Properties

Label 45.45.217...129.1
Degree $45$
Signature $[45, 0]$
Discriminant $2.172\times 10^{100}$
Root discriminant \(169.71\)
Ramified primes $7,61$
Class number not computed
Class group not computed
Galois group $C_3\times C_{15}$ (as 45T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)
 
gp: K = bnfinit(y^45 - 12*y^44 - 50*y^43 + 1120*y^42 - 626*y^41 - 44264*y^40 + 106520*y^39 + 966733*y^38 - 3644562*y^37 - 12564205*y^36 + 68449875*y^35 + 92270622*y^34 - 830216501*y^33 - 202049055*y^32 + 6933616229*y^31 - 3181542881*y^30 - 41084005259*y^29 + 40234212427*y^28 + 174510130968*y^27 - 250382990469*y^26 - 527354195438*y^25 + 1015613064703*y^24 + 1094857977723*y^23 - 2871848912941*y^22 - 1403630755463*y^21 + 5763542313829*y^20 + 629825101852*y^19 - 8167242308412*y^18 + 1257688359124*y^17 + 7983130477206*y^16 - 2778691581109*y^15 - 5144196139934*y^14 + 2589633668016*y^13 + 2011525458240*y^12 - 1316532612469*y^11 - 396963769511*y^10 + 358975449036*y^9 + 14178066293*y^8 - 45168898819*y^7 + 5258463166*y^6 + 1537215457*y^5 - 288043322*y^4 - 5291206*y^3 + 2457345*y^2 - 9309*y - 6089, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)
 

\( x^{45} - 12 x^{44} - 50 x^{43} + 1120 x^{42} - 626 x^{41} - 44264 x^{40} + 106520 x^{39} + 966733 x^{38} - 3644562 x^{37} - 12564205 x^{36} + 68449875 x^{35} + \cdots - 6089 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[45, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(217\!\cdots\!129\) \(\medspace = 7^{30}\cdot 61^{42}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(169.71\)
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))
 
