Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1)
gp: K = bnfinit(y^36 - y^35 - 36*y^34 + 36*y^33 + 593*y^32 - 593*y^31 - 5919*y^30 + 5919*y^29 + 39961*y^28 - 39961*y^27 - 192880*y^26 + 192880*y^25 + 685907*y^24 - 685907*y^23 - 1824913*y^22 + 1824913*y^21 + 3651272*y^20 - 3651272*y^19 - 5475703*y^18 + 5475703*y^17 + 6085132*y^16 - 6085132*y^15 - 4909788*y^14 + 4909788*y^13 + 2786656*y^12 - 2786656*y^11 - 1061566*y^10 + 1061566*y^9 + 253044*y^8 - 253044*y^7 - 33780*y^6 + 33780*y^5 + 2073*y^4 - 2073*y^3 - 36*y^2 + 36*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1)
\( x^{36} - x^{35} - 36 x^{34} + 36 x^{33} + 593 x^{32} - 593 x^{31} - 5919 x^{30} + 5919 x^{29} + 39961 x^{28} + \cdots + 1 \)
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: | $[36, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2987052991985699215206951151365900403178222386352058024870316677\)
\(\medspace = 3^{18}\cdot 37^{35}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.97\) | 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: | $3^{1/2}37^{35/36}\approx 57.969718630268325$ | ||
| Ramified primes: |
\(3\), \(37\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{37}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(111=3\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(2,·)$, $\chi_{111}(4,·)$, $\chi_{111}(5,·)$, $\chi_{111}(7,·)$, $\chi_{111}(8,·)$, $\chi_{111}(10,·)$, $\chi_{111}(14,·)$, $\chi_{111}(16,·)$, $\chi_{111}(17,·)$, $\chi_{111}(20,·)$, $\chi_{111}(23,·)$, $\chi_{111}(25,·)$, $\chi_{111}(28,·)$, $\chi_{111}(29,·)$, $\chi_{111}(32,·)$, $\chi_{111}(34,·)$, $\chi_{111}(35,·)$, $\chi_{111}(40,·)$, $\chi_{111}(46,·)$, $\chi_{111}(49,·)$, $\chi_{111}(50,·)$, $\chi_{111}(56,·)$, $\chi_{111}(58,·)$, $\chi_{111}(59,·)$, $\chi_{111}(64,·)$, $\chi_{111}(67,·)$, $\chi_{111}(68,·)$, $\chi_{111}(70,·)$, $\chi_{111}(73,·)$, $\chi_{111}(80,·)$, $\chi_{111}(85,·)$, $\chi_{111}(89,·)$, $\chi_{111}(92,·)$, $\chi_{111}(98,·)$, $\chi_{111}(100,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $35$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(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: |
$a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+a^{11}+63206a^{10}-11a^{9}-37180a^{8}+44a^{7}+13013a^{6}-77a^{5}-2366a^{4}+55a^{3}+169a^{2}-11a-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a+1$, $a^{35}-36a^{33}+593a^{31}-5920a^{29}+39991a^{27}-193284a^{25}+689130a^{23}-1841840a^{21}+3712775a^{19}-5633804a^{17}+6374082a^{15}-5281696a^{13}+3115658a^{11}-1253240a^{9}+321708a^{7}-a^{6}-47328a^{5}+6a^{4}+3281a^{3}-8a^{2}-68a-1$, $a^{35}-35a^{33}+560a^{31}-a^{30}-5425a^{29}+30a^{28}+35524a^{27}-404a^{26}-166230a^{25}+3224a^{24}+572977a^{23}-16951a^{22}-1477796a^{21}+61754a^{20}+2867710a^{19}-159601a^{18}-4175117a^{17}+294594a^{16}+4511187a^{15}-385796a^{14}-3546491a^{13}+351480a^{12}+1966759a^{11}-215138a^{10}-734601a^{9}+83732a^{8}+172373a^{7}-19020a^{6}-22749a^{5}+2184a^{4}+1386a^{3}-96a^{2}-24a