}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{13}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{20}-\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{8}a^{16}-\frac{1}{8}a^{14}+\frac{1}{8}a^{13}+\frac{1}{8}a^{12}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{2}a^{8}+\frac{1}{8}a^{7}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{21}-\frac{1}{8}a^{19}-\frac{1}{8}a^{18}-\frac{1}{8}a^{16}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}+\frac{1}{8}a^{9}+\frac{3}{8}a^{8}-\frac{1}{2}a^{7}-\frac{3}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{8}a^{22}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{3}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{23}-\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{24}-\frac{1}{8}a^{16}-\frac{3}{8}a^{8}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{25}-\frac{1}{8}a^{17}+\frac{1}{8}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{181424}a^{26}+\frac{5153}{181424}a^{25}-\frac{10609}{181424}a^{24}+\frac{400}{11339}a^{23}+\frac{45}{1564}a^{22}-\frac{929}{90712}a^{21}-\frac{89}{2668}a^{20}+\frac{2355}{45356}a^{19}-\frac{6887}{181424}a^{18}-\frac{10173}{181424}a^{17}-\frac{7519}{181424}a^{16}+\frac{395}{45356}a^{15}+\frac{9661}{90712}a^{14}-\frac{19939}{90712}a^{13}+\frac{213}{3128}a^{12}-\frac{11233}{45356}a^{11}+\frac{39543}{181424}a^{10}-\frac{41403}{181424}a^{9}-\frac{8767}{181424}a^{8}-\frac{22453}{45356}a^{7}-\frac{15647}{90712}a^{6}-\frac{10265}{45356}a^{5}-\frac{3879}{22678}a^{4}+\frac{4739}{45356}a^{3}-\frac{7839}{45356}a^{2}-\frac{10335}{22678}a+\frac{5421}{22678}$, $\frac{1}{362848}a^{27}-\frac{1}{362848}a^{26}-\frac{13233}{362848}a^{25}-\frac{7075}{181424}a^{24}+\frac{8055}{181424}a^{23}+\frac{1631}{45356}a^{22}+\frac{11321}{181424}a^{21}+\frac{9589}{181424}a^{20}+\frac{18709}{362848}a^{19}-\frac{5645}{362848}a^{18}-\frac{7413}{362848}a^{17}-\frac{12437}{181424}a^{16}-\frac{2637}{181424}a^{15}+\frac{471}{7888}a^{14}-\frac{239}{2668}a^{13}-\frac{20745}{90712}a^{12}+\frac{8657}{362848}a^{11}-\frac{38839}{362848}a^{10}+\frac{4993}{362848}a^{9}+\frac{4265}{45356}a^{8}-\frac{67147}{181424}a^{7}+\frac{659}{7888}a^{6}+\frac{24295}{90712}a^{5}+\frac{8157}{90712}a^{4}-\frac{1073}{3128}a^{3}+\frac{1837}{11339}a^{2}+\frac{1389}{45356}a-\frac{11877}{45356}$, $\frac{1}{362848}a^{28}-\frac{2649}{362848}a^{25}-\frac{10363}{181424}a^{24}+\frac{875}{181424}a^{23}-\frac{2687}{181424}a^{22}-\frac{2339}{90712}a^{21}-\frac{941}{362848}a^{20}+\frac{2441}{22678}a^{19}-\frac{8853}{90712}a^{18}-\frac{45161}{362848}a^{17}+\frac{431}{7888}a^{16}+\frac{4249}{90712}a^{15}+\frac{12369}{181424}a^{14}+\frac{9387}{90712}a^{13}-\frac{44511}{362848}a^{12}+\frac{1123}{181424}a^{11}+\frac{1511}{90712}a^{10}-\frac{1945}{21344}a^{9}-\frac{25501}{90712}a^{8}-\frac{978}{11339}a^{7}-\frac{49223}{181424}a^{6}-\frac{9347}{45356}a^{5}+\frac{212}{493}a^{4}+\frac{17513}{90712}a^{3}-\frac{10009}{22678}a^{2}-\frac{3215}{11339}a+\frac{21801}{45356}$, $\frac{1}{69\cdots 88}a^{29}-\frac{73\cdots 69}{69\cdots 88}a^{28}-\frac{44\cdots 01}{69\cdots 88}a^{27}-\frac{14\cdots 97}{34\cdots 44}a^{26}-\frac{36\cdots 85}{34\cdots 44}a^{25}+\frac{54\cdots 51}{10\cdots 