Ramified primes:   \(7\), \(61\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $45$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(427=7\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{427}(256,·)$, $\chi_{427}(1,·)$, $\chi_{427}(386,·)$, $\chi_{427}(260,·)$, $\chi_{427}(134,·)$, $\chi_{427}(135,·)$, $\chi_{427}(9,·)$, $\chi_{427}(142,·)$, $\chi_{427}(15,·)$, $\chi_{427}(16,·)$, $\chi_{427}(22,·)$, $\chi_{427}(408,·)$, $\chi_{427}(25,·)$, $\chi_{427}(156,·)$, $\chi_{427}(291,·)$, $\chi_{427}(422,·)$, $\chi_{427}(424,·)$, $\chi_{427}(169,·)$, $\chi_{427}(302,·)$, $\chi_{427}(179,·)$, $\chi_{427}(74,·)$, $\chi_{427}(137,·)$, $\chi_{427}(184,·)$, $\chi_{427}(57,·)$, $\chi_{427}(58,·)$, $\chi_{427}(317,·)$, $\chi_{427}(198,·)$, $\chi_{427}(330,·)$, $\chi_{427}(205,·)$, $\chi_{427}(81,·)$, $\chi_{427}(86,·)$, $\chi_{427}(347,·)$, $\chi_{427}(95,·)$, $\chi_{427}(352,·)$, $\chi_{427}(225,·)$, $\chi_{427}(123,·)$, $\chi_{427}(361,·)$, $\chi_{427}(144,·)$, $\chi_{427}(239,·)$, $\chi_{427}(240,·)$, $\chi_{427}(400,·)$, $\chi_{427}(375,·)$, $\chi_{427}(379,·)$, $\chi_{427}(253,·)$, $\chi_{427}(382,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{11}a^{27}+\frac{3}{11}a^{26}-\frac{1}{11}a^{25}-\frac{1}{11}a^{24}-\frac{1}{11}a^{23}-\frac{5}{11}a^{22}+\frac{1}{11}a^{21}-\frac{5}{11}a^{20}+\frac{1}{11}a^{18}-\frac{4}{11}a^{17}+\frac{2}{11}a^{16}-\frac{4}{11}a^{15}-\frac{3}{11}a^{14}-\frac{4}{11}a^{13}-\frac{2}{11}a^{12}-\frac{3}{11}a^{11}-\frac{1}{11}a^{10}+\frac{1}{11}a^{9}+\frac{3}{11}a^{8}+\frac{5}{11}a^{7}-\frac{5}{11}a^{6}+\frac{2}{11}a^{5}+\frac{4}{11}a^{3}+\frac{5}{11}a^{2}+\frac{2}{11}a-\frac{3}{11}$, $\frac{1}{11}a^{28}+\frac{1}{11}a^{26}+\frac{2}{11}a^{25}+\frac{2}{11}a^{24}-\frac{2}{11}a^{23}+\frac{5}{11}a^{22}+\frac{3}{11}a^{21}+\frac{4}{11}a^{20}+\frac{1}{11}a^{19}+\frac{4}{11}a^{18}+\frac{3}{11}a^{17}+\frac{1}{11}a^{16}-\frac{2}{11}a^{15}+\frac{5}{11}a^{14}-\frac{1}{11}a^{13}+\frac{3}{11}a^{12}-\frac{3}{11}a^{11}+\frac{4}{11}a^{10}-\frac{4}{11}a^{8}+\frac{2}{11}a^{7}-\frac{5}{11}a^{6}+\frac{5}{11}a^{5}+\frac{4}{11}a^{4}+\frac{4}{11}a^{3}-\frac{2}{11}a^{2}+\frac{2}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{29}-\frac{1}{11}a^{26}+\frac{3}{11}a^{25}-\frac{1}{11}a^{24}-\frac{5}{11}a^{23}-\frac{3}{11}a^{22}+\frac{3}{11}a^{21}-\frac{5}{11}a^{20}+\frac{4}{11}a^{19}+\frac{2}{11}a^{18}+\frac{5}{11}a^{17}-\frac{4}{11}a^{16}-\frac{2}{11}a^{15}+\frac{2}{11}a^{14}-\frac{4}{11}a^{13}-\frac{1}{11}a^{12}-\frac{4}{11}a^{11}+\frac{1}{11}a^{10}-\frac{5}{11}a^{9}-\frac{1}{11}a^{8}+\frac{1}{11}a^{7}-\frac{1}{11}a^{6}+\frac{2}{