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Properties

Label 34.34.1961958480...3537.1
Degree $34$
Signature $[34, 0]$
Discriminant $17^{17}\cdot 137^{32}$
Root discriminant $422.92$
Ramified primes $17, 137$
Class number Not computed
Class group Not computed
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3442499, 14551367854, -1939629457334, 13237346619564, -28596481432406, 2611013192693, 67476562563917, -61734986214223, -50760905017747, 88907255266315, 6114426062492, -60519652443494, 13271416308100, 23730007824939, -9647819216362, -5634679707055, 3369418132953, 775123720834, -719304361398, -43046875822, 100217214155, -4180942935, -9256973842, 1043586159, 557622095, -96252370, -20624715, 4971981, 391482, -148451, -777, 2354, -92, -15, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 15*x^33 - 92*x^32 + 2354*x^31 - 777*x^30 - 148451*x^29 + 391482*x^28 + 4971981*x^27 - 20624715*x^26 - 96252370*x^25 + 557622095*x^24 + 1043586159*x^23 - 9256973842*x^22 - 4180942935*x^21 + 100217214155*x^20 - 43046875822*x^19 - 719304361398*x^18 + 775123720834*x^17 + 3369418132953*x^16 - 5634679707055*x^15 - 9647819216362*x^14 + 23730007824939*x^13 + 13271416308100*x^12 - 60519652443494*x^11 + 6114426062492*x^10 + 88907255266315*x^9 - 50760905017747*x^8 - 61734986214223*x^7 + 67476562563917*x^6 + 2611013192693*x^5 - 28596481432406*x^4 + 13237346619564*x^3 - 1939629457334*x^2 + 14551367854*x + 3442499)
 
gp: K = bnfinit(x^34 - 15*x^33 - 92*x^32 + 2354*x^31 - 777*x^30 - 148451*x^29 + 391482*x^28 + 4971981*x^27 - 20624715*x^26 - 96252370*x^25 + 557622095*x^24 + 1043586159*x^23 - 9256973842*x^22 - 4180942935*x^21 + 100217214155*x^20 - 43046875822*x^19 - 719304361398*x^18 + 775123720834*x^17 + 3369418132953*x^16 - 5634679707055*x^15 - 9647819216362*x^14 + 23730007824939*x^13 + 13271416308100*x^12 - 60519652443494*x^11 + 6114426062492*x^10 + 88907255266315*x^9 - 50760905017747*x^8 - 61734986214223*x^7 + 67476562563917*x^6 + 2611013192693*x^5 - 28596481432406*x^4 + 13237346619564*x^3 - 1939629457334*x^2 + 14551367854*x + 3442499, 1)
 

Normalized defining polynomial

\( x^{34} - 15 x^{33} - 92 x^{32} + 2354 x^{31} - 777 x^{30} - 148451 x^{29} + 391482 x^{28} + 4971981 x^{27} - 20624715 x^{26} - 96252370 x^{25} + 557622095 x^{24} + 1043586159 x^{23} - 9256973842 x^{22} - 4180942935 x^{21} + 100217214155 x^{20} - 43046875822 x^{19} - 719304361398 x^{18} + 775123720834 x^{17} + 3369418132953 x^{16} - 5634679707055 x^{15} - 9647819216362 x^{14} + 23730007824939 x^{13} + 13271416308100 x^{12} - 60519652443494 x^{11} + 6114426062492 x^{10} + 88907255266315 x^{9} - 50760905017747 x^{8} - 61734986214223 x^{7} + 67476562563917 x^{6} + 2611013192693 x^{5} - 28596481432406 x^{4} + 13237346619564 x^{3} - 1939629457334 x^{2} + 14551367854 x + 3442499 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(196195848028617931431732592057946012741534882637292625123919185528221161924968101160363537=17^{17}\cdot 137^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $422.92$
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:  $17, 137$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2329=17\cdot 137\)