16}a^{24}-\frac{10\cdots 73}{34\cdots 44}a^{23}-\frac{21\cdots 61}{34\cdots 44}a^{22}-\frac{18\cdots 27}{69\cdots 88}a^{21}-\frac{60\cdots 05}{40\cdots 64}a^{20}-\frac{86\cdots 97}{69\cdots 88}a^{19}-\frac{16\cdots 97}{34\cdots 44}a^{18}-\frac{50\cdots 41}{47\cdots 28}a^{17}-\frac{41\cdots 55}{34\cdots 44}a^{16}+\frac{99\cdots 28}{10\cdots 17}a^{15}+\frac{11\cdots 21}{43\cdots 68}a^{14}+\frac{10\cdots 65}{69\cdots 88}a^{13}+\frac{92\cdots 29}{40\cdots 64}a^{12}+\frac{71\cdots 17}{69\cdots 88}a^{11}-\frac{42\cdots 95}{17\cdots 72}a^{10}+\frac{83\cdots 41}{99\cdots 52}a^{9}+\frac{36\cdots 13}{34\cdots 44}a^{8}-\frac{15\cdots 03}{86\cdots 36}a^{7}+\frac{15\cdots 03}{17\cdots 72}a^{6}-\frac{95\cdots 17}{23\cdots 64}a^{5}+\frac{19\cdots 19}{43\cdots 68}a^{4}-\frac{24\cdots 15}{86\cdots 36}a^{3}+\frac{12\cdots 55}{29\cdots 84}a^{2}-\frac{11\cdots 19}{43\cdots 68}a+\frac{13\cdots 98}{10\cdots 17}$ 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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $15$
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:   $\frac{63\cdots 45}{23\cdots 72}a^{29}-\frac{91\cdots 87}{23\cdots 72}a^{28}+\frac{73\cdots 57}{23\cdots 72}a^{27}-\frac{15\cdots 87}{59\cdots 68}a^{26}+\frac{14\cdots 65}{11\cdots 36}a^{25}-\frac{20\cdots 91}{29\cdots 84}a^{24}-\frac{20\cdots 49}{11\cdots 36}a^{23}-\frac{13\cdots 55}{11\cdots 36}a^{22}-\frac{86\cdots 83}{23\cdots 72}a^{21}+\frac{49\cdots 45}{23\cdots 72}a^{20}-\frac{12\cdots 87}{23\cdots 72}a^{19}+\frac{12\cdots 09}{59\cdots 68}a^{18}+\frac{36\cdots 77}{16\cdots 32}a^{17}-\frac{21\cdots 77}{11\cdots 36}a^{16}+\frac{16\cdots 11}{59\cdots 68}a^{15}-\frac{18\cdots 11}{29\cdots 84}a^{14}+\frac{17\cdots 25}{23\cdots 72}a^{13}-\frac{95\cdots 93}{23\cdots 72}a^{12}+\frac{24\cdots 43}{23\cdots 72}a^{11}+\frac{31\cdots 39}{11\cdots 36}a^{10}-\frac{77\cdots 81}{14\cdots 56}a^{9}+\frac{48\cdots 55}{11\cdots 36}a^{8}-\frac{40\cdots 61}{59\cdots 68}a^{7}-\frac{18\cdots 15}{25\cdots 16}a^{6}+\frac{79\cdots 01}{81\cdots 16}a^{5}-\frac{27\cdots 73}{29\cdots 84}a^{4}-\frac{19\cdots 43}{29\cdots 84}a^{3}+\frac{59\cdots 05}{29\cdots 84}a^{2}+\frac{14\cdots 09}{43\cdots 38}a+\frac{46\cdots 95}{74\cdots 46}$, $\frac{14\cdots 53}{69\cdots 88}a^{29}-\frac{23\cdots 47}{69\cdots 88}a^{28}+\frac{58\cdots 41}{23\cdots 72}a^{27}-\frac{53\cdots 65}{86\cdots 36}a^{26}+\frac{30\cdots 11}{34\cdots 44}a^{25}-\frac{12\cdots 15}{17\cdots 72}a^{24}-\frac{50\cdots 69}{34\cdots 44}a^{23}-\frac{75\cdots 41}{88\cdots 84}a^{22}-\frac{80\cdots 97}{30\cdots 56}a^{21}+\frac{56\cdots 85}{23\cdots 72}a^{20}-\frac{15\cdots 31}{40\cdots 64}a^{19}+\frac{15\cdots 61}{86\cdots 36}a^{18}+\frac{74\cdots 95}{47\cdots 28}a^{17}-\frac{72\cdots 99}{34\cdots 44}a^{16}+\frac{34\cdots 63}{17\cdots 72}a^{15}-\frac{45\cdots 75}{86\cdots 36}a^{14}+\frac{41\cdots 13}{69\cdots 88}a^{13}-\frac{23\cdots 89}{69\cdots 88}a^{12}+\frac{30\cdots 23}{69\cdots 88}a^{11}+\frac{79\cdots 73}{34\cdots 44}a^{10}-\frac{44\cdots 41}{99\cdots 52}a^{9}+\frac{11\cdots 89}{34\cdots 44}a^{8}-\frac{58\cdots 11}{17\cdots 72}a^{7}-\frac{17\cdots 53}{17\cdots 72}a^{6}+\frac{20\cdots 09}{23\cdots 64}a^{5}-\frac{63\cdots 17}{86\cdots 