11}a^{5}+\frac{4}{11}a^{4}+\frac{5}{11}a^{3}-\frac{3}{11}a^{2}-\frac{4}{11}a+\frac{3}{11}$, $\frac{1}{11}a^{30}-\frac{5}{11}a^{26}-\frac{2}{11}a^{25}+\frac{5}{11}a^{24}-\frac{4}{11}a^{23}-\frac{2}{11}a^{22}-\frac{4}{11}a^{21}-\frac{1}{11}a^{20}+\frac{2}{11}a^{19}-\frac{5}{11}a^{18}+\frac{3}{11}a^{17}-\frac{2}{11}a^{15}+\frac{4}{11}a^{14}-\frac{5}{11}a^{13}+\frac{5}{11}a^{12}-\frac{2}{11}a^{11}+\frac{5}{11}a^{10}+\frac{4}{11}a^{8}+\frac{4}{11}a^{7}-\frac{3}{11}a^{6}-\frac{5}{11}a^{5}+\frac{5}{11}a^{4}+\frac{1}{11}a^{3}+\frac{1}{11}a^{2}+\frac{5}{11}a-\frac{3}{11}$, $\frac{1}{11}a^{31}+\frac{2}{11}a^{26}+\frac{2}{11}a^{24}+\frac{4}{11}a^{23}+\frac{4}{11}a^{22}+\frac{4}{11}a^{21}-\frac{1}{11}a^{20}-\frac{5}{11}a^{19}-\frac{3}{11}a^{18}+\frac{2}{11}a^{17}-\frac{3}{11}a^{16}-\frac{5}{11}a^{15}+\frac{2}{11}a^{14}-\frac{4}{11}a^{13}-\frac{1}{11}a^{12}+\frac{1}{11}a^{11}-\frac{5}{11}a^{10}-\frac{2}{11}a^{9}-\frac{3}{11}a^{8}+\frac{3}{11}a^{6}+\frac{4}{11}a^{5}+\frac{1}{11}a^{4}-\frac{1}{11}a^{3}-\frac{3}{11}a^{2}-\frac{4}{11}a-\frac{4}{11}$, $\frac{1}{11}a^{32}+\frac{5}{11}a^{26}+\frac{4}{11}a^{25}-\frac{5}{11}a^{24}-\frac{5}{11}a^{23}+\frac{3}{11}a^{22}-\frac{3}{11}a^{21}+\frac{5}{11}a^{20}-\frac{3}{11}a^{19}+\frac{5}{11}a^{17}+\frac{2}{11}a^{16}-\frac{1}{11}a^{15}+\frac{2}{11}a^{14}-\frac{4}{11}a^{13}+\frac{5}{11}a^{12}+\frac{1}{11}a^{11}-\frac{5}{11}a^{9}+\frac{5}{11}a^{8}+\frac{4}{11}a^{7}+\frac{3}{11}a^{6}-\frac{3}{11}a^{5}-\frac{1}{11}a^{4}-\frac{3}{11}a^{2}+\frac{3}{11}a-\frac{5}{11}$, $\frac{1}{11}a^{33}-\frac{3}{11}a^{23}+\frac{3}{11}a^{13}+\frac{4}{11}a^{11}-\frac{1}{11}a^{3}-\frac{4}{11}a+\frac{4}{11}$, $\frac{1}{11}a^{34}-\frac{3}{11}a^{24}+\frac{3}{11}a^{14}+\frac{4}{11}a^{12}-\frac{1}{11}a^{4}-\frac{4}{11}a^{2}+\frac{4}{11}a$, $\frac{1}{11}a^{35}-\frac{3}{11}a^{25}+\frac{3}{11}a^{15}+\frac{4}{11}a^{13}-\frac{1}{11}a^{5}-\frac{4}{11}a^{3}+\frac{4}{11}a^{2}$, $\frac{1}{11}a^{36}-\frac{3}{11}a^{26}+\frac{3}{11}a^{16}+\frac{4}{11}a^{14}-\frac{1}{11}a^{6}-\frac{4}{11}a^{4}+\frac{4}{11}a^{3}$, $\frac{1}{11}a^{37}-\frac{2}{11}a^{26}-\frac{3}{11}a^{25}-\frac{3}{11}a^{24}-\frac{3}{11}a^{23}-\frac{4}{11}a^{22}+\frac{3}{11}a^{21}-\frac{4}{11}a^{20}+\frac{3}{11}a^{18}+\frac{2}{11}a^{17}-\frac{5}{11}a^{16}+\frac{3}{11}a^{15}+\frac{2}{11}a^{14}-\frac{1}{11}a^{13}+\frac{5}{11}a^{12}+\frac{2}{11}a^{11}-\frac{3}{11}a^{10}+\frac{3}{11}a^{9}-\frac{2}{11}a^{8}+\frac{3}{11}a^{7}-\frac{4}{11}a^{6}+\frac{2}{11}a^{5}+\frac{4}{11}a^{4}+\frac{1}{11}a^{3}+\frac{4}{11}a^{2}-\frac{5}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{38}+\frac{3}{11}a^{26}-\