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a+1$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}+a^{23}-95680a^{22}-23a^{21}+283360a^{20}+230a^{19}-615296a^{18}-1311a^{17}+980628a^{16}+4692a^{15}-1136959a^{14}-10948a^{13}+940562a^{12}+16744a^{11}-537395a^{10}-16446a^{9}+201342a^{8}+9876a^{7}-45402a^{6}-3315a^{5}+5244a^{4}+531a^{3}-207a^{2}-27a-1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}+a^{14}-10948a^{13}-14a^{12}+16744a^{11}+77a^{10}-16445a^{9}-210a^{8}+9867a^{7}+294a^{6}-3289a^{5}-196a^{4}+506a^{3}+49a^{2}-23a-3$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}+a^{21}-63756a^{20}-21a^{19}+168245a^{18}+189a^{17}-319770a^{16}-952a^{15}+436050a^{14}+2940a^{13}-419900a^{12}-5733a^{11}+277134a^{10}+7007a^{9}-119340a^{8}-5148a^{7}+30940a^{6}+2079a^{5}-4200a^{4}-385a^{3}+225a^{2}+21a-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24753a^{12}-27468a^{10}+19359a^{8}-8120a^{6}+1821a^{4}-180a^{2}+5$, $a^{12}-12a^{10}+54a^{8}-111a^{6}+99a^{4}-27a^{2}+1$, $a^{33}-a^{32}-33a^{31}+32a^{30}+495a^{29}-464a^{28}-4465a^{27}+4032a^{26}+27000a^{25}-23400a^{24}-115507a^{23}+95680a^{22}+359536a^{21}-283360a^{20}-824735a^{19}+615297a^{18}+1396671a^{17}-980646a^{16}-1732742a^{15}+1137094a^{14}+1547782a^{13}-941108a^{12}-967486a^{11}+538682a^{10}+405406a^{9}-203124a^{8}-106896a^{7}+46788a^{6}+16140a^{5}-5784a^{4}-1208a^{3}+288a^{2}+33a-1$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+a^{5}+5440a^{4}-5a^{3}-256a^{2}+5a+2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}+a^{10}-72930a^{9}-10a^{8}+30888a^{7}+35a^{6}-7371a^{5}-50a^{4}+819a^{3}+25a^{2}-27a-2$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}-2778446a^{16}+2778446a^{14}-1998724a^{12}+999362a^{10}-329460a^{8}+65892a^{6}-6936a^{4}+289a^{2}-2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}+a^{6}-14756a^{5}-6a^{4}+1240a^{3}+9a^{2}-31a-2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+a^{15}+3740a^{14}-15a^{13}-8008a^{12}+90a^{11}+11011a^{10}-275a^{9}-9438a^{8}+450a^{7}+4719a^{6}-378a^{5}-1210a^{4}+140a^{3}+121a^{2}-15a-2$, $a^{20}-20a^{18}+a^{17}+170a^{16}-17a^{15}-800a^{14}+119a^{13}+2275a^{12}-442a^{11}-4004a^{10}+935a^{9}+4290a^{8}-1121a^{7}-2640a^{6}+707a^{5}+825a^{4}-190a^{3}-100a^{2}+10a+2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+a^{7}+30940a^{6}-7a^{5}-4200a^{4}+14a^{3}+225a^{2}-7a-2$, $a^{21}-21a^{19}+189a^{17}+a^{16}-952a^{15}-16a^{14}+2940a^{13}+104a^{12}-5733a^{11}-352a^{10}+7007a^{9}+660a^{8}-5148a^{7}-672a^{6}+2079a^{5}+336a^{4}-385a^{3}-64a^{2}+21a+2$, $a^{29}-29a^{27}+377a^{25}+a^{24}-2900a^{23}-24a^{22}+14674a^{21}+252a^{20}-51359a^{19}-1520a^{18}+127281a^{17}+5814a^{16}-224808a^{15}-14688a^{14}+281010a^{13}+24752a^{12}-243542a^{11}-27456a^{10}+140998a^{9}+19305a^{8}-51272a^{7}-8008a^{6}+10556a^{5}+1716a^{4}-1015a^{3}-144a^{2}+29a+2$, $a^{35}-36a^{33}+a^{32}+593a^{31}-33a^{30}-5919a^{29}+494a^{28}+39961a^{27}-4436a^{26}-192880a^{25}+26623a^{24}+685907a^{23}-112607a^{22}-1824913a^{21}+344863a^{20}+3651272a^{19}-773397a^{18}-5475703a^{17}