36}a^{4}-\frac{53\cdots 57}{50\cdots 08}a^{3}+\frac{24\cdots 51}{86\cdots 36}a^{2}+\frac{22\cdots 43}{74\cdots 46}a-\frac{64\cdots 05}{21\cdots 34}$, $\frac{62\cdots 99}{69\cdots 88}a^{29}-\frac{65\cdots 91}{69\cdots 88}a^{28}+\frac{68\cdots 63}{69\cdots 88}a^{27}+\frac{27\cdots 41}{86\cdots 36}a^{26}+\frac{67\cdots 39}{17\cdots 72}a^{25}-\frac{28\cdots 07}{43\cdots 68}a^{24}-\frac{23\cdots 21}{34\cdots 44}a^{23}-\frac{13\cdots 71}{34\cdots 44}a^{22}-\frac{94\cdots 89}{69\cdots 88}a^{21}+\frac{16\cdots 13}{69\cdots 88}a^{20}-\frac{10\cdots 93}{69\cdots 88}a^{19}+\frac{68\cdots 33}{10\cdots 17}a^{18}+\frac{24\cdots 65}{23\cdots 64}a^{17}-\frac{44\cdots 99}{11\cdots 36}a^{16}+\frac{12\cdots 93}{17\cdots 72}a^{15}-\frac{28\cdots 45}{17\cdots 72}a^{14}+\frac{10\cdots 55}{69\cdots 88}a^{13}-\frac{16\cdots 77}{69\cdots 88}a^{12}-\frac{33\cdots 07}{69\cdots 88}a^{11}+\frac{47\cdots 83}{34\cdots 44}a^{10}-\frac{17\cdots 42}{10\cdots 59}a^{9}+\frac{27\cdots 31}{34\cdots 44}a^{8}+\frac{57\cdots 65}{17\cdots 72}a^{7}-\frac{45\cdots 89}{86\cdots 36}a^{6}+\frac{96\cdots 55}{23\cdots 64}a^{5}-\frac{22\cdots 51}{86\cdots 36}a^{4}-\frac{13\cdots 58}{10\cdots 17}a^{3}+\frac{70\cdots 69}{86\cdots 36}a^{2}+\frac{39\cdots 49}{37\cdots 73}a+\frac{51\cdots 31}{43\cdots 68}$, $\frac{25\cdots 95}{69\cdots 88}a^{29}-\frac{34\cdots 05}{69\cdots 88}a^{28}+\frac{29\cdots 23}{69\cdots 88}a^{27}-\frac{12\cdots 67}{17\cdots 72}a^{26}+\frac{58\cdots 95}{34\cdots 44}a^{25}-\frac{13\cdots 51}{17\cdots 72}a^{24}-\frac{82\cdots 83}{34\cdots 44}a^{23}-\frac{54\cdots 25}{34\cdots 44}a^{22}-\frac{36\cdots 81}{69\cdots 88}a^{21}+\frac{16\cdots 15}{69\cdots 88}a^{20}-\frac{53\cdots 73}{69\cdots 88}a^{19}+\frac{50\cdots 37}{17\cdots 72}a^{18}+\frac{16\cdots 19}{47\cdots 28}a^{17}-\frac{76\cdots 61}{34\cdots 44}a^{16}+\frac{40\cdots 43}{10\cdots 16}a^{15}-\frac{17\cdots 11}{21\cdots 34}a^{14}+\frac{66\cdots 07}{69\cdots 88}a^{13}-\frac{34\cdots 43}{69\cdots 88}a^{12}+\frac{79\cdots 69}{69\cdots 88}a^{11}+\frac{13\cdots 73}{34\cdots 44}a^{10}-\frac{68\cdots 25}{99\cdots 52}a^{9}+\frac{60\cdots 51}{11\cdots 36}a^{8}-\frac{82\cdots 99}{17\cdots 72}a^{7}-\frac{20\cdots 93}{17\cdots 72}a^{6}+\frac{33\cdots 83}{23\cdots 64}a^{5}-\frac{10\cdots 45}{86\cdots 36}a^{4}-\frac{18\cdots 29}{86\cdots 36}a^{3}+\frac{26\cdots 05}{86\cdots 36}a^{2}+\frac{68\cdots 27}{37\cdots 73}a-\frac{16\cdots 98}{10\cdots 17}$, $\frac{26\cdots 17}{69\cdots 88}a^{29}-\frac{32\cdots 41}{69\cdots 88}a^{28}+\frac{29\cdots 53}{69\cdots 88}a^{27}+\frac{42\cdots 99}{86\cdots 36}a^{26}+\frac{28\cdots 83}{17\cdots 72}a^{25}-\frac{54\cdots 85}{86\cdots 36}a^{24}-\frac{92\cdots 51}{34\cdots 44}a^{23}-\frac{56\cdots 37}{34\cdots 44}a^{22}-\frac{56\cdots 41}{10\cdots 64}a^{21}+\frac{13\cdots 55}{69\cdots 88}a^{20}-\frac{48\cdots 91}{69\cdots 88}a^{19}+\frac{10\cdots 81}{37\cdots 32}a^{18}+\frac{90\cdots 61}{23\cdots 64}a^{17}-\frac{72\cdots 29}{34\cdots 44}a^{16}+\frac{58\cdots 09}{17\cdots 72}a^{15}-\frac{13\cdots 95}{17\cdots 72}a^{14}+\frac{58\cdots 17}{69\cdots 88}a^{13}-\frac{10\cdots 49}{30\cdots 56}a^{12}-\frac{12\cdots 41}{69\cdots 88}a^{11}+\frac{14\cdots 81}{34\cdots 44}a^{10}-\frac{82\cdots 