↑

Properties

Label 44.0.11656732006...0625.3
Degree $44$
Signature $[0, 22]$
Discriminant $3^{22}\cdot 5^{22}\cdot 23^{42}$
Root discriminant $77.25$
Ramified primes $3, 5, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{22}$ (as 44T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17592186044416, -4398046511104, -3298534883328, 1924145348608, 343597383680, -566935683072, 55834574848, 127775277056, -45902462976, -20468203520, 16592666624, 968884224, -4390387712, 855375872, 883752960, -434782208, -112242688, 136756224, -6128384, -32656960, 9696336, 5740156, -3859123, 1435039, 606021, -510265, -23939, 133551, -27403, -26537, 13485, 3263, -4187, 231, 989, -305, -171, 119, 13, -33, 5, 7, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^44 - x^43 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416)
 
gp: K = bnfinit(x^44 - x^43 - 3*x^42 + 7*x^41 + 5*x^40 - 33*x^39 + 13*x^38 + 119*x^37 - 171*x^36 - 305*x^35 + 989*x^34 + 231*x^33 - 4187*x^32 + 3263*x^31 + 13485*x^30 - 26537*x^29 - 27403*x^28 + 133551*x^27 - 23939*x^26 - 510265*x^25 + 606021*x^24 + 1435039*x^23 - 3859123*x^22 + 5740156*x^21 + 9696336*x^20 - 32656960*x^19 - 6128384*x^18 + 136756224*x^17 - 112242688*x^16 - 434782208*x^15 + 883752960*x^14 + 855375872*x^13 - 4390387712*x^12 + 968884224*x^11 + 16592666624*x^10 - 20468203520*x^9 - 45902462976*x^8 + 127775277056*x^7 + 55834574848*x^6 - 566935683072*x^5 + 343597383680*x^4 + 1924145348608*x^3 - 3298534883328*x^2 - 4398046511104*x + 17592186044416, 1)
 

Normalized defining polynomial

\( x^{44} - x^{43} - 3 x^{42} + 7 x^{41} + 5 x^{40} - 33 x^{39} + 13 x^{38} + 119 x^{37} - 171 x^{36} - 305 x^{35} + 989 x^{34} + 231 x^{33} - 4187 x^{32} + 3263 x^{31} + 13485 x^{30} - 26537 x^{29} - 27403 x^{28} + 133551 x^{27} - 23939 x^{26} - 510265 x^{25} + 606021 x^{24} + 1435039 x^{23} - 3859123 x^{22} + 5740156 x^{21} + 9696336 x^{20} - 32656960 x^{19} - 6128384 x^{18} + 136756224 x^{17} - 112242688 x^{16} - 434782208 x^{15} + 883752960 x^{14} + 855375872 x^{13} - 4390387712 x^{12} + 968884224 x^{11} + 16592666624 x^{10} - 20468203520 x^{9} - 45902462976 x^{8} + 127775277056 x^{7} + 55834574848 x^{6} - 566935683072 x^{5} + 343597383680 x^{4} + 1924145348608 x^{3} - 3298534883328 x^{2} - 4398046511104 x + 17592186044416 \)

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

Invariants

Degree:  $44$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 22]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(116567320065927752512435466812933331534234135648947894549180382317450046539306640625=3^{22}\cdot 5^{22}\cdot 23^{42}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.25$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(345=3\cdot 5\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{345}(256,·)$, $\chi_{345}(1,·)$, $\chi_{345}(134,·)$, $\chi_{345}(136,·)$, $\chi_{345}(269,·)$, $\chi_{345}(14,·)$, $\chi_{345}(271,·)$, $\chi_{345}(16,·)$, $\chi_{345}(149,·)$, $\chi_{345}(151,·)$, $\chi_{345}(284,·)$, $\chi_{345}(29,·)$, $\chi_{345}(286,·)$, $\chi_{345}(31,·)$, $\chi_{345}(164,·)$, $\chi_{345}(166,·)$, $\chi_{345}(44,·)$, $\chi_{345}(301,·)$, $\chi_{345}(179,·)$, $\chi_{345}(181,·)$, $\chi_{345}(314,·)$, $\chi_{345}(59,·)$, $\chi_{345}(316,·)$, $\chi_{345}(61,·)$, $\chi_{345}(194,·)$, $\chi_{345}(196,·)$, $\chi_{345}(329,·)$, $\chi_{345}(74,·)$, $\chi_{345}(331,·)$, $\chi_{345}(76,·)$, $\chi_{345}(209,·)$, $\chi_{345}(211,·)$, $\chi_{345}(344,·)$, $\chi_{345}(89,·)$, $\chi_{345}(91,·)$, $\chi_{345}(