Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768)
gp: K = bnfinit(y^30 - 58*y^28 + 1512*y^26 - 23400*y^24 + 239200*y^22 - 1700160*y^20 + 8614144*y^18 - 31380096*y^16 + 81861120*y^14 - 150492160*y^12 + 189190144*y^10 - 154791936*y^8 + 76038144*y^6 - 19496960*y^4 + 1966080*y^2 - 32768, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768)
\( x^{30} - 58 x^{28} + 1512 x^{26} - 23400 x^{24} + 239200 x^{22} - 1700160 x^{20} + 8614144 x^{18} + \cdots - 32768 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[30, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(20159382829191092591451779536401274948781988965475418112\)
\(\medspace = 2^{45}\cdot 31^{28}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(69.74\) | 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: | $2^{3/2}31^{14/15}\approx 69.74010138658973$ | ||
| Ramified primes: |
\(2\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(248=2^{3}\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{248}(1,·)$, $\chi_{248}(5,·)$, $\chi_{248}(129,·)$, $\chi_{248}(9,·)$, $\chi_{248}(109,·)$, $\chi_{248}(205,·)$, $\chi_{248}(45,·)$, $\chi_{248}(81,·)$, $\chi_{248}(237,·)$, $\chi_{248}(149,·)$, $\chi_{248}(25,·)$, $\chi_{248}(101,·)$, $\chi_{248}(157,·)$, $\chi_{248}(69,·)$, $\chi_{248}(133,·)$, $\chi_{248}(33,·)$, $\chi_{248}(113,·)$, $\chi_{248}(165,·)$, $\chi_{248}(97,·)$, $\chi_{248}(41,·)$, $\chi_{248}(193,·)$, $\chi_{248}(225,·)$, $\chi_{248}(173,·)$, $\chi_{248}(221,·)$, $\chi_{248}(49,·)$, $\chi_{248}(245,·)$, $\chi_{248}(233,·)$, $\chi_{248}(121,·)$, $\chi_{248}(125,·)$, $\chi_{248}(169,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$, $\frac{1}{16384}a^{28}$, $\frac{1}{16384}a^{29}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $29$ | 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: |
$\frac{1}{16}a^{8}-a^{6}+5a^{4}-8a^{2}+2$, $\frac{1}{256}a^{16}-\frac{1}{8}a^{14}+\frac{13}{8}a^{12}-11a^{10}+\frac{165}{4}a^{8}-84a^{6}+84a^{4}-32a^{2}+2$, $\frac{1}{16384}a^{28}-\frac{27}{8192}a^{26}+\frac{325}{4096}a^{24}-\frac{575}{512}a^{22}+\frac{5313}{512}a^{20}-\frac{33649}{512}a^{18}+\frac{74613}{256}a^{16}-\frac{14535}{16}a^{14}+\frac{62985}{32}a^{12}-\frac{46189}{16}a^{10}+\frac{21879}{8}a^{8}-1547a^{6}+455a^{4}-\frac{105}{2}a^{2}+1$, $\frac{1}{4}a^{4}-2a^{2}+2$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}-\frac{12597}{128}a^{16}+\frac{12597}{32}a^{14}-\frac{8619}{8}a^{12}+\frac{31603}{16}a^{10}-\frac{9295}{4}a^{8}+\frac{13013}{8}a^{6}-\frac{1183}{2}a^{4}+\frac{169}{2}a^{2}-2$, $\frac{1}{1024}a^{20}-\frac{5}{128}a^{18}+\frac{85}{128}a^{16}-\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}-\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