35}{12\cdots 44}a^{9}+\frac{13\cdots 25}{34\cdots 44}a^{8}+\frac{10\cdots 99}{17\cdots 72}a^{7}-\frac{17\cdots 48}{10\cdots 17}a^{6}+\frac{30\cdots 77}{23\cdots 64}a^{5}-\frac{85\cdots 17}{86\cdots 36}a^{4}-\frac{48\cdots 87}{10\cdots 17}a^{3}+\frac{33\cdots 23}{86\cdots 36}a^{2}+\frac{27\cdots 05}{37\cdots 73}a-\frac{69\cdots 21}{43\cdots 68}$, $\frac{42\cdots 97}{34\cdots 44}a^{29}-\frac{50\cdots 79}{34\cdots 44}a^{28}+\frac{23\cdots 79}{17\cdots 72}a^{27}+\frac{86\cdots 73}{34\cdots 44}a^{26}+\frac{17\cdots 83}{34\cdots 44}a^{25}-\frac{15\cdots 97}{86\cdots 36}a^{24}-\frac{82\cdots 29}{86\cdots 36}a^{23}-\frac{91\cdots 85}{17\cdots 72}a^{22}-\frac{61\cdots 93}{34\cdots 44}a^{21}+\frac{21\cdots 27}{34\cdots 44}a^{20}-\frac{33\cdots 83}{17\cdots 72}a^{19}+\frac{31\cdots 01}{34\cdots 44}a^{18}+\frac{62\cdots 75}{47\cdots 28}a^{17}-\frac{12\cdots 15}{17\cdots 72}a^{16}+\frac{15\cdots 45}{17\cdots 72}a^{15}-\frac{42\cdots 39}{17\cdots 72}a^{14}+\frac{80\cdots 85}{34\cdots 44}a^{13}-\frac{20\cdots 09}{34\cdots 44}a^{12}-\frac{25\cdots 59}{43\cdots 68}a^{11}+\frac{57\cdots 37}{34\cdots 44}a^{10}-\frac{22\cdots 53}{99\cdots 52}a^{9}+\frac{95\cdots 23}{86\cdots 36}a^{8}+\frac{75\cdots 39}{17\cdots 72}a^{7}-\frac{11\cdots 25}{17\cdots 72}a^{6}+\frac{14\cdots 79}{34\cdots 48}a^{5}-\frac{24\cdots 41}{86\cdots 36}a^{4}-\frac{17\cdots 81}{86\cdots 36}a^{3}+\frac{15\cdots 41}{10\cdots 17}a^{2}+\frac{56\cdots 31}{14\cdots 92}a-\frac{52\cdots 01}{43\cdots 68}$, $\frac{53\cdots 53}{97\cdots 48}a^{29}+\frac{66\cdots 85}{33\cdots 12}a^{28}+\frac{10\cdots 91}{97\cdots 48}a^{27}+\frac{15\cdots 43}{48\cdots 24}a^{26}+\frac{73\cdots 13}{12\cdots 56}a^{25}+\frac{24\cdots 13}{24\cdots 12}a^{24}-\frac{83\cdots 91}{48\cdots 24}a^{23}-\frac{20\cdots 03}{48\cdots 24}a^{22}-\frac{17\cdots 27}{97\cdots 48}a^{21}-\frac{15\cdots 79}{57\cdots 44}a^{20}+\frac{32\cdots 39}{97\cdots 48}a^{19}-\frac{36\cdots 53}{48\cdots 24}a^{18}+\frac{45\cdots 79}{15\cdots 57}a^{17}+\frac{51\cdots 05}{48\cdots 24}a^{16}-\frac{19\cdots 25}{61\cdots 28}a^{15}+\frac{35\cdots 99}{24\cdots 12}a^{14}-\frac{63\cdots 83}{97\cdots 48}a^{13}+\frac{84\cdots 67}{97\cdots 48}a^{12}-\frac{35\cdots 15}{57\cdots 44}a^{11}+\frac{46\cdots 37}{24\cdots 12}a^{10}+\frac{77\cdots 35}{44\cdots 31}a^{9}-\frac{30\cdots 09}{48\cdots 24}a^{8}+\frac{17\cdots 05}{30\cdots 14}a^{7}-\frac{52\cdots 27}{30\cdots 14}a^{6}-\frac{21\cdots 17}{24\cdots 12}a^{5}+\frac{29\cdots 69}{26\cdots 36}a^{4}-\frac{20\cdots 13}{15\cdots 57}a^{3}+\frac{26\cdots 61}{12\cdots 56}a^{2}+\frac{57\cdots 43}{21\cdots 32}a+\frac{10\cdots 39}{26\cdots 36}$, $\frac{49\cdots 33}{97\cdots 48}a^{29}-\frac{32\cdots 19}{97\cdots 48}a^{28}+\frac{48\cdots 91}{97\cdots 48}a^{27}+\frac{21\cdots 35}{48\cdots 24}a^{26}+\frac{22\cdots 95}{12\cdots 56}a^{25}+\frac{71\cdots 33}{24\cdots 12}a^{24}-\frac{28\cdots 23}{48\cdots 24}a^{23}-\frac{11\cdots 83}{48\cdots 24}a^{22}-\frac{81\cdots 99}{97\cdots 48}a^{21}+\frac{58\cdots 69}{97\cdots 48}a^{20}-\frac{18\cdots 41}{97\cdots 48}a^{19}+\frac{16\cdots 07}{48\cdots 24}a^{18}+\frac{50\cdots 61}{61\cdots 28}a^{17}-\frac{11\cdots 71}{48\cdots 24}a^{16}-\frac{19\cdots 53}{15\cdots 