Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
gp: K = bnfinit(y^30 + 3*y^28 - y^27 + 9*y^26 - 6*y^25 + 28*y^24 - 27*y^23 + 90*y^22 - 109*y^21 + 297*y^20 + 507*y^19 + 1000*y^18 + 1224*y^17 + 2493*y^16 + 2672*y^15 + 6255*y^14 + 5523*y^13 + 16093*y^12 + 10314*y^11 + 42756*y^10 + 14849*y^9 + 5157*y^8 + 1791*y^7 + 622*y^6 + 216*y^5 + 75*y^4 + 26*y^3 + 9*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
\( x^{30} + 3 x^{28} - x^{27} + 9 x^{26} - 6 x^{25} + 28 x^{24} - 27 x^{23} + 90 x^{22} - 109 x^{21} + \cdots + 1 \)
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: | $[0, 15]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-159386923550435671074967363509984324121230045171\)
\(\medspace = -\,3^{40}\cdot 11^{27}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(37.45\) | 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: | $3^{4/3}11^{9/10}\approx 37.44683262965639$ | ||
| Ramified primes: |
\(3\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $30$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(99=3^{2}\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(70,·)$, $\chi_{99}(7,·)$, $\chi_{99}(73,·)$, $\chi_{99}(10,·)$, $\chi_{99}(76,·)$, $\chi_{99}(13,·)$, $\chi_{99}(79,·)$, $\chi_{99}(16,·)$, $\chi_{99}(82,·)$, $\chi_{99}(19,·)$, $\chi_{99}(85,·)$, $\chi_{99}(25,·)$, $\chi_{99}(91,·)$, $\chi_{99}(28,·)$, $\chi_{99}(94,·)$, $\chi_{99}(31,·)$, $\chi_{99}(97,·)$, $\chi_{99}(34,·)$, $\chi_{99}(37,·)$, $\chi_{99}(40,·)$, $\chi_{99}(43,·)$, $\chi_{99}(46,·)$, $\chi_{99}(49,·)$, $\chi_{99}(52,·)$, $\chi_{99}(58,·)$, $\chi_{99}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{16384}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{14317073}a^{21}-\frac{4597228}{14317073}a^{20}+\frac{48209}{14317073}a^{19}+\frac{525388}{14317073}a^{18}+\frac{4741855}{14317073}a^{17}+\frac{1527955}{14317073}a^{16}-\frac{616896}{14317073}a^{15}-\frac{157990}{14317073}a^{14}-\frac{3378643}{14317073}a^{13}+\frac{142926}{14317073}a^{12}+\frac{4339134}{14317073}a^{11}-\frac{5254364}{14317073}a^{10}+\frac{1444706}{14317073}a^{9}+\frac{4338513}{14317073}a^{8}-\frac{4728591}{14317073}a^{7}-\frac{2746240}{14317073}a^{6}-\frac{4207213}{14317073}a^{5}-\frac{3510129}{14317073}a^{4}+\frac{4441674}{14317073}a^{3}-\frac{6323174}{14317073}a^{2}+\frac{2518078}{14317073}a+\frac{5222950}{14317073}$, $\frac{1}{14317073}a^{22}+\frac{5255288}{14317073}a^{11}+\frac{439665}{14317073}$, $\frac{1}{14317073}a^{23}+\frac{5255288}{14317073}a^{12}+\frac{439665}{14317073}a$, $\frac{1}{14317073}a^{24}+\frac{5255288}{14317073}a^{13}+\frac{439665}{14317073}a^{2}$, $\frac{1}{14317073}a^{25}+\frac{5255288}{14317073}a^{14}+\frac{439665}{14317073}a^{3}$, $\frac{1}{14317073}a^{26}+\frac{5255288}{14317073}a^{15}+\frac{439665}{14317073}a^{4}$, $\frac{1}{14317073}a^{27}+\frac{5255288}{14317073}a^{16}+\frac{439665}{14317073}a^{5}$, $\frac{1}{14317073}a^{28}+\frac{5255288}{14317073}a^{17}+\frac{439665}{14317073}a^{6}$, $\frac{1}{14317073}a^{29}+\frac{5255288}{14317073}a^{18}+\frac{439665}{14317073}a^{7}$
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
$C_{93}$, which has order $93$ (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: | $14$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1799175}{14317073} a^{28} + \frac{599725}{14317073} a^{27} - \frac{5397525}{14317073} a^{26} + \frac{3598350}{14317073} a^{25} - \frac{16792300}{14317073} a^{24} + \frac{16192575}{14317073} a^{23} - \frac{53975250}{14317073} a^{22} + \frac{65370025}{14317073} a^{21} - \frac{178118325}{14317073} a^{20} + \frac{250084998}{14317073} a^{19} - \frac{599725000}{14317073} a^{18} - \frac{734063400}{14317073} a^{17} - \frac{1495114425}{14317073} a^{16} - \frac{1602465200}{14317073} a^{15} - \frac{3751279875}{14317073} a^{14} - \frac{3312281175}{14317073} a^{13} - \frac{9651374425}{14317073} a^{12} - \frac{6185563650}{14317073} a^{11} - \frac{25641842100}{14317073} a^{10} - \frac{8905316525}{14317073} a^{9} - \frac{70740393066}{14317073} a^{8} - \frac{1074107475}{14317073} a^{7} - \frac{373028950}{14317073} a^{6} - \frac{129540600}{14317073} a^{5} - \frac{44979375}{14317073} a^{4} - \frac{15592850}{14317073} a^{3} - \frac{5397525}{14317073} a^{2} - \frac{1799175}{14317073} a - \frac{599725}{14317073} \)
(order $22$)
| 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: |
$\frac{25109}{14317073}a^{27}+\frac{297}{14317073}a^{23}+\frac{23198697}{14317073}a^{16}+\frac{259579}{14317073}a^{12}-\frac{2833695252}{14317073}a^{5}-\frac{41224371}{14317073}a$, $\frac{3000}{14317073}a^{25}+\frac{2766627}{14317073}a^{14}-\frac{341785468}{14317073}a^{3}-1$, $\frac{25109}{14317073}a^{27}+\frac{3000}{14317073}a^{25}+\frac{23198697}{14317073}a^{16}+\frac{2766627}{14317073}a^{14}-\frac{2833695252}{14317073}a^{5}-\frac{341785468}{14317073}a^{3}$, $\frac{1000}{14317073}a^{24}+\frac{297}{14317073}a^{23}+\frac{922209}{14317073}a^{13}+\frac{259579}{14317073}a^{12}-\frac{118700847}{14317073}a^{2}-\frac{41224371}{14317073}a-1$, $\frac{1799175}{14317073}a^{29}-\frac{599725}{14317073}a^{28}+\frac{5397525}{14317073}a^{27}-\frac{3598350}{14317073}a^{26}+\frac{16792300}{14317073}a^{25}-\frac{16192575}{14317073}a^{24}+\frac{53975250}{14317073}a^{23}-\frac{65370025}{14317073}a^{22}+\frac{178118325}{14317073}a^{21}-\frac{250084998}{14317073}a^{20}+\frac{599725000}{14317073}a^{19}+\frac{734063400}{14317073}a^{18}+\frac{1495114425}{14317073}a^{17}+\frac{1602465200}{14317073}a^{16}+\frac{3751279875}{14317073}a^{15}+\frac{3312281175}{14317073}a^{14}+\frac{9651374425}{14317073}a^{13}+\frac{6185563650}{14317073}a^{12}+\frac{25641842100}{14317073}a^{11}+\frac{8905316525}{14317073}a^{10}+\frac{70740393066}{14317073}a^{9}+\frac{1074107475}{14317073}a^{8}+\frac{373028950}{14317073}a^{7}+\frac{129540600}{14317073}a^{6}+\frac{44979375}{14317073}a^{5}+\frac{15592850}{14317073}a^{4}+\frac{5397525}{14317073}a^{3}+\frac{1799175}{14317073}a^{2}+\frac{599725}{14317073}a$, $\frac{703}{14317073}a^{23}+\frac{662630}{14317073}a^{12}-\frac{63159403}{14317073}a$, $\frac{8703}{14317073}a^{26}-\frac{2000}{14317073}a^{24}+\frac{8040302}{14317073}a^{15}-\frac{1844418}{14317073}a^{13}-\frac{984132033}{14317073}a^{4}+\frac{223084621}{14317073}a^{2}-1$, $\frac{1799175}{14317073}a^{28}-\frac{638240}{14317073}a^{27}+\frac{5397525}{14317073}a^{26}-\frac{3598350}{14317073}a^{25}+\frac{16791300}{14317073}a^{24}-\frac{16192575}{14317073}a^{23}+\frac{53975250}{14317073}a^{22}-\frac{65370025}{14317073}a^{21}+\frac{178118325}{14317073}a^{20}-\frac{250084998}{14317073}a^{19}+\frac{599725000}{14317073}a^{18}+\frac{734063400}{14317073}a^{17}+\frac{1459523960}{14317073}a^{16}+\frac{1602465200}{14317073}a^{15}+\frac{3751279875}{14317073}a^{14}+\frac{3311358966}{14317073}a^{13}+\frac{9651374425}{14317073}a^{12}+\frac{6185563650}{14317073}a^{11}+\frac{25641842100}{14317073}a^{10}+\frac{8905316525}{14317073}a^{9}+\frac{70740393066}{14317073}a^{8}+\frac{1074107475}{14317073}a^{7}+\frac{373028950}{14317073}a^{6}+\frac{4471013603}{14317073}a^{5}+\frac{44979375}{14317073}a^{4}+\frac{15592850}{14317073}a^{3}+\frac{124098372}{14317073}a^{2}+\frac{1799175}{14317073}a+\frac{599725}{14317073}$, $\frac{1799175}{14317073}a^{28}-\frac{552507}{14317073}a^{27}+\frac{5397525}{14317073}a^{26}-\frac{3598350}{14317073}a^{25}+\frac{16791300}{14317073}a^{24}-\frac{16192575}{14317073}a^{23}+\frac{53975250}{14317073}a^{22}-\frac{