224,·)$, $\chi_{345}(226,·)$, $\chi_{345}(104,·)$, $\chi_{345}(106,·)$, $\chi_{345}(239,·)$, $\chi_{345}(241,·)$, $\chi_{345}(119,·)$, $\chi_{345}(121,·)$, $\chi_{345}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{15436492} a^{23} + \frac{1}{4} a^{22} - \frac{1}{4} a^{21} + \frac{1}{4} a^{20} - \frac{1}{4} a^{19} + \frac{1}{4} a^{18} - \frac{1}{4} a^{17} + \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1435039}{3859123}$, $\frac{1}{61745968} a^{24} - \frac{1}{61745968} a^{23} - \frac{1}{16} a^{22} - \frac{3}{16} a^{21} + \frac{7}{16} a^{20} + \frac{5}{16} a^{19} - \frac{1}{16} a^{18} - \frac{3}{16} a^{17} + \frac{7}{16} a^{16} + \frac{5}{16} a^{15} - \frac{1}{16} a^{14} - \frac{3}{16} a^{13} + \frac{7}{16} a^{12} + \frac{5}{16} a^{11} - \frac{1}{16} a^{10} - \frac{3}{16} a^{9} + \frac{7}{16} a^{8} + \frac{5}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{16} a^{5} + \frac{7}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{16} a^{2} + \frac{1435039}{15436492} a + \frac{606021}{3859123}$, $\frac{1}{246983872} a^{25} - \frac{1}{246983872} a^{24} - \frac{3}{246983872} a^{23} + \frac{29}{64} a^{22} - \frac{25}{64} a^{21} - \frac{27}{64} a^{20} - \frac{1}{64} a^{19} - \frac{19}{64} a^{18} + \frac{23}{64} a^{17} - \frac{11}{64} a^{16} - \frac{17}{64} a^{15} - \frac{3}{64} a^{14} + \frac{7}{64} a^{13} + \frac{5}{64} a^{12} + \frac{31}{64} a^{11} + \frac{13}{64} a^{10} - \frac{9}{64} a^{9} + \frac{21}{64} a^{8} + \frac{15}{64} a^{7} + \frac{29}{64} a^{6} - \frac{25}{64} a^{5} - \frac{27}{64} a^{4} - \frac{1}{64} a^{3} + \frac{1435039}{61745968} a^{2} + \frac{606021}{15436492} a - \frac{510265}{3859123}$, $\frac{1}{987935488} a^{26} - \frac{1}{987935488} a^{25} - \frac{3}{987935488} a^{24} + \frac{7}{987935488} a^{23} + \frac{103}{256} a^{22} + \frac{37}{256} a^{21} + \frac{63}{256} a^{20} + \frac{45}{256} a^{19} - \frac{41}{256} a^{18} + \frac{117}{256} a^{17} + \frac{47}{256} a^{16} - \frac{3}{256} a^{15} + \frac{71}{256} a^{14} - \frac{59}{256} a^{13} + \frac{31}{256} a^{12} - \frac{51}{256} a^{11} - \frac{73}{256} a^{10} + \frac{21}{256} a^{9} + \frac{15}{256} a^{8} - \frac{99}{256} a^{7} + \frac{39}{256} a^{6} + \frac{101}{256} a^{5} - \frac{1}{256} a^{4} + \frac{1435039}{246983872} a^{3} + \frac{606021}{61745968} a^{2} - \frac{510265}{15436492} a - \frac{23939}{3859123}$, $\frac{1}{3951741952} a^{27} - \frac{1}{3951741952} a^{26} - \frac{3}{3951741952} a^{25} + \frac{7}{3951741952} a^{24} + \frac{5}{3951741952} a^{23} + \frac{37}{1024} a^{22} - \frac{449}{1024} a^{21} + \frac{301}{1024} a^{20} + \frac{471}{1024} a^{19} + \frac{373}{1024} a^{18} - \frac{209}{1024} a^{17} - \frac{259}{1024} a^{16} + \frac{71}{1024} a^{15} - \frac{59}{1024} a^{14} - \frac{225}{1024} a^{13} + \frac{461}{1024} a^{12} + \frac{439}{1024} a^{11} - \frac{235}{1024} a^{10} - \frac{497}{1024} a^{9} + \frac{413}{1024} a^{8} - \frac{473}{1024} a^{7} - \frac{155}{1024} a^{6} - \frac{1}{1024} a^{5} + \frac{1435039}{987935488} a^{4} + \frac{606021}{246983872} a^{3} - \frac{510265}{61745968} a^{2} - \frac{23939}{15436492} a + \frac{133551}{3859123}$, $\frac{1}{15806967808} a^{28} - \frac{1}{15806967808} a^{27} - \frac{3}{15806967808} a^{26} + \frac{7}{15806967808} a^{25} + \frac{5}{15806967808} a^{24} - \frac{33}{15806967808} a^{23} + \frac{575}{4096} a^{22} - \frac{723}{4096} a^{21} - \frac{1577}{4096} a^{20} + \frac{373}{4096} a^{19} + \frac{1839}{4096} a^{18} + \frac{765}{4096} a^{17} + \frac{71}{4096} a^{16} + \frac{965}{4096} a^{15} - \frac{1249}{4096} a^{14} + \frac{1485}{4096} a^{13} - \frac{585}{4096} a^{12} - \frac{1259}{4096} a^{11} - \frac{497}{4096} a^{10} + \frac{1437}{4096} a^{9} + \frac{551}{4096} a^{8} + \frac{1893}{4096} a^{7} - \frac{1}{4096} a^{6} + \frac{1435039}{3951741952} a^{5} + \frac{606021}{987935488} a^{4} - \frac{510265}{246983872} a^{3} - \frac{23939}{61745968} a^{2} + \frac{133551}{15436492} a - \frac{27403}{3859123}$, $\frac{1}{63227871232} a^{29} - \frac{1}{63227871232} a^{28} - \frac{3}{63227871232} a^{27} + \frac{7}{63227871232} a^{26} + \frac{5}{63227871232} a^{25} - \frac{33}{63227871232} a^{24} + \frac{13}{63227871232} a^{23} - \frac{723}{16384} a^{22} - \frac{1577}{16384} a^{21} + \frac{4469}{16384} a^{20} + \frac{1839}{16384} a^{19} - \frac{3331}{16384} a^{18} - \frac{4025}{16384} a^{17} + \frac{965}{16384} a^{16} - \frac{1249}{16384} a^{15} - \frac{2611}{16384} a^{14} + \frac{7607}{16384} a^{13} + \frac{2837}{16384} a^{12} - \frac{497}{16384} a^{11} + \frac{5533}{16384} a^{10} - \frac{3545}{16384} a^{9} - \frac{2203}{16384} a^{8} - \frac{1}{16384} a^{7} + \frac{1435039}{15806967808} a^{6} + \frac{606021}{3951741952} a^{5} - \frac{510265}{987935488} a^{4} - \frac{23939}{246983872} a^{3} + \frac{133551}{61745968} a^{2} - \frac{27403}{15436492} a - \frac{26537}{3859123}$, $\frac{1}{252911484928} a^{30} - \frac{1}{252911484928} a^{29} - \frac{3}{252911484928} a^{28} + \frac{7}{252911484928} a^{27} + \frac{5}{252911484928} a^{26} - \frac{33}{252911484928} a^{25} + \frac{13}{252911484928} a^{24} + \frac{119}{252911484928} a^{23} + \frac{31191}{65536} a^{22} - \frac{28299}{65536} a^{21} - \frac{30929}{65536} a^{20} + \frac{13053}{65536} a^{19} - \frac{20409}{65536} a^{18} - \frac{31803}{65536} a^{17} - \frac{17633}{65536} a^{16} + \frac{13773}{65536} a^{15} - \frac{8777}{65536} a^{14} + \frac{19221}{65536} a^{13} + \frac{15887}{65536} a^{12} - \frac{27235}{65536} a^{11} + \frac{29223}{65536} a^{10} + \frac{14181}{65536} a^{9} - \frac{1}{65536} a^{8} + \frac{1435039}{63227871232} a^{7} + \frac{606021}{15806967808} a^{6} - \frac{510265}{3951741952} a^{5} - \frac{23939}{987935488} a^{4} + \frac{133551}{246983872} a^{3} - \frac{27403}{61745968} a^{2} - \frac{26537}{15436492} a + \frac{13485}{3859123}$, $\frac{1}{1011645939712} a^{31} - \frac{1}{1011645939712} a^{30} - \frac{3}{1011645939712} a^{29} + \frac{7}{1011645939712} a^{28} + \frac{5}{1011645939712} a^{27} - \frac{33}{1011645939712} a^{26} + \frac{13}{1011645939712} a^{25} + \frac{119}{1011645939712} a^{24} - \frac{171}{1011645939712} a^{23} + \frac{102773}{262144} a^{22} + \frac{34607}{262144} a^{21} + \frac{78589}{262144} a^{20} + \frac{45127}{262144} a^{19} - \frac{97339}{262144} a^{18} - \frac{83169}{262144} a^{17} - \frac{51763}{262144} a^{16} + \frac{122295}{262144} a^{15} + \frac{84757}{262144} a^{14} - \frac{49649}{262144} a^{13} - \frac{27235}{262144} a^{12} - \frac{36313}{262144} a^{11} - \frac{116891}{262144} a^{10} - \frac{1}{262144} a^{9} + \frac{1435039}{252911484928} a^{8} + \frac{606021}{63227871232} a^{7} - \frac{510265}{15806967808} a^{6} - \frac{23939}{3951741952} a^{5} + \frac{133551}{987935488} a^{4} - \frac{27403}{246983872} a^{3} - \frac{26537}{61745968} a^{2} + \frac{13485}{15436492} a + \frac{3263}{3859123}$, $\frac{1}{4046583758848} a^{32} - \frac{1}{4046583758848} a^{31} - \frac{3}{4046583758848} a^{30} + \frac{7}{4046583758848} a^{29} + \frac{5}{4046583758848} a^{28} - \frac{33}{4046583758848} a^{27} + \frac{13}{4046583758848} a^{26} + \frac{119}{4046583758848} a^{25} - \frac{171}{4046583758848} a^{24} - \frac{305}{4046583758848} a^{23} - \frac{227537}{1048576} a^{22} - \frac{183555}{1048576} a^{21} + \frac{45127}{1048576} a^{20} - \frac{359483}{1048576} a^{19} + \frac{178975}{1048576} a^{18} + \frac{210381}{1048576} a^{17} + \frac{122295}{1048576} a^{16} + \frac{84757}{1048576} a^{15} + \frac{474639}{1048576} a^{14} + \frac{234909}{1048576} a^{13} - \frac{36313}{1048576} a^{12} + \frac{145253}{1048576} a^{11} - \frac{1}{1048576} a^{10} + \frac{1435039}{1011645939712} a^{9} + \frac{606021}{252911484928} a^{8} - \frac{510265}{63227871232} a^{7} - \frac{23939}{15806967808} a^{6} + \frac{133551}{3951741952} a^{5} - \frac{27403}{987935488} a^{4} - \frac{26537}{246983872} a^{3} + \frac{13485}{61745968} a^{2} + \frac{3263}{15436492} a - \frac{4187}{3859123}$, $\frac{1}{16186335035392} a^{33} - \frac{1}{16186335035392} a^{32} - \frac{3}{16186335035392} a^{31} + \frac{7}{16186335035392} a^{30} + \frac{5}{16186335035392} a^{29} - \frac{33}{16186335035392} a^{28} + \frac{13}{16186335035392} a^{27} + \frac{119}{16186335035392} a^{26} - \frac{171}{16186335035392} a^{25} - \frac{305}{16186335035392} a^{24} + \frac{989}{16186335035392} a^{23} + \frac{1913597}{4194304} a^{22} - \frac{1003449}{4194304} a^{21} + \frac{1737669}{4194304} a^{20} - \frac{1918177}{4194304} a^{19} - \frac{838195}{4194304} a^{18} + \frac{122295}{4194304} a^{17} - \frac{963819}{4194304} a^{16} + \frac{474639}{4194304} a^{15} - \frac{813667}{4194304} a^{14} - \frac{1084889}{4194304} a^{13} + \frac{145253}{4194304} a^{12} - \frac{1}{4194304} a^{11} + \frac{1435039}{4046583758848} a^{10} + \frac{606021}{1011645939712} a^{9} - \frac{510265}{252911484928} a^{8} - \frac{23939}{63227871232} a^{7} + \frac{133551}{15806967808} a^{6} - \frac{27403}{3951741952} a^{5} - \frac{26537}{987935488} a^{4} + \frac{13485}{246983872} a^{3} + \frac{3263}{61745968} a^{2} - \frac{4187}{15436492} a + \frac{231}{3859123}$, $\frac{1}{64745340141568} a^{34} - \frac{1}{64745340141568} a^{33} - \frac{3}{64745340141568} a^{32} + \frac{7}{64745340141568} a^{31} + \frac{5}{64745340141568} a^{30} - \frac{33}{64745340141568} a^{29} + \frac{13}{64745340141568} a^{28} + \frac{119}{64745340141568} a^{27} - \frac{171}{64745340141568} a^{26} - \frac{305}{64745340141568} a^{25} + \frac{989}{64745340141568} a^{24} + \frac{231}{64745340141568} a^{23} - \frac{1003449}{16777216} a^{22} - \frac{6650939}{16777216} a^{21} - \frac{6112481}{16777216} a^{20} - \frac{838195}{16777216} a^{19} - \frac{8266313}{16777216} a^{18} - \frac{5158123}{16777216} a^{17} + \frac{4668943}{16777216} a^{16} - \frac{813667}{16777216} a^{15} - \frac{1084889}{16777216} a^{14} + \frac{4339557}{16777216} a^{13} - \frac{1}{16777216} a^{12} + \frac{1435039}{16186335035392} a^{11} + \frac{606021}{4046583758848} a^{10} - \frac{510265}{1011645939712} a^{9} - \frac{23939}{252911484928} a^{8} + \frac{133551}{63227871232} a^{7} - \frac{27403}{15806967808} a^{6} - \frac{26537}{3951741952} a^{5} + \frac{13485}{987935488} a^{4} + \frac{3263}{246983872} a^{3} - \frac{4187}{61745968} a^{2} + \frac{231}{15436492} a + \frac{989}{3859123}$, $\frac{1}{258981360566272} a^{35} - \frac{1}{258981360566272} a^{34} - \frac{3}{258981360566272} a^{33} + \frac{7}{258981360566272} a^{32} + \frac{5}{258981360566272} a^{31} - \frac{33}{258981360566272} a^{30} + \frac{13}{258981360566272} a^{29} + \frac{119}{258981360566272} a^{28} - \frac{171}{258981360566272} a^{27} - \frac{305}{258981360566272} a^{26} + \frac{989}{258981360566272} a^{25} + \frac{231}{258981360566272} a^{24} - \frac{4187}{258981360566272} a^{23} - \frac{6650939}{67108864} a^{22} + \frac{10664735}{67108864} a^{21} + \frac{15939021}{67108864} a^{20} + \frac{8510903}{67108864} a^{19} - \frac{5158123}{67108864} a^{18} - \frac{28885489}{67108864} a^{17} - \frac{17590883}{67108864} a^{16} - \frac{1084889}{67108864} a^{15} + \frac{4339557}{67108864} a^{14} - \frac{1}{67108864} a^{13} + \frac{1435039}{64745340141568} a^{12} + \frac{606021}{16186335035392} a^{11} - \frac{510265}{4046583758848} a^{10} - \frac{23939}{1011645939712} a^{9} + \frac{133551}{252911484928} a^{8} - \frac{27403}{63227871232} a^{7} - \frac{26537}{15806967808} a^{6} + \frac{13485}{3951741952} a^{5} + \frac{3263}{987935488} a^{4} - \frac{4187}{246983872} a^{3} + \frac{231}{61745968} a^{2} + \frac{989}{15436492} a - \frac{305}{3859123}$, $\frac{1}{1035925442265088} a^{36} - \frac{1}{1035925442265088} a^{35} - \frac{3}{1035925442265088} a^{34} + \frac{7}{1035925442265088} a^{33} + \frac{5}{1035925442265088} a^{32} - \frac{33}{1035925442265088} a^{31} + \frac{13}{1035925442265088} a^{30} + \frac{119}{1035925442265088} a^{29} - \frac{171}{1035925442265088} a^{28} - \frac{305}{1035925442265088} a^{27} + \frac{989}{1035925442265088} a^{26} + \frac{231}{1035925442265088} a^{25} - \frac{4187}{1035925442265088} a^{24} + \frac{3263}{1035925442265088} a^{23} + \frac{10664735}{268435456} a^{22} + \frac{15939021}{268435456} a^{21} - \frac{58597961}{268435456} a^{20} - \frac{5158123}{268435456} a^{19} - \frac{28885489}{268435456} a^{18} + \frac{49517981}{268435456} a^{17} + \frac{66023975}{268435456} a^{16} + \frac{4339557}{268435456} a^{15} - \frac{1}{268435456} a^{14} + \frac{1435039}{258981360566272} a^{13} + \frac{606021}{64745340141568} a^{12} - \frac{510265}{16186335035392} a^{11} - \frac{23939}{4046583758848} a^{10} + \frac{133551}{1011645939712} a^{9} - \frac{27403}{252911484928} a^{8} - \frac{26537}{63227871232} a^{7} + \frac{13485}{15806967808} a^{6} + \frac{3263}{3951741952} a^{5} - \frac{4187}{987935488} a^{4} + \frac{231}{246983872} a^{3} + \frac{989}{61745968} a^{2} - \frac{305}{15436492} a - \frac{171}{3859123}$, $\frac{1}{4143701769060352} a^{37} - \frac{1}{4143701769060352} a^{36} - \frac{3}{4143701769060352} a^{35} + \frac{7}{4143701769060352} a^{34} + \frac{5}{4143701769060352} a^{33} - \frac{33}{4143701769060352} a^{32} + \frac{13}{4143701769060352} a^{31} + \frac{119}{4143701769060352} a^{30} - \frac{171}{4143701769060352} a^{29} - \frac{305}{4143701769060352} a^{28} + \frac{989}{4143701769060352} a^{27} + \frac{231}{4143701769060352} a^{26} - \frac{4187}{4143701769060352} a^{25} + \frac{3263}{4143701769060352} a^{24} + \frac{13485}{4143701769060352} a^{23} + \frac{284374477}{1073741824} a^{22} - \frac{327033417}{1073741824} a^{21} + \frac{263277333}{1073741824} a^{20} - \frac{28885489}{1073741824} a^{19} + \frac{49517981}{1073741824} a^{18} + \frac{66023975}{1073741824} a^{17} - \frac{264095899}{1073741824} a^{16} - \frac{1}{1073741824} a^{15} + \frac{1435039}{1035925442265088} a^{14} + \frac{606021}{258981360566272} a^{13} - \frac{510265}{64745340141568} a^{12} - \frac{23939}{16186335035392} a^{11} + \frac{133551}{4046583758848} a^{10} - \frac{27403}{1011645939712} a^{9} - \frac{26537}{252911484928} a^{8} + \frac{13485}{63227871232} a^{7} + \frac{3263}{15806967808} a^{6} - \frac{4187}{3951741952} a^{5} + \frac{231}{987935488} a^{4} + \frac{989}{246983872} a^{3} - \frac{305}{61745968} a^{2} - \frac{171}{15436492} a + \frac{119}{3859123}$, $\frac{1}{16574807076241408} a^{38} - \frac{1}{16574807076241408} a^{37} - \frac{3}{16574807076241408} a^{36} + \frac{7}{16574807076241408} a^{35} + \frac{5}{16574807076241408} a^{34} - \frac{33}{16574807076241408} a^{33} + \frac{13}{16574807076241408} a^{32} + \frac{119}{16574807076241408} a^{31} - \frac{171}{16574807076241408} a^{30} - \frac{305}{16574807076241408} a^{29} + \frac{989}{16574807076241408} a^{28} + \frac{231}{16574807076241408} a^{27} - \frac{4187}{16574807076241408} a^{26} + \frac{3263}{16574807076241408} a^{25} + \frac{13485}{16574807076241408} a^{24} - \frac{26537}{16574807076241408} a^{23} + \frac{746708407}{4294967296} a^{22} - \frac{1884206315}{4294967296} a^{21} - \frac{1102627313}{4294967296} a^{20} + \frac{49517981}{4294967296} a^{19} + \frac{66023975}{4294967296} a^{18} - \frac{264095899}{4294967296} a^{17} - \frac{1}{4294967296} a^{16} + \frac{1435039}{4143701769060352} a^{15} + \frac{606021}{1035925442265088} a^{14} - \frac{510265}{258981360566272} a^{13} - \frac{23939}{64745340141568} a^{12} + \frac{133551}{16186335035392} a^{11} - \frac{27403}{4046583758848} a^{10} - \frac{26537}{1011645939712} a^{9} + \frac{13485}{252911484928} a^{8} + \frac{3263}{63227871232} a^{7} - \frac{4187}{15806967808} a^{6} + \frac{231}{3951741952} a^{5} + \frac{989}{987935488} a^{4} - \frac{305}{246983872} a^{3} - \frac{171}{61745968} a^{2} + \frac{119}{15436492} a + \frac{13}{3859123}$, $\frac{1}{66299228304965632} a^{39} - \frac{1}{66299228304965632} a^{38} - \frac{3}{66299228304965632} a^{37} + \frac{7}{66299228304965632} a^{36} + \frac{5}{66299228304965632} a^{35} - \frac{33}{66299228304965632} a^{34} + \frac{13}{66299228304965632} a^{33} + \frac{119}{66299228304965632} a^{32} - \frac{171}{66299228304965632} a^{31} - \frac{305}{66299228304965632} a^{30} + \frac{989}{66299228304965632} a^{29} + \frac{231}{66299228304965632} a^{28} - \frac{4187}{66299228304965632} a^{27} + \frac{3263}{66299228304965632} a^{26} + \frac{13485}{66299228304965632} a^{25} - \frac{26537}{66299228304965632} a^{24} - \frac{27403}{66299228304965632} a^{23} - \frac{6179173611}{17179869184} a^{22} + \frac{3192339983}{17179869184} a^{21} + \frac{4344485277}{17179869184} a^{20} + \frac{66023975}{17179869184} a^{19} - \frac{264095899}{17179869184} a^{18} - \frac{1}{17179869184} a^{17} + \frac{1435039}{16574807076241408} a^{16} + \frac{606021}{4143701769060352} a^{15} - \frac{510265}{1035925442265088} a^{14} - \frac{23939}{258981360566272} a^{13} + \frac{133551}{64745340141568} a^{12} - \frac{27403}{16186335035392} a^{11} - \frac{26537}{4046583758848} a^{10} + \frac{13485}{1011645939712} a^{9} + \frac{3263}{252911484928} a^{8} - \frac{4187}{63227871232} a^{7} + \frac{231}{15806967808} a^{6} + \frac{989}{3951741952} a^{5} - \frac{305}{987935488} a^{4} - \frac{171}{246983872} a^{3} + \frac{119}{61745968} a^{2} + \frac{13}{15436492} a - \frac{33}{3859123}$, $\frac{1}{265196913219862528} a^{40} - \frac{1}{265196913219862528} a^{39} - \frac{3}{265196913219862528} a^{38} + \frac{7}{265196913219862528} a^{37} + \frac{5}{265196913219862528} a^{36} - \frac{33}{265196913219862528} a^{35} + \frac{13}{265196913219862528} a^{34} + \frac{119}{265196913219862528} a^{33} - \frac{171}{265196913219862528} a^{32} - \frac{305}{265196913219862528} a^{31} + \frac{989}{265196913219862528} a^{30} + \frac{231}{265196913219862528} a^{29} - \frac{4187}{265196913219862528} a^{28} + \frac{3263