57}a^{15}-\frac{19\cdots 57}{24\cdots 12}a^{14}+\frac{16\cdots 33}{97\cdots 48}a^{13}+\frac{83\cdots 59}{97\cdots 48}a^{12}-\frac{10\cdots 59}{97\cdots 48}a^{11}+\frac{20\cdots 97}{24\cdots 12}a^{10}-\frac{31\cdots 75}{44\cdots 31}a^{9}-\frac{17\cdots 93}{48\cdots 24}a^{8}+\frac{29\cdots 61}{30\cdots 14}a^{7}-\frac{17\cdots 91}{30\cdots 14}a^{6}+\frac{14\cdots 03}{24\cdots 12}a^{5}+\frac{71\cdots 11}{61\cdots 28}a^{4}-\frac{42\cdots 69}{15\cdots 57}a^{3}+\frac{82\cdots 57}{71\cdots 68}a^{2}+\frac{12\cdots 51}{21\cdots 32}a-\frac{55\cdots 51}{61\cdots 28}$, $\frac{12\cdots 89}{69\cdots 88}a^{29}-\frac{12\cdots 55}{69\cdots 88}a^{28}+\frac{82\cdots 17}{40\cdots 64}a^{27}+\frac{12\cdots 69}{17\cdots 72}a^{26}+\frac{28\cdots 19}{34\cdots 44}a^{25}-\frac{19\cdots 25}{17\cdots 72}a^{24}-\frac{46\cdots 69}{34\cdots 44}a^{23}-\frac{28\cdots 95}{34\cdots 44}a^{22}-\frac{19\cdots 87}{69\cdots 88}a^{21}+\frac{24\cdots 57}{69\cdots 88}a^{20}-\frac{22\cdots 19}{69\cdots 88}a^{19}+\frac{22\cdots 77}{17\cdots 72}a^{18}+\frac{10\cdots 83}{47\cdots 28}a^{17}-\frac{21\cdots 63}{34\cdots 44}a^{16}+\frac{27\cdots 59}{17\cdots 72}a^{15}-\frac{73\cdots 27}{21\cdots 34}a^{14}+\frac{22\cdots 29}{69\cdots 88}a^{13}-\frac{44\cdots 57}{69\cdots 88}a^{12}-\frac{39\cdots 33}{69\cdots 88}a^{11}+\frac{84\cdots 03}{34\cdots 44}a^{10}-\frac{17\cdots 69}{58\cdots 56}a^{9}+\frac{51\cdots 73}{34\cdots 44}a^{8}+\frac{10\cdots 69}{17\cdots 72}a^{7}-\frac{14\cdots 71}{17\cdots 72}a^{6}+\frac{14\cdots 01}{23\cdots 64}a^{5}-\frac{35\cdots 55}{86\cdots 36}a^{4}-\frac{23\cdots 97}{86\cdots 36}a^{3}+\frac{11\cdots 47}{86\cdots 36}a^{2}+\frac{12\cdots 89}{21\cdots 34}a+\frac{91\cdots 91}{10\cdots 17}$, $\frac{20\cdots 71}{34\cdots 44}a^{29}-\frac{24\cdots 77}{34\cdots 44}a^{28}+\frac{23\cdots 75}{34\cdots 44}a^{27}+\frac{11\cdots 01}{86\cdots 36}a^{26}+\frac{47\cdots 23}{17\cdots 72}a^{25}-\frac{59\cdots 31}{86\cdots 36}a^{24}-\frac{66\cdots 51}{17\cdots 72}a^{23}-\frac{45\cdots 51}{17\cdots 72}a^{22}-\frac{31\cdots 77}{34\cdots 44}a^{21}+\frac{68\cdots 43}{34\cdots 44}a^{20}-\frac{43\cdots 93}{34\cdots 44}a^{19}+\frac{47\cdots 20}{10\cdots 17}a^{18}+\frac{14\cdots 47}{23\cdots 64}a^{17}-\frac{38\cdots 19}{17\cdots 72}a^{16}+\frac{12\cdots 17}{18\cdots 16}a^{15}-\frac{10\cdots 45}{86\cdots 36}a^{14}+\frac{46\cdots 87}{34\cdots 44}a^{13}-\frac{18\cdots 83}{34\cdots 44}a^{12}+\frac{27\cdots 73}{34\cdots 44}a^{11}+\frac{11\cdots 01}{17\cdots 72}a^{10}-\frac{50\cdots 31}{49\cdots 76}a^{9}+\frac{11\cdots 29}{17\cdots 72}a^{8}+\frac{28\cdots 75}{86\cdots 36}a^{7}-\frac{15\cdots 21}{86\cdots 36}a^{6}+\frac{26\cdots 07}{11\cdots 32}a^{5}-\frac{16\cdots 17}{10\cdots 17}a^{4}-\frac{21\cdots 33}{43\cdots 68}a^{3}+\frac{17\cdots 11}{43\cdots 68}a^{2}+\frac{35\cdots 87}{74\cdots 46}a-\frac{22\cdots 41}{10\cdots 17}$, $\frac{15\cdots 05}{69\cdots 88}a^{29}-\frac{48\cdots 23}{69\cdots 88}a^{28}+\frac{22\cdots 65}{69\cdots 88}a^{27}-\frac{41\cdots 55}{86\cdots 36}a^{26}+\frac{39\cdots 99}{34\cdots 44}a^{25}-\frac{39\cdots 17}{17\cdots 72}a^{24}-\frac{43\cdots 21}{34\cdots 44}a^{23}-\frac{10\cdots 73}{15\cdots 28}a^{22}-\frac{10\cdots 59}{69\cdots 88}a^{21}+\frac{45\cdots 97}{69\cdots 88}a^{20}-\frac{60\cdots 03}{69\cdots 88}a^{19}+\frac{23\cdots 57}{86\cdots 36}a^{18}-\frac{60\cdots 65}{47\cdots 28}a^{17}-\frac{13\cdots 95}{34\cdots 44}a^{16}+\frac{10\cdots 43}{17\cdots 72}a^{15}-\frac{45\cdots 15}{43\cdots 68}a^{14}+\frac{10\cdots 17}{69\cdots 88}a^{13}-\frac{10\cdots 17}{69\cdots 88}a^{12}+\frac{56\cdots 95}{69\cdots 88}a^{11}+\frac{19\cdots 25}{34\cdots 44}a^{10}-\frac{10\cdots 45}{99\cdots 52}a^{9}+\frac{46\cdots 17}{34\cdots 44}a^{8}-\frac{16\cdots 33}{17\cdots 72}a^{7}+\frac{37\cdots 17}{17\cdots 72}a^{6}+\frac{55\cdots 57}{23\cdots 64}a^{5}-\frac{30\cdots 11}{86\cdots 36}a^{4}+\frac{17\cdots 03}{86\cdots 36}a^{3}-\frac{33\cdots 37}{86\cdots 36}a^{2}-\frac{65\cdots 59}{21\cdots 34}a+\frac{23\cdots 98}{10\cdots 17}$, $\frac{11\cdots 37}{69\cdots 88}a^{29}-\frac{16\cdots 77}{69\cdots 88}a^{28}+\frac{13\cdots 85}{69\cdots 88}a^{27}-\frac{22\cdots 37}{17\cdots 72}a^{26}+\frac{12\cdots 41}{17\cdots 72}a^{25}-\frac{70\cdots 79}{17\cdots 72}a^{24}-\frac{38\cdots 55}{34\cdots 44}a^{23}-\frac{24\cdots 21}{34\cdots 44}a^{22}-\frac{15\cdots 59}{69\cdots 88}a^{21}+\frac{89\cdots 91}{69\cdots 88}a^{20}-\frac{22\cdots 67}{69\cdots 88}a^{19}+\frac{22\cdots 25}{17\cdots 72}a^{18}+\frac{34\cdots 97}{23\cdots 64}a^{17}-\frac{41\cdots 83}{34\cdots 44}a^{16}+\frac{29\cdots 91}{17\cdots 72}a^{15}-\frac{65\cdots 45}{17\cdots 72}a^{14}+\frac{30\cdots 37}{69\cdots 88}a^{13}-\frac{15\cdots 39}{69\cdots 88}a^{12}+\frac{22\cdots 35}{69\cdots 88}a^{11}+\frac{65\cdots 51}{34\cdots 44}a^{10}-\frac{41\cdots 35}{12\cdots 44}a^{9}+\frac{84\cdots 43}{34\cdots 44}a^{8}-\frac{36\cdots 49}{17\cdots 72}a^{7}-\frac{28\cdots 27}{43\cdots 68}a^{6}+\frac{16\cdots 97}{23\cdots 64}a^{5}-\frac{50\cdots 69}{86\cdots 36}a^{4}-\frac{16\cdots 29}{21\cdots 34}a^{3}+\frac{51\cdots 61}{29\cdots 84}a^{2}+\frac{25\cdots 17}{94\cdots 58}a-\frac{18\cdots 21}{43\cdots 68}$, $\frac{68\cdots 73}{94\cdots 56}a^{29}-\frac{10\cdots 27}{94\cdots 56}a^{28}+\frac{80\cdots 05}{94\cdots 56}a^{27}-\frac{30\cdots 03}{23\cdots 64}a^{26}+\frac{15\cdots 49}{47\cdots 28}a^{25}-\frac{21\cdots 39}{10\cdots 68}a^{24}-\frac{21\cdots 61}{47\cdots 28}a^{23}-\frac{14\cdots 15}{47\cdots 28}a^{22}-\frac{91\cdots 59}{94\cdots 56}a^{21}+\frac{60\cdots 13}{94\cdots 56}a^{20}-\frac{13\cdots 19}{94\cdots 56}a^{19}+\frac{14\cdots 65}{23\cdots 64}a^{18}+\frac{27\cdots 57}{47\cdots 28}a^{17}-\frac{25\cdots 71}{47\cdots 28}a^{16}+\frac{17\cdots 95}{23\cdots 64}a^{15}-\frac{20\cdots 55}{11\cdots 32}a^{14}+\frac{19\cdots 77}{94\cdots 56}a^{13}-\frac{11\cdots 49}{94\cdots 56}a^{12}+\frac{35\cdots 95}{94\cdots 56}a^{11}+\frac{30\cdots 79}{47\cdots 28}a^{10}-\frac{66\cdots 15}{47\cdots 56}a^{9}+\frac{53\cdots 65}{47\cdots 28}a^{8}-\frac{52\cdots 63}{23\cdots 64}a^{7}-\frac{44\cdots 73}{23\cdots 64}a^{6}+\frac{61\cdots 09}{23\cdots 64}a^{5}-\frac{29\cdots 67}{11\cdots 32}a^{4}-\frac{14\cdots 71}{11\cdots 32}a^{3}+\frac{40\cdots 07}{69\cdots 96}a^{2}+\frac{30\cdots 39}{29\cdots 58}a-\frac{91\cdots 33}{29\cdots 58}$, $\frac{26\cdots 51}{34\cdots 44}a^{29}-\frac{90\cdots 65}{86\cdots 36}a^{28}+\frac{30\cdots 37}{34\cdots 44}a^{27}-\frac{74\cdots 01}{17\cdots 72}a^{26}+\frac{11\cdots 