65370025}{14317073}a^{21}+\frac{178118325}{14317073}a^{20}-\frac{250084998}{14317073}a^{19}+\frac{599725000}{14317073}a^{18}+\frac{734063400}{14317073}a^{17}+\frac{1538745192}{14317073}a^{16}+\frac{1602465200}{14317073}a^{15}+\frac{3751279875}{14317073}a^{14}+\frac{3311358966}{14317073}a^{13}+\frac{9651374425}{14317073}a^{12}+\frac{6185563650}{14317073}a^{11}+\frac{25641842100}{14317073}a^{10}+\frac{8905316525}{14317073}a^{9}+\frac{70740393066}{14317073}a^{8}+\frac{1074107475}{14317073}a^{7}+\frac{373028950}{14317073}a^{6}-\frac{5196064436}{14317073}a^{5}+\frac{44979375}{14317073}a^{4}+\frac{15592850}{14317073}a^{3}+\frac{124098372}{14317073}a^{2}+\frac{1799175}{14317073}a+\frac{599725}{14317073}$, $\frac{4972266}{14317073}a^{29}+\frac{14916798}{14317073}a^{27}-\frac{4972266}{14317073}a^{26}+\frac{44753394}{14317073}a^{25}-\frac{29833596}{14317073}a^{24}+\frac{139224151}{14317073}a^{23}-\frac{134251182}{14317073}a^{22}+\frac{447503831}{14317073}a^{21}-\frac{541976994}{14317073}a^{20}+\frac{1476763002}{14317073}a^{19}+\frac{2520938862}{14317073}a^{18}+\frac{4972266000}{14317073}a^{17}+\frac{6086053584}{14317073}a^{16}+\frac{12395859138}{14317073}a^{15}+\frac{13288661379}{14317073}a^{14}+\frac{31101523830}{14317073}a^{13}+\frac{27462487748}{14317073}a^{12}+\frac{80018676738}{14317073}a^{11}+\frac{51283808052}{14317073}a^{10}+\frac{212594205096}{14317073}a^{9}+\frac{73833177834}{14317073}a^{8}+\frac{25641975762}{14317073}a^{7}+\frac{8905328406}{14317073}a^{6}+\frac{3092749452}{14317073}a^{5}+\frac{1074009456}{14317073}a^{4}+\frac{31134482}{14317073}a^{3}+\frac{129278916}{14317073}a^{2}-\frac{18409009}{14317073}a+\frac{14916798}{14317073}$, $\frac{135951}{14317073}a^{28}-\frac{297}{14317073}a^{23}+\frac{125618626}{14317073}a^{17}-\frac{259579}{14317073}a^{12}-\frac{15334468543}{14317073}a^{6}+\frac{41224371}{14317073}a+1$, $\frac{319120}{14317073}a^{29}-\frac{1726848}{14317073}a^{28}+\frac{599725}{14317073}a^{27}-\frac{5397525}{14317073}a^{26}+\frac{3598350}{14317073}a^{25}-\frac{16792300}{14317073}a^{24}+\frac{16192575}{14317073}a^{23}-\frac{53975250}{14317073}a^{22}+\frac{65370025}{14317073}a^{21}-\frac{178118325}{14317073}a^{20}+\frac{250084998}{14317073}a^{19}-\frac{304856981}{14317073}a^{18}-\frac{667233936}{14317073}a^{17}-\frac{1495114425}{14317073}a^{16}-\frac{1602465200}{14317073}a^{15}-\frac{3751279875}{14317073}a^{14}-\frac{3312281175}{14317073}a^{13}-\frac{9651374425}{14317073}a^{12}-\frac{6185563650}{14317073}a^{11}-\frac{25641842100}{14317073}a^{10}-\frac{8905316525}{14317073}a^{9}-\frac{70740393066}{14317073}a^{8}-\frac{37068649597}{14317073}a^{7}-\frac{8532329238}{14317073}a^{6}-\frac{129540600}{14317073}a^{5}-\frac{44979375}{14317073}a^{4}-\frac{15592850}{14317073}a^{3}-\frac{5397525}{14317073}a^{2}-\frac{1799175}{14317073}a-\frac{599725}{14317073}$, $\frac{72327}{14317073}a^{28}-\frac{47218}{14317073}a^{27}+\frac{8703}{14317073}a^{26}+\frac{66829464}{14317073}a^{17}-\frac{43630767}{14317073}a^{16}+\frac{8040302}{14317073}a^{15}-\frac{8159300288}{14317073}a^{6}+\frac{5325605036}{14317073}a^{5}-\frac{984132033}{14317073}a^{4}$, $\frac{1799175}{14317073}a^{28}-\frac{624834}{14317073}a^{27}+\frac{5397525}{14317073}a^{26}-\frac{3598350}{14317073}a^{25}+\frac{16792300}{14317073}a^{24}-\frac{16191872}{14317073}a^{23}+\frac{53975250}{14317073}a^{22}-\frac{65370025}{14317073}a^{21}+\frac{178118325}{14317073}a^{20}-\frac{250084998}{14317073}a^{19}+\frac{599725000}{14317073}a^{18}+\frac{734063400}{14317073}a^{17}+\frac{1471915728}{14317073}a^{16}+\frac{1602465200}{14317073}a^{15}+\frac{3751279875}{14317073}a^{14}+\frac{3312281175}{14317073}a^{13}+\frac{9652037055}{14317073}a^{12}+\frac{6185563650}{14317073}a^{11}+\frac{25641842100}{14317073}a^{10}+\frac{8905316525}{14317073}a^{9}+\frac{70740393066}{14317073}a^{8}+\frac{1074107475}{14317073}a^{7}+\frac{373028950}{14317073}a^{6}+\frac{2963235852}{14317073}a^{5}+\frac{44979375}{14317073}a^{4}+\frac{15592850}{14317073}a^{3}+\frac{5397525}{14317073}a^{2}-\frac{61360228}{14317073}a+\frac{599725}{14317073}$