frac{5}{11}a^{25}-\frac{5}{11}a^{24}+\frac{5}{11}a^{23}+\frac{4}{11}a^{22}-\frac{2}{11}a^{21}+\frac{1}{11}a^{20}+\frac{3}{11}a^{19}+\frac{4}{11}a^{18}-\frac{2}{11}a^{17}-\frac{4}{11}a^{16}+\frac{5}{11}a^{15}+\frac{4}{11}a^{14}-\frac{3}{11}a^{13}-\frac{2}{11}a^{12}+\frac{2}{11}a^{11}+\frac{1}{11}a^{10}-\frac{2}{11}a^{8}-\frac{5}{11}a^{7}+\frac{3}{11}a^{6}-\frac{3}{11}a^{5}+\frac{1}{11}a^{4}+\frac{1}{11}a^{3}+\frac{5}{11}a^{2}-\frac{5}{11}a+\frac{5}{11}$, $\frac{1}{11}a^{39}-\frac{3}{11}a^{26}-\frac{2}{11}a^{25}-\frac{3}{11}a^{24}-\frac{4}{11}a^{23}+\frac{2}{11}a^{22}-\frac{2}{11}a^{21}-\frac{4}{11}a^{20}+\frac{4}{11}a^{19}-\frac{5}{11}a^{18}-\frac{3}{11}a^{17}-\frac{1}{11}a^{16}+\frac{5}{11}a^{15}-\frac{5}{11}a^{14}-\frac{1}{11}a^{13}-\frac{3}{11}a^{12}-\frac{1}{11}a^{11}+\frac{3}{11}a^{10}-\frac{5}{11}a^{9}-\frac{3}{11}a^{8}-\frac{1}{11}a^{7}+\frac{1}{11}a^{6}-\frac{5}{11}a^{5}+\frac{1}{11}a^{4}+\frac{4}{11}a^{3}+\frac{2}{11}a^{2}-\frac{1}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{40}-\frac{4}{11}a^{26}+\frac{5}{11}a^{25}+\frac{4}{11}a^{24}-\frac{1}{11}a^{23}+\frac{5}{11}a^{22}-\frac{1}{11}a^{21}-\frac{5}{11}a^{19}-\frac{2}{11}a^{17}+\frac{5}{11}a^{15}+\frac{1}{11}a^{14}-\frac{4}{11}a^{13}+\frac{4}{11}a^{12}+\frac{5}{11}a^{11}+\frac{3}{11}a^{10}-\frac{3}{11}a^{8}+\frac{5}{11}a^{7}+\frac{2}{11}a^{6}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{3}{11}a^{3}+\frac{3}{11}a^{2}+\frac{4}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{41}-\frac{5}{11}a^{26}-\frac{5}{11}a^{24}+\frac{1}{11}a^{23}+\frac{1}{11}a^{22}+\frac{4}{11}a^{21}-\frac{3}{11}a^{20}+\frac{2}{11}a^{18}-\frac{5}{11}a^{17}+\frac{2}{11}a^{16}-\frac{4}{11}a^{15}-\frac{5}{11}a^{14}-\frac{1}{11}a^{13}-\frac{3}{11}a^{12}+\frac{2}{11}a^{11}-\frac{4}{11}a^{10}+\frac{1}{11}a^{9}-\frac{5}{11}a^{8}-\frac{2}{11}a^{6}+\frac{1}{11}a^{5}+\frac{3}{11}a^{4}-\frac{3}{11}a^{3}+\frac{2}{11}a^{2}-\frac{1}{11}a-\frac{1}{11}$, $\frac{1}{72479}a^{42}+\frac{164}{72479}a^{41}-\frac{116}{6589}a^{40}+\frac{1909}{72479}a^{39}+\frac{2341}{72479}a^{38}+\frac{2170}{72479}a^{37}+\frac{1597}{72479}a^{36}-\frac{694}{72479}a^{35}+\frac{1508}{72479}a^{34}+\frac{479}{72479}a^{33}-\frac{537}{72479}a^{32}-\frac{2509}{72479}a^{31}+\frac{730}{72479}a^{30}+\frac{2025}{72479}a^{29}-\frac{18}{6589}a^{28}+\frac{2295}{72479}a^{27}-\frac{3726}{72479}a^{26}+\frac{368}{6589}a^{25}-\frac{25759}{72479}a^{24}+\frac{14412}{72479}a^{23}+\frac{29725}{72479}a^{22}+\frac{14184}{72479}a^{21}+\frac{30047}{72479}a^{20}-\frac{14225}{72479}a^{19}-\frac{13096}{72479}a^{18}-\frac{21892}{72479}a^{17}-\frac{15034}{72479