75}{34\cdots 44}a^{25}-\frac{30\cdots 59}{17\cdots 72}a^{24}-\frac{20\cdots 71}{37\cdots 73}a^{23}-\frac{70\cdots 75}{21\cdots 34}a^{22}-\frac{37\cdots 45}{34\cdots 44}a^{21}+\frac{96\cdots 55}{17\cdots 72}a^{20}-\frac{48\cdots 47}{34\cdots 44}a^{19}+\frac{26\cdots 57}{44\cdots 92}a^{18}+\frac{34\cdots 91}{47\cdots 28}a^{17}-\frac{23\cdots 37}{43\cdots 68}a^{16}+\frac{34\cdots 35}{50\cdots 08}a^{15}-\frac{30\cdots 61}{17\cdots 72}a^{14}+\frac{64\cdots 87}{34\cdots 44}a^{13}-\frac{14\cdots 03}{17\cdots 72}a^{12}+\frac{18\cdots 69}{34\cdots 44}a^{11}+\frac{72\cdots 59}{86\cdots 36}a^{10}-\frac{14\cdots 55}{99\cdots 52}a^{9}+\frac{16\cdots 39}{17\cdots 72}a^{8}+\frac{13\cdots 83}{86\cdots 36}a^{7}-\frac{52\cdots 45}{17\cdots 72}a^{6}+\frac{31\cdots 41}{11\cdots 32}a^{5}-\frac{59\cdots 25}{25\cdots 04}a^{4}-\frac{55\cdots 63}{86\cdots 36}a^{3}+\frac{33\cdots 27}{43\cdots 68}a^{2}+\frac{40\cdots 43}{21\cdots 34}a-\frac{32\cdots 59}{43\cdots 68}$, $\frac{14\cdots 83}{34\cdots 44}a^{29}-\frac{19\cdots 63}{34\cdots 44}a^{28}+\frac{16\cdots 23}{34\cdots 44}a^{27}-\frac{26\cdots 56}{10\cdots 17}a^{26}+\frac{16\cdots 93}{86\cdots 36}a^{25}-\frac{18\cdots 03}{21\cdots 34}a^{24}-\frac{20\cdots 47}{10\cdots 16}a^{23}-\frac{29\cdots 39}{17\cdots 72}a^{22}-\frac{19\cdots 33}{34\cdots 44}a^{21}+\frac{75\cdots 01}{34\cdots 44}a^{20}-\frac{35\cdots 09}{34\cdots 44}a^{19}+\frac{27\cdots 37}{86\cdots 36}a^{18}+\frac{19\cdots 55}{59\cdots 16}a^{17}-\frac{12\cdots 73}{75\cdots 64}a^{16}+\frac{48\cdots 43}{86\cdots 36}a^{15}-\frac{19\cdots 97}{21\cdots 34}a^{14}+\frac{41\cdots 03}{34\cdots 44}a^{13}-\frac{27\cdots 17}{34\cdots 44}a^{12}+\frac{49\cdots 45}{11\cdots 36}a^{11}+\frac{51\cdots 01}{17\cdots 72}a^{10}-\frac{87\cdots 41}{12\cdots 44}a^{9}+\frac{12\cdots 33}{17\cdots 72}a^{8}-\frac{22\cdots 37}{86\cdots 36}a^{7}+\frac{77\cdots 77}{21\cdots 34}a^{6}+\frac{15\cdots 11}{11\cdots 32}a^{5}-\frac{61\cdots 05}{43\cdots 68}a^{4}+\frac{27\cdots 58}{10\cdots 17}a^{3}-\frac{15\cdots 09}{43\cdots 68}a^{2}-\frac{75\cdots 47}{10\cdots 17}a+\frac{32\cdots 15}{21\cdots 34}$ Copy content Toggle raw display (assuming GRH)
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:  \( 145673548066.95038 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 2 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 145673548066.95038 \cdot 2}{2\cdot\sqrt{14026461290181639205847118647072000000000000000}}\cr\approx \mathstrut & 0.735334238100197 \end{aligned}\] (assuming GRH)

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^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16) 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^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16, 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^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16); 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 = polynomial_ring(QQ); K, a = number_field(x^30 - x^29 + 11*x^28 + 4*x^27 + 44*x^26 - 6*x^25 - 74*x^24 - 446*x^23 - 1543*x^22 + 183*x^21 - 1721*x^20 + 7116*x^19 + 11884*x^18 - 3168*x^17 + 7664*x^16 - 18756*x^15 + 17109*x^14 - 3363*x^13 - 2431*x^12 + 11594*x^11 - 15080*x^10 + 6868*x^9 + 3884*x^8 - 3768*x^7 + 2512*x^6 - 1864*x^5 - 1696*x^4 + 656*x^3 + 448*x^2 + 16*x - 16); 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