}{265196913219862528} a^{27} + \frac{13485}{265196913219862528} a^{26} - \frac{26537}{265196913219862528} a^{25} - \frac{27403}{265196913219862528} a^{24} + \frac{133551}{265196913219862528} a^{23} + \frac{20372209167}{68719476736} a^{22} + \frac{4344485277}{68719476736} a^{21} - \frac{17113845209}{68719476736} a^{20} - \frac{264095899}{68719476736} a^{19} - \frac{1}{68719476736} a^{18} + \frac{1435039}{66299228304965632} a^{17} + \frac{606021}{16574807076241408} a^{16} - \frac{510265}{4143701769060352} a^{15} - \frac{23939}{1035925442265088} a^{14} + \frac{133551}{258981360566272} a^{13} - \frac{27403}{64745340141568} a^{12} - \frac{26537}{16186335035392} a^{11} + \frac{13485}{4046583758848} a^{10} + \frac{3263}{1011645939712} a^{9} - \frac{4187}{252911484928} a^{8} + \frac{231}{63227871232} a^{7} + \frac{989}{15806967808} a^{6} - \frac{305}{3951741952} a^{5} - \frac{171}{987935488} a^{4} + \frac{119}{246983872} a^{3} + \frac{13}{61745968} a^{2} - \frac{33}{15436492} a + \frac{5}{3859123}$, $\frac{1}{1060787652879450112} a^{41} - \frac{1}{1060787652879450112} a^{40} - \frac{3}{1060787652879450112} a^{39} + \frac{7}{1060787652879450112} a^{38} + \frac{5}{1060787652879450112} a^{37} - \frac{33}{1060787652879450112} a^{36} + \frac{13}{1060787652879450112} a^{35} + \frac{119}{1060787652879450112} a^{34} - \frac{171}{1060787652879450112} a^{33} - \frac{305}{1060787652879450112} a^{32} + \frac{989}{1060787652879450112} a^{31} + \frac{231}{1060787652879450112} a^{30} - \frac{4187}{1060787652879450112} a^{29} + \frac{3263}{1060787652879450112} a^{28} + \frac{13485}{1060787652879450112} a^{27} - \frac{26537}{1060787652879450112} a^{26} - \frac{27403}{1060787652879450112} a^{25} + \frac{133551}{1060787652879450112} a^{24} - \frac{23939}{1060787652879450112} a^{23} + \frac{4344485277}{274877906944} a^{22} - \frac{85833321945}{274877906944} a^{21} + \frac{68455380837}{274877906944} a^{20} - \frac{1}{274877906944} a^{19} + \frac{1435039}{265196913219862528} a^{18} + \frac{606021}{66299228304965632} a^{17} - \frac{510265}{16574807076241408} a^{16} - \frac{23939}{4143701769060352} a^{15} + \frac{133551}{1035925442265088} a^{14} - \frac{27403}{258981360566272} a^{13} - \frac{26537}{64745340141568} a^{12} + \frac{13485}{16186335035392} a^{11} + \frac{3263}{4046583758848} a^{10} - \frac{4187}{1011645939712} a^{9} + \frac{231}{252911484928} a^{8} + \frac{989}{63227871232} a^{7} - \frac{305}{15806967808} a^{6} - \frac{171}{3951741952} a^{5} + \frac{119}{987935488} a^{4} + \frac{13}{246983872} a^{3} - \frac{33}{61745968} a^{2} + \frac{5}{15436492} a + \frac{7}{3859123}$, $\frac{1}{4243150611517800448} a^{42} - \frac{1}{4243150611517800448} a^{41} - \frac{3}{4243150611517800448} a^{40} + \frac{7}{4243150611517800448} a^{39} + \frac{5}{4243150611517800448} a^{38} - \frac{33}{4243150611517800448} a^{37} + \frac{13}{4243150611517800448} a^{36} + \frac{119}{4243150611517800448} a^{35} - \frac{171}{4243150611517800448} a^{34} - \frac{305}{4243150611517800448} a^{33} + \frac{989}{4243150611517800448} a^{32} + \frac{231}{4243150611517800448} a^{31} - \frac{4187}{4243150611517800448} a^{30} + \frac{3263}{4243150611517800448} a^{29} + \frac{13485}{4243150611517800448} a^{28} - \frac{26537}{4243150611517800448} a^{27} - \frac{27403}{4243150611517800448} a^{26} + \frac{133551}{4243150611517800448} a^{25} - \frac{23939}{4243150611517800448} a^{24} - \frac{510265}{4243150611517800448} a^{23} - \frac{360711228889}{1099511627776} a^{22} + \frac{343333287