}a^{16}+\frac{9639}{72479}a^{15}-\frac{999}{6589}a^{14}-\frac{25147}{72479}a^{13}-\frac{20939}{72479}a^{12}-\frac{2097}{6589}a^{11}+\frac{2281}{72479}a^{10}-\frac{34447}{72479}a^{9}+\frac{22698}{72479}a^{8}+\frac{24945}{72479}a^{7}-\frac{14909}{72479}a^{6}-\frac{34317}{72479}a^{5}+\frac{340}{6589}a^{4}+\frac{6243}{72479}a^{3}+\frac{1601}{72479}a^{2}-\frac{23213}{72479}a+\frac{9033}{72479}$, $\frac{1}{72479}a^{43}-\frac{1816}{72479}a^{41}+\frac{325}{72479}a^{40}-\frac{1052}{72479}a^{39}+\frac{408}{72479}a^{38}+\frac{1523}{72479}a^{37}+\frac{958}{72479}a^{36}-\frac{298}{6589}a^{35}-\frac{3040}{72479}a^{34}-\frac{25}{72479}a^{33}-\frac{98}{72479}a^{32}-\frac{2901}{72479}a^{31}+\frac{907}{72479}a^{30}-\frac{2848}{72479}a^{29}+\frac{1822}{72479}a^{28}+\frac{2056}{72479}a^{27}-\frac{10843}{72479}a^{26}-\frac{24142}{72479}a^{25}-\frac{4428}{72479}a^{24}-\frac{21104}{72479}a^{23}-\frac{17801}{72479}a^{22}-\frac{287}{6589}a^{21}-\frac{33128}{72479}a^{20}-\frac{25880}{72479}a^{19}+\frac{4194}{72479}a^{18}-\frac{28929}{72479}a^{17}-\frac{28605}{72479}a^{16}-\frac{10425}{72479}a^{15}-\frac{21748}{72479}a^{14}+\frac{17989}{72479}a^{13}-\frac{35118}{72479}a^{12}+\frac{3183}{72479}a^{11}-\frac{13191}{72479}a^{10}-\frac{7712}{72479}a^{9}-\frac{27454}{72479}a^{8}-\frac{20709}{72479}a^{7}+\frac{18952}{72479}a^{6}-\frac{15045}{72479}a^{5}+\frac{25427}{72479}a^{4}-\frac{20723}{72479}a^{3}-\frac{28806}{72479}a^{2}-\frac{32011}{72479}a+\frac{1113}{72479}$, $\frac{1}{80\!\cdots\!73}a^{44}-\frac{32\!\cdots\!40}{73\!\cdots\!43}a^{43}+\frac{36\!\cdots\!24}{80\!\cdots\!73}a^{42}-\frac{99\!\cdots\!62}{80\!\cdots\!73}a^{41}-\frac{21\!\cdots\!59}{80\!\cdots\!73}a^{40}+\frac{14\!\cdots\!42}{80\!\cdots\!73}a^{39}-\frac{15\!\cdots\!27}{80\!\cdots\!73}a^{38}-\frac{27\!\cdots\!86}{80\!\cdots\!73}a^{37}+\frac{10\!\cdots\!01}{80\!\cdots\!73}a^{36}-\frac{19\!\cdots\!83}{80\!\cdots\!73}a^{35}+\frac{21\!\cdots\!42}{80\!\cdots\!73}a^{34}+\frac{11\!\cdots\!30}{80\!\cdots\!73}a^{33}+\frac{19\!\cdots\!10}{73\!\cdots\!43}a^{32}-\frac{26\!\cdots\!38}{80\!\cdots\!73}a^{31}+\frac{26\!\cdots\!60}{80\!\cdots\!73}a^{30}-\frac{32\!\cdots\!28}{73\!\cdots\!43}a^{29}-\frac{31\!\cdots\!21}{80\!\cdots\!73}a^{28}-\frac{26\!\cdots\!64}{80\!\cdots\!73}a^{27}-\frac{17\!\cdots\!65}{80\!\cdots\!73}a^{26}+\frac{91\!\cdots\!80}{80\!\cdots\!73}a^{25}+\frac{24\!\cdots\!71}{80\!\cdots\!73}a^{24}-\frac{17\!\cdots\!56}{80\!\cdots\!73}a^{23}-\frac{11\!\cdots\!58}{80\!\cdots\!73}a^{22}-\frac{97\!\cdots\!98}{80\!\cdots\!73}a^{21}+\frac{23\!\cdots\!67}{80\!\cdots\!73}a^{20}+\frac{18\!\cdots\!31}{80\!