$D_{30}$ (as 30T14):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.524.1, 5.1.17161.1, 6.2.34322000.3, 10.2.920312253125.1, 15.1.677952124826430464.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 30 sibling: deg 30
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 $30$ R $30$ ${\href{/padicField/11.5.0.1}{5} }^{6}$ $30$ ${\href{/padicField/17.2.0.1}{2} }^{15}$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{15}$ ${\href{/padicField/29.2.0.1}{2} }^{14}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{14}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{15}$ $15^{2}$ $30$ ${\href{/padicField/47.2.0.1}{2} }^{15}$ ${\href{/padicField/53.6.0.1}{6} }^{5}$ $15^{2}$

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 2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(5\) Copy content Toggle raw display 5.5.2.5a1.2$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 24 x + 14$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
5.5.2.5a1.2$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 24 x + 14$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
5.5.2.5a1.2$x^{10} + 8 x^{6} + 6 x^{5} + 16 x^{2} + 24 x + 14$$2$$5$$5$$C_{10}$$$[\ ]_{2}^{5}$$
\(131\) Copy content Toggle raw display $\Q_{131}$$x + 129$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{131}$$x + 129$$1$$1$$0$Trivial$$[\ ]$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$
131.1.2.1a1.1$x^{2} + 131$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*60 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.655.2t1.a.a$1$ $ 5 \cdot 131 $ \(\Q(\sqrt{-655}) \) $C_2$ (as 2T1) $1$ $-1$
1.131.2t1.a.a$1$ $ 131 $ \(\Q(\sqrt{-131}) \) $C_2$ (as 2T1) $1$ $-1$
*60 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
*60 2.13100.6t3.b.a$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 6.0.4496182000.1 $D_{6}$ (as 6T3) $1$ $0$
*60 2.524.3t2.a.a$2$ $ 2^{2} \cdot 131 $ 3.1.524.1 $S_3$ (as 3T2) $1$ $0$
*60 2.131.5t2.a.b$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
*60 2.131.5t2.a.a$2$ $ 131 $ 5.1.17161.1 $D_{5}$ (as 5T2) $1$ $0$
*60 2.3275.10t3.a.b$2$ $ 5^{2} \cdot 131 $ 10.0.120560905159375.1 $D_{10}$ (as 10T3) $1$ $0$
*60 2.3275.10t3.a.a$2$ $ 5^{2} \cdot 131 $ 10.0.120560905159375.1 $D_{10}$ (as 10T3) $1$ $0$
*60 2.524.15t2.a.a$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
*60 2.13100.30t14.a.c$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
*60 2.524.15t2.a.c$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
*60 2.13100.30t14.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
*60 2.524.15t2.a.b$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
*60 2.13100.30t14.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$
*60 2.524.15t2.a.d$2$ $ 2^{2} \cdot 131 $ 15.1.677952124826430464.1 $D_{15}$ (as 15T2) $1$ $0$
*60 2.13100.30t14.a.d$2$ $ 2^{2} \cdot 5^{2} \cdot 131 $ 30.2.14026461290181639205847118647072000000000000000.1 $D_{30}$ (as 30T14) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers

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