| 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: | \( 15853905121.091976 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 15853905121.091976 \cdot 93}{22\cdot\sqrt{159386923550435671074967363509984324121230045171}}\cr\approx \mathstrut & 0.157640645261730 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*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^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*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^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*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^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*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 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ |
Intermediate fields
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.0.8732691.1, \(\Q(\zeta_{11})\), 15.15.10943023107606534329121.1 |
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 | $30$ | R | $15^{2}$ | $30$ | R | $30$ | ${\href{/padicField/17.10.0.1}{10} }^{3}$ | ${\href{/padicField/19.10.0.1}{10} }^{3}$ | ${\href{/padicField/23.3.0.1}{3} }^{10}$ | $30$ | $15^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{6}$ | $30$ | ${\href{/padicField/43.6.0.1}{6} }^{5}$ | $15^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{6}$ | $15^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ |
| 3.15.20.65 | $x^{15} + 30 x^{14} + 360 x^{13} + 2175 x^{12} + 6840 x^{11} + 11016 x^{10} + 13050 x^{9} + 21060 x^{8} + 9720 x^{7} + 24084 x^{6} + 55728 x^{5} + 167184 x^{4} + 79137 x^{3} + 474822 x^{2} + 138024$ | $3$ | $5$ | $20$ | $C_{15}$ | $[2]^{5}$ | |
|
\(11\)
| Deg $30$ | $10$ | $3$ | $27$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.99.6t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.99.6t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 6.0.8732691.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.a | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.a | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.b | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.b | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.c | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.c | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.d | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.d | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.c | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.e | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.e | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.f | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.f | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.11.5t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})^+\) | $C_5$ (as 5T1) | $0$ | $1$ |
| * | 1.11.10t1.a.d | $1$ | $ 11 $ | \(\Q(\zeta_{11})\) | $C_{10}$ (as 10T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.g | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.g | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
| * | 1.99.15t1.a.h | $1$ | $ 3^{2} \cdot 11 $ | 15.15.10943023107606534329121.1 | $C_{15}$ (as 15T1) | $0$ | $1$ |
| * | 1.99.30t1.a.h | $1$ | $ 3^{2} \cdot 11 $ | 30.0.159386923550435671074967363509984324121230045171.1 | $C_{30}$ (as 30T1) | $0$ | $-1$ |
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 *.