781}{1099511627776} a^{21} - \frac{1}{1099511627776} a^{20} + \frac{1435039}{1060787652879450112} a^{19} + \frac{606021}{265196913219862528} a^{18} - \frac{510265}{66299228304965632} a^{17} - \frac{23939}{16574807076241408} a^{16} + \frac{133551}{4143701769060352} a^{15} - \frac{27403}{1035925442265088} a^{14} - \frac{26537}{258981360566272} a^{13} + \frac{13485}{64745340141568} a^{12} + \frac{3263}{16186335035392} a^{11} - \frac{4187}{4046583758848} a^{10} + \frac{231}{1011645939712} a^{9} + \frac{989}{252911484928} a^{8} - \frac{305}{63227871232} a^{7} - \frac{171}{15806967808} a^{6} + \frac{119}{3951741952} a^{5} + \frac{13}{987935488} a^{4} - \frac{33}{246983872} a^{3} + \frac{5}{61745968} a^{2} + \frac{7}{15436492} a - \frac{3}{3859123}$, $\frac{1}{16972602446071201792} a^{43} - \frac{1}{16972602446071201792} a^{42} - \frac{3}{16972602446071201792} a^{41} + \frac{7}{16972602446071201792} a^{40} + \frac{5}{16972602446071201792} a^{39} - \frac{33}{16972602446071201792} a^{38} + \frac{13}{16972602446071201792} a^{37} + \frac{119}{16972602446071201792} a^{36} - \frac{171}{16972602446071201792} a^{35} - \frac{305}{16972602446071201792} a^{34} + \frac{989}{16972602446071201792} a^{33} + \frac{231}{16972602446071201792} a^{32} - \frac{4187}{16972602446071201792} a^{31} + \frac{3263}{16972602446071201792} a^{30} + \frac{13485}{16972602446071201792} a^{29} - \frac{26537}{16972602446071201792} a^{28} - \frac{27403}{16972602446071201792} a^{27} + \frac{133551}{16972602446071201792} a^{26} - \frac{23939}{16972602446071201792} a^{25} - \frac{510265}{16972602446071201792} a^{24} + \frac{606021}{16972602446071201792} a^{23} + \frac{1442844915557}{4398046511104} a^{22} - \frac{1}{4398046511104} a^{21} + \frac{1435039}{4243150611517800448} a^{20} + \frac{606021}{1060787652879450112} a^{19} - \frac{510265}{265196913219862528} a^{18} - \frac{23939}{66299228304965632} a^{17} + \frac{133551}{16574807076241408} a^{16} - \frac{27403}{4143701769060352} a^{15} - \frac{26537}{1035925442265088} a^{14} + \frac{13485}{258981360566272} a^{13} + \frac{3263}{64745340141568} a^{12} - \frac{4187}{16186335035392} a^{11} + \frac{231}{4046583758848} a^{10} + \frac{989}{1011645939712} a^{9} - \frac{305}{252911484928} a^{8} - \frac{171}{63227871232} a^{7} + \frac{119}{15806967808} a^{6} + \frac{13}{3951741952} a^{5} - \frac{33}{987935488} a^{4} + \frac{5}{246983872} a^{3} + \frac{7}{61745968} a^{2} - \frac{3}{15436492} a - \frac{1}{3859123}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $21$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{989}{16186335035392} a^{34} - \frac{6568471697}{16186335035392} a^{11} \) (order $46$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{22}$ (as 44T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 44
The 44 conjugacy class representatives for $C_2\times C_{22}$
Character table for $C_2\times C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{345}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{-23})\), \(\Q(\zeta_{23})^+\), 22.22.341419566026798986253349758444608447265625.1, \(\Q(\zeta_{23})\), 22.0.14844328957686912445797815584548193359375.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{4}$ R R $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ $22^{2}$ R $22^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{4}$ $22^{2}$ $22^{2}$ $22^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{44}$ $22^{2}$ $22^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
23Data not computed