+1269578a^{16}+6085132a^{15}-1508868a^{14}-4909788a^{13}+1269578a^{12}+2786656a^{11}-729146a^{10}-1061565a^{9}+270216a^{8}+253035a^{7}-59244a^{6}-33753a^{5}+6648a^{4}+2043a^{3}-288a^{2}-26a+1$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+a^{7}+30940a^{6}-7a^{5}-4200a^{4}+15a^{3}+225a^{2}-10a-2$, $a^{4}-3a^{2}+1$, $a^{24}-a^{23}-24a^{22}+23a^{21}+252a^{20}-230a^{19}-1520a^{18}+1311a^{17}+5814a^{16}-4692a^{15}-14688a^{14}+10948a^{13}+24752a^{12}-16744a^{11}-27456a^{10}+16445a^{9}+19305a^{8}-9867a^{7}-8008a^{6}+3289a^{5}+1716a^{4}-506a^{3}-144a^{2}+23a+2$, $a^{34}-34a^{32}+527a^{30}-a^{29}-4930a^{28}+29a^{27}+31059a^{26}-376a^{25}-139230a^{24}+2875a^{23}+457470a^{22}-14399a^{21}-1118261a^{20}+49609a^{19}+2042995a^{18}-120157a^{17}-2778616a^{16}+205446a^{15}+2779246a^{14}-245444a^{13}-2000998a^{12}+199873a^{11}+1003354a^{10}-106447a^{9}-333697a^{8}+34925a^{7}+68429a^{6}-6566a^{5}-7682a^{4}+655a^{3}+377a^{2}-28a-4$
| 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: | \( 181157165602858830000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{36}\cdot(2\pi)^{0}\cdot 181157165602858830000 \cdot 1}{2\cdot\sqrt{2987052991985699215206951151365900403178222386352058024870316677}}\cr\approx \mathstrut & 0.113889556472583 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1)
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 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1, 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 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1);
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 - x^35 - 36*x^34 + 36*x^33 + 593*x^32 - 593*x^31 - 5919*x^30 + 5919*x^29 + 39961*x^28 - 39961*x^27 - 192880*x^26 + 192880*x^25 + 685907*x^24 - 685907*x^23 - 1824913*x^22 + 1824913*x^21 + 3651272*x^20 - 3651272*x^19 - 5475703*x^18 + 5475703*x^17 + 6085132*x^16 - 6085132*x^15 - 4909788*x^14 + 4909788*x^13 + 2786656*x^12 - 2786656*x^11 - 1061566*x^10 + 1061566*x^9 + 253044*x^8 - 253044*x^7 - 33780*x^6 + 33780*x^5 + 2073*x^4 - 2073*x^3 - 36*x^2 + 36*x + 1);
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)
| A cyclic group of order 36 |
| The 36 conjugacy class representatives for $C_{36}$ |
| Character table for $C_{36}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{37}) \), 3.3.1369.1, 4.4.455877.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.12.129701946277226641077.1, \(\Q(\zeta_{37})^+\) |
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 | $36$ | R | $36$ | ${\href{/padicField/7.9.0.1}{9} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{12}$ | $36$ | $36$ | $36$ | ${\href{/padicField/23.12.0.1}{12} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{9}$ | R | ${\href{/padicField/41.9.0.1}{9} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{9}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | $18^{2}$ | $36$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $2$ | $9$ | $9$ | |||
| Deg $18$ | $2$ | $9$ | $9$ | ||||
|
\(37\)
| Deg $36$ | $36$ | $1$ | $35$ |