\cdots\!73}a^{19}-\frac{64\!\cdots\!33}{80\!\cdots\!73}a^{18}+\frac{21\!\cdots\!57}{80\!\cdots\!73}a^{17}+\frac{32\!\cdots\!42}{80\!\cdots\!73}a^{16}+\frac{23\!\cdots\!68}{73\!\cdots\!43}a^{15}+\frac{31\!\cdots\!64}{80\!\cdots\!73}a^{14}+\frac{35\!\cdots\!61}{80\!\cdots\!73}a^{13}+\frac{75\!\cdots\!78}{80\!\cdots\!73}a^{12}+\frac{11\!\cdots\!65}{80\!\cdots\!73}a^{11}+\frac{12\!\cdots\!16}{80\!\cdots\!73}a^{10}+\frac{35\!\cdots\!13}{73\!\cdots\!43}a^{9}+\frac{21\!\cdots\!46}{80\!\cdots\!73}a^{8}+\frac{16\!\cdots\!48}{80\!\cdots\!73}a^{7}-\frac{19\!\cdots\!85}{80\!\cdots\!73}a^{6}-\frac{40\!\cdots\!83}{80\!\cdots\!73}a^{5}+\frac{26\!\cdots\!64}{80\!\cdots\!73}a^{4}+\frac{16\!\cdots\!45}{80\!\cdots\!73}a^{3}+\frac{13\!\cdots\!20}{80\!\cdots\!73}a^{2}+\frac{40\!\cdots\!46}{80\!\cdots\!73}a-\frac{38\!\cdots\!31}{80\!\cdots\!73}$ Copy content Toggle raw display

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

not computed

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:  $44$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

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

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)
 
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^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089, 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^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089);
 
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^45 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089);
 
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

$C_3\times C_{15}$ (as 45T2):

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)
 
An abelian group of order 45
The 45 conjugacy class representatives for $C_3\times C_{15}$
Character table for $C_3\times C_{15}$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.3721.1, 3.3.182329.2, 3.3.182329.1, 5.5.13845841.1, 9.9.6061320523197289.1, 15.15.749787887447605250170677896929.1, 15.15.9876832533361318095112441.1, 15.15.2789960729192539135885092454472809.2, 15.15.2789960729192539135885092454472809.1

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 $15^{3}$ $15^{3}$ $15^{3}$ R ${\href{/padicField/11.3.0.1}{3} }^{15}$ ${\href{/padicField/13.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ $15^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$ ${\href{/padicField/41.5.0.1}{5} }^{9}$ $15^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{15}$ $15^{3}$ $15^{3}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display Deg $45$$3$$15$$30$
\(61\) Copy content Toggle raw display Deg $45$$15$$3$$42$