}-330a^{6}+\frac{825}{4}a^{4}-50a^{2}+3$, $\frac{1}{32}a^{10}-\frac{5}{8}a^{8}+\frac{35}{8}a^{6}-\frac{25}{2}a^{4}+\frac{25}{2}a^{2}-2$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{3741}{128}a^{14}-\frac{4011}{32}a^{12}+\frac{693}{2}a^{10}-603a^{8}+\frac{5013}{8}a^{6}-\frac{1405}{4}a^{4}+83a^{2}-2$, $\frac{1}{512}a^{18}-\frac{9}{128}a^{16}+\frac{135}{128}a^{14}-\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}-\frac{891}{8}a^{8}+\frac{693}{4}a^{6}-135a^{4}+\frac{81}{2}a^{2}-2$, $\frac{1}{16384}a^{28}-\frac{7}{2048}a^{26}+\frac{351}{4096}a^{24}-\frac{325}{256}a^{22}+\frac{12649}{1024}a^{20}-\frac{10621}{128}a^{18}+\frac{12597}{32}a^{16}-\frac{169727}{128}a^{14}+\frac{25137}{8}a^{12}-\frac{81757}{16}a^{10}+\frac{87153}{16}a^{8}-\frac{28063}{8}a^{6}+\frac{4655}{4}a^{4}-133a^{2}+1$, $\frac{1}{256}a^{16}-\frac{1}{8}a^{14}+\frac{13}{8}a^{12}-11a^{10}+\frac{165}{4}a^{8}-84a^{6}+84a^{4}-32a^{2}+3$, $\frac{1}{16384}a^{28}-\frac{27}{8192}a^{26}+\frac{325}{4096}a^{24}-\frac{575}{512}a^{22}+\frac{5313}{512}a^{20}-\frac{33649}{512}a^{18}+\frac{74613}{256}a^{16}-\frac{14535}{16}a^{14}+\frac{62985}{32}a^{12}-\frac{46189}{16}a^{10}+\frac{21879}{8}a^{8}-1547a^{6}+455a^{4}-\frac{105}{2}a^{2}$, $\frac{1}{16384}a^{28}-\frac{27}{8192}a^{26}+\frac{325}{4096}a^{24}-\frac{575}{512}a^{22}+\frac{5313}{512}a^{20}-\frac{33649}{512}a^{18}+\frac{18653}{64}a^{16}-\frac{116265}{128}a^{14}+\frac{15735}{8}a^{12}-\frac{92103}{32}a^{10}+\frac{10827}{4}a^{8}-\frac{5999}{4}a^{6}+420a^{4}-45a^{2}$, $\frac{1}{512}a^{18}-\frac{17}{256}a^{16}+\frac{15}{16}a^{14}-\frac{57}{8}a^{12}+\frac{253}{8}a^{10}-\frac{333}{4}a^{8}+126a^{6}-100a^{4}+33a^{2}-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{1}{1024}a^{20}-\frac{5313}{64}a^{19}+\frac{5}{128}a^{18}+\frac{100947}{256}a^{17}-\frac{85}{128}a^{16}-\frac{10659}{8}a^{15}+\frac{25}{4}a^{14}+\frac{101745}{32}a^{13}-\frac{2275}{64}a^{12}-\frac{20995}{4}a^{11}+\frac{1001}{8}a^{10}+\frac{46189}{8}a^{9}-\frac{2145}{8}a^{8}-3978a^{7}+330a^{6}+1547a^{5}-\frac{825}{4}a^{4}-280a^{3}+50a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{20995}{4}a^{11}+\frac{46189}{8}a^{9}-3978a^{7}+1547a^{5}-280a^{3}+\frac{1}{2}a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{20995}{4}a^{11}+\frac{46189}{8}a^{9}-3978a^{7}+\frac{1}{8}a^{6}+1547a^{5}-\frac{3}{2}a^{4}-280a^{3}+\frac{9}{2}a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{1}{64}a^{12}-\frac{20995}{4}a^{11}+\frac{3}{8}a^{10}+\frac{46189}{8}a^{9}-\frac{27}{8}a^{8}-3978a^{7}+14a^{6}+1547a^{5}-\frac{105}{4}a^{4}-280a^{3}+18a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{1}{8192}a^{26}+\frac{351}{4096}a^{25}-\frac{13}{2048}a^{24}-\frac{325}{256}a^{23}+\frac{299}{2048}a^{22}+\frac{6325}{512}a^{21}-\frac{1001}{512}a^{20}-\frac{5313}{64}a^{19}+\frac{8645}{512}a^{18}+\frac{100947}{256}a^{17}-\frac{12597}{128}a^{16}-\frac{10659}{8}a^{15}+\frac{12597}{32}a^{14}+\frac{101745}{32}a^{13}-\frac{8619}{8}a^{12}-\frac{20995}{4}a^{11}+\frac{31603}{16}a^{10}+\frac{46189}{8}a^{9}-\frac{9295}{4}a^{8}-3978a^{7}+\frac{13013}{8}a^{6}+1547a^{5}-\frac{1183}{2}a^{4}-280a^{3}+\frac{169}{2}a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{1}{16384}a^{28}-\frac{7}{2048}a^{27}+\frac{27}{8192}a^{26}+\frac{351}{4096}a^{25}-\frac{325}{4096}a^{24}-\frac{325}{256}a^{23}+\frac{575}{512}a^{22}+\frac{6325}{512}a^{21}-\frac{5313}{512}a^{20}-\frac{5313}{64}a^{19}+\frac{33649}{512}a^{18}+\frac{100947}{256}a^{17}-\frac{74613}{256}a^{16}-\frac{10659}{8}a^{15}+\frac{14535}{16}a^{14}+\frac{101745}{32}a^{13}-\frac{62985}{32}a^{12}-\frac{20995}{4}a^{11}+\frac{46189}{16}a^{10}+\frac{46189}{8}a^{9}-\frac{21879}{8}a^{8}-3978a^{7}+1547a^{6}+1547a^{5}-455a^{4}-280a^{3}+\frac{105}{2}a^{2}+15a-1$, $\frac{1}{1024}a^{21}-\frac{1}{1024}a^{20}-\frac{21}{512}a^{19}+\frac{5}{128}a^{18}+\frac{189}{256}a^{17}-\frac{85}{128}a^{16}-\frac{119}{16}a^{15}+\frac{25}{4}a^{14}+\frac{735}{16}a^{13}-\frac{2275}{64}a^{12}-\frac{5733}{32}a^{11}+\frac{1001}{8}a^{10}+\frac{7007}{16}a^{9}-\frac{2145}{8}a^{8}-\frac{1287}{2}a^{7}+330a^{6}+\frac{2079}{4}a^{5}-\frac{825}{4}a^{4}-\frac{385}{2}a^{3}+50a^{2}+21a-3$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{1}{8}a^{7}+\frac{147}{4}a^{6}-\frac{7}{4}a^{5}-49a^{4}+7a^{3}+\frac{49}{2}a^{2}-7a-1$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{20995}{4}a^{11}+\frac{46189}{8}a^{9}-3978a^{7}+1547a^{5}-\frac{1}{4}a^{4}-280a^{3}+2a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}+\frac{1}{256}a^{16}-\frac{10659}{8}a^{15}-\frac{1}{8}a^{14}+\frac{101745}{32}a^{13}+\frac{13}{8}a^{12}-\frac{20995}{4}a^{11}-11a^{10}+\frac{46189}{8}a^{9}+\frac{165}{4}a^{8}-3978a^{7}-84a^{6}+1547a^{5}+84a^{4}-280a^{3}-32a^{2}+15a+2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{1}{128}a^{14}+\frac{101745}{32}a^{13}-\frac{7}{32}a^{12}-\frac{20995}{4}a^{11}+\frac{77}{32}a^{10}+\frac{46189}{8}a^{9}-\frac{105}{8}a^{8}-3978a^{7}+\frac{147}{4}a^{6}+1547a^{5}-49a^{4}-280a^{3}+\frac{49}{2}a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}-\frac{1}{512}a^{18}+\frac{100947}{256}a^{17}+\frac{9}{128}a^{16}-\frac{10659}{8}a^{15}-\frac{135}{128}a^{14}+\frac{101745}{32}a^{13}+\frac{273}{32}a^{12}-\frac{20995}{4}a^{11}-\frac{1287}{32}a^{10}+\frac{46189}{8}a^{9}+\frac{891}{8}a^{8}-3978a^{7}-\frac{693}{4}a^{6}+1547a^{5}+135a^{4}-280a^{3}-\frac{81}{2}a^{2}+15a+2$, $\frac{1}{16384}a^{29}+\frac{1}{16384}a^{28}-\frac{7}{2048}a^{27}-\frac{7}{2048}a^{26}+\frac{351}{4096}a^{25}+\frac{175}{2048}a^{24}-\frac{325}{256}a^{23}-\frac{161}{128}a^{22}+\frac{6325}{512}a^{21}+\frac{12397}{1024}a^{20}-\frac{5313}{64}a^{19}-\frac{10241}{128}a^{18}+\frac{100947}{256}a^{17}+\frac{47481}{128}a^{16}-\frac{10659}{8}a^{15}-\frac{4845}{4}a^{14}+\frac{101745}{32}a^{13}+\frac{88179}{32}a^{12}-\frac{20995}{4}a^{11}-\frac{17017}{4}a^{10}+\frac{46189}{8}a^{9}+\frac{17017}{4}a^{8}-3978a^{7}-2548a^{6}+1547a^{5}+\frac{3185}{4}a^{4}-280a^{3}-98a^{2}+15a+2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{20995}{4}a^{11}+\frac{46189}{8}a^{9}-\frac{1}{16}a^{8}-3978a^{7}+a^{6}+1547a^{5}-5a^{4}-280a^{3}+8a^{2}+15a-2$, $\frac{1}{16384}a^{29}-\frac{7}{2048}a^{27}+\frac{351}{4096}a^{25}-\frac{325}{256}a^{23}+\frac{6325}{512}a^{21}-\frac{5313}{64}a^{19}+\frac{100947}{256}a^{17}-\frac{10659}{8}a^{15}+\frac{101745}{32}a^{13}-\frac{20995}{4}a^{11}+\frac{46189}{8}a^{9}-3978a^{7}+1547a^{5}-280a^{3}+15a-1$
| 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: | \( 1001916921345060900 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{30}\cdot(2\pi)^{0}\cdot 1001916921345060900 \cdot 1}{2\cdot\sqrt{20159382829191092591451779536401274948781988965475418112}}\cr\approx \mathstrut & 0.119801696911587 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768)
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^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768, 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^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768);
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^30 - 58*x^28 + 1512*x^26 - 23400*x^24 + 239200*x^22 - 1700160*x^20 + 8614144*x^18 - 31380096*x^16 + 81861120*x^14 - 150492160*x^12 + 189190144*x^10 - 154791936*x^8 + 76038144*x^6 - 19496960*x^4 + 1966080*x^2 - 32768);
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 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 3.3.961.1, 5.5.923521.1, 6.6.472842752.1, 10.10.27947533514866688.1, \(\Q(\zeta_{31})^+\) |
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 | $30$ | ${\href{/padicField/5.6.0.1}{6} }^{5}$ | $15^{2}$ | $30$ | $30$ | $15^{2}$ | $30$ | ${\href{/padicField/23.5.0.1}{5} }^{6}$ | ${\href{/padicField/29.10.0.1}{10} }^{3}$ | R | ${\href{/padicField/37.6.0.1}{6} }^{5}$ | $15^{2}$ | $30$ | ${\href{/padicField/47.5.0.1}{5} }^{6}$ | $30$ | $30$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
| 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ | |
|
\(31\)
| 31.15.14.1 | $x^{15} + 31$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |
| 31.15.14.1 | $x^{15} + 31$ | $15$ | $1$ | $14$ | $C_{15}$ | $[\ ]_{15}$ |