Dirichlet character group:    $\lbrace$$\chi_{2329}(256,·)$, $\chi_{2329}(1,·)$, $\chi_{2329}(1155,·)$, $\chi_{2329}(2311,·)$, $\chi_{2329}(2056,·)$, $\chi_{2329}(1293,·)$, $\chi_{2329}(526,·)$, $\chi_{2329}(16,·)$, $\chi_{2329}(1682,·)$, $\chi_{2329}(1429,·)$, $\chi_{2329}(407,·)$, $\chi_{2329}(1563,·)$, $\chi_{2329}(800,·)$, $\chi_{2329}(290,·)$, $\chi_{2329}(1444,·)$, $\chi_{2329}(681,·)$, $\chi_{2329}(171,·)$, $\chi_{2329}(1580,·)$, $\chi_{2329}(1837,·)$, $\chi_{2329}(2226,·)$, $\chi_{2329}(1716,·)$, $\chi_{2329}(1718,·)$, $\chi_{2329}(1854,·)$, $\chi_{2329}(324,·)$, $\chi_{2329}(1990,·)$, $\chi_{2329}(1869,·)$, $\chi_{2329}(1956,·)$, $\chi_{2329}(2143,·)$, $\chi_{2329}(50,·)$, $\chi_{2329}(1767,·)$, $\chi_{2329}(2177,·)$, $\chi_{2329}(1903,·)$, $\chi_{2329}(2041,·)$, $\chi_{2329}(1019,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{127} a^{27} - \frac{34}{127} a^{26} - \frac{21}{127} a^{25} + \frac{38}{127} a^{24} - \frac{6}{127} a^{23} + \frac{36}{127} a^{22} + \frac{9}{127} a^{21} + \frac{33}{127} a^{20} + \frac{18}{127} a^{19} + \frac{9}{127} a^{18} + \frac{36}{127} a^{17} + \frac{29}{127} a^{16} - \frac{29}{127} a^{15} + \frac{41}{127} a^{14} + \frac{16}{127} a^{13} - \frac{32}{127} a^{12} - \frac{60}{127} a^{11} - \frac{62}{127} a^{10} + \frac{22}{127} a^{9} + \frac{53}{127} a^{8} - \frac{5}{127} a^{7} - \frac{2}{127} a^{6} - \frac{34}{127} a^{5} - \frac{35}{127} a^{4} - \frac{27}{127} a^{3} + \frac{12}{127} a^{2} - \frac{34}{127} a + \frac{28}{127}$, $\frac{1}{4699} a^{28} - \frac{14}{4699} a^{27} - \frac{701}{4699} a^{26} - \frac{1398}{4699} a^{25} + \frac{2151}{4699} a^{24} - \frac{1227}{4699} a^{23} - \frac{922}{4699} a^{22} - \frac{1057}{4699} a^{21} + \frac{2075}{4699} a^{20} - \frac{393}{4699} a^{19} + \frac{724}{4699} a^{18} - \frac{1664}{4699} a^{17} + \frac{805}{4699} a^{16} + \frac{350}{4699} a^{15} - \frac{1958}{4699} a^{14} + \frac{669}{4699} a^{13} + \frac{62}{4699} a^{12} + \frac{1913}{4699} a^{11} - \frac{710}{4699} a^{10} - \frac{1920}{4699} a^{9} - \frac{1993}{4699} a^{8} + \frac{2311}{4699} a^{7} - \frac{455}{4699} a^{6} + \frac{936}{4699} a^{5} - \frac{1997}{4699} a^{4} - \frac{28}{127} a^{3} + \frac{1730}{4699} a^{2} - \frac{1287}{4699} a - \frac{1853}{4699}$, $\frac{1}{4699} a^{29} - \frac{9}{4699} a^{27} + \frac{887}{4699} a^{26} + \frac{1523}{4699} a^{25} + \frac{1544}{4699} a^{24} + \frac{67}{4699} a^{23} - \frac{793}{4699} a^{22} - \frac{32}{4699} a^{21} + \frac{1573}{4699} a^{20} + \frac{1808}{4699} a^{19} - \frac{2332}{4699} a^{18} + \frac{79}{4699} a^{17} - \frac{220}{4699} a^{16} + \frac{685}{4699} a^{15} + \frac{267}{4699} a^{14} + \frac{141}{4699} a^{13} - \frac{2140}{4699} a^{12} + \frac{986}{4699} a^{11} - \frac{1130}{4699} a^{10} + \frac{61}{4699} a^{9} - \frac{2022}{4699} a^{8} - \frac{735}{4699} a^{7} + \frac{2188}{4699} a^{6} - \frac{289}{4699} a^{5} + \frac{1013}{4699} a^{4} + \frac{842}{4699} a^{3} + \frac{696}{4699} a^{2} + \frac{1626}{4699} a - \frac{1078}{4699}$, $\frac{1}{192659} a^{30} + \frac{4}{192659} a^{29} + \frac{6}{192659} a^{28} + \frac{234}{192659} a^{27} + \frac{69481}{192659} a^{26} + \frac{4611}{192659} a^{25} + \frac{8945}{192659} a^{24} + \frac{2308}{192659} a^{23} - \frac{12890}{192659} a^{22} + \frac{19519}{192659} a^{21} + \frac{21095}{192659} a^{20} - \frac{83505}{192659} a^{19} + \frac{77831}{192659} a^{18} + \frac{75}{5207} a^{17} - \frac{14020}{192659} a^{16} + \frac{95244}{192659} a^{15} - \frac{49547}{192659} a^{14} - \frac{26247}{192659} a^{13} + \frac{62768}{192659} a^{12} - \frac{5158}{192659} a^{11} + \frac{14824}{192659} a^{10} - \frac{95920}{192659} a^{9} + \frac{85380}{192659} a^{8} + \frac{12453}{192659} a^{7} - \frac{44538}{192659} a^{6} - \frac{47449}{192659} a^{5} - \frac{24913}{192659} a^{4} - \frac{85069}{192659} a^{3} - \frac{2718}{192659} a^{2} - \frac{18837}{192659} a - \frac{29406}{192659}$, $\frac{1}{211346923} a^{31} + \frac{371}{211346923} a^{30} - \frac{14680}{211346923} a^{29} + \frac{12809}{211346923} a^{28} - \frac{514991}{211346923} a^{27} + \frac{45967443}{211346923} a^{26} - \frac{39120058}{211346923} a^{25} - \frac{31952245}{211346923} a^{24} - \frac{6448192}{211346923} a^{23} + \frac{65716967}{211346923} a^{22} + \frac{53509443}{211346923} a^{21} - \frac{51299558}{211346923} a^{20} + \frac{30619650}{211346923} a^{19} - \frac{101441296}{211346923} a^{18} + \frac{33851391}{211346923} a^{17} - \frac{75397691}{211346923} a^{16} + \frac{27071090}{211346923} a^{15} - \frac{101535280}{211346923} a^{14} - \frac{65342304}{211346923} a^{13} + \frac{28067220}{211346923} a^{12} - \frac{57914297}{211346923} a^{11} + \frac{75530174}{211346923} a^{10} + \frac{1404153}{5712079} a^{9} - \frac{11851548}{211346923} a^{8} - \frac{48752844}{211346923} a^{7} - \frac{78235614}{211346923} a^{6} - \frac{37097991}{211346923} a^{5} - \frac{40756320}{211346923} a^{4} + \frac{80054731}{211346923} a^{3} + \frac{26923763}{211346923} a^{2} + \frac{34228795}{211346923} a + \frac{61680746}{211346923}$, $\frac{1}{458898274030885187123794199} a^{32} - \frac{603917726091112987}{458898274030885187123794199} a^{31} + \frac{340259797942049037427}{458898274030885187123794199} a^{30} + \frac{43132682794905705581537}{458898274030885187123794199} a^{29} - \frac{10848111127054140020310}{458898274030885187123794199} a^{28} + \frac{797240860360696499112276}{458898274030885187123794199} a^{27} + \frac{174285641512248652582795454}{458898274030885187123794199} a^{26} + \frac{141823006741919408968788909}{458898274030885187123794199} a^{25} - \frac{17448607677658044303202364}{458898274030885187123794199} a^{24} - \frac{2625334129928105200425949}{458898274030885187123794199} a^{23} + \frac{5234287451406801702944326}{12402656054888788841183627} a^{22} + \frac{80942031411380032513348399}{458898274030885187123794199} a^{21} - \frac{155000008482367432483050376}{458898274030885187123794199} a^{20} + \frac{178860930385109892748991478}{458898274030885187123794199} a^{19} + \frac{39610048259579896456481400}{458898274030885187123794199} a^{18} - \frac{85686734751405785338195860}{458898274030885187123794199} a^{17} + \frac{101302999823572167393408081}{458898274030885187123794199} a^{16} - \frac{92311950848735787738761861}{458898274030885187123794199} a^{15} - \frac{119622524812752147493500740}{458898274030885187123794199} a^{14} + \frac{179408398868779147423393879}{458898274030885187123794199} a^{13} - \frac{120759675871213306346083354}{458898274030885187123794199} a^{12} + \frac{225990238969110047518999396}{458898274030885187123794199} a^{11} - \frac{145564381418518820289227779}{458898274030885187123794199} a^{10} - \frac{77907811196735200418722655}{458898274030885187123794199} a^{9} - \frac{56293415039008524236397866}{458898274030885187123794199} a^{8} - \frac{120501834248320764353387969}{458898274030885187123794199} a^{7} - \frac{204895939194068318956057617}{458898274030885187123794199} a^{6} + \frac{152526322489819640266695991}{458898274030885187123794199} a^{5} + \frac{173278277751224543639399176}{458898274030885187123794199} a^{4} - \frac{69785783695808662396533295}{458898274030885187123794199} a^{3} + \frac{72864623903314688089784391}{458898274030885187123794199} a^{2} + \frac{110972345017525582143888608}{458898274030885187123794199} a + \frac{16151544685442333319415663}{458898274030885187123794199}$, $\frac{1}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{33} - \frac{16509233601229629741228537419156373687344793837211653242851729381089657498301543573418866412263767270285294434890388501362887765}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{32} + \frac{15558963621448814161364345658748930833036600253139082249092417608553196446837618117725856779013289831851425293328977448755632869701806761340655832}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{31} + \frac{5130064481205234157662343225500197371889569417302234192808008730301107658075999203011547292226394435562517030619654553217401141135924253110943930775}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{30} - \frac{673897411198530173365399551459279715325797097991057348145300863019772373279576303009629621203139559197711807237850469639147843908513138597622913426452}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{29} - \frac{29210364411199346807035120843533483687959981657544425379582333951042723820857112684316400944188687706474912266940345695177902750128457747202711525985}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{28} + \frac{8060747360085098645704557509928267382161170596743816320587804076362890759204035861319675430315644819105502230074478519777371948674558981314313435909393}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{27} - \frac{6306737368989626124910089605354053744312981386119119094749374509312973533552580864677278789080235926357147104017719531003244295297984063532068342335108360}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{26} + \frac{7116032683482109991230016599962260939582788665831399927657882557409796889511918720213169136230009053248174608012629967643397442292376943894004554971671813}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{25} - \frac{4110692807450362021653787892419221601206847783829023969468434344615924603709670430499568105844888389041009433327702628192382447955429639378671871767637333}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{24} - \frac{2452445527628883947262573083997504904824456266642754023800726776640872937311332870094491163668491516584087372148787352843339560176445771745905112941985414}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{23} + \frac{2168816950276523006830351602738581970815752304367486410941152818216431636270491891058779901901161879104387181030852889779448370204746250456241464954483620}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{22} - \frac{2671189927657622311327565360534369627456946348861367656716673288582223279495576061202427370961427385596075999924239402912075844579425342793965812824759413}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{21} + \frac{10575485402235889722361934546000623261222934733310476848803322188758780434469925995829728765616835955619657649562428784387420691111719850834437554739473610}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{20} + \frac{2796261607572327247543464339391190898923636430539582785159162792095138219866097532143952773465385033828688496265263207383466112920737412732199563030127881}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{19} + \frac{5387341167303758424400918226352101573575981386781256180454581750874430888160277557062003982047609707804870166597538865994791928798783567742761619695760201}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{18} + \frac{10513123494729514373963495782121568306648460357304383208948300528355039549698779837845937865969481084382614912642149819726388578506696749635626920973528291}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{17} + \frac{3120801778243833801865844992326072517439185100135903488108856426821422044925009340143429152664186634685098404455927193716487707521923449205422519488604192}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{16} - \frac{4179695250842998707633022131835582235763796137397883364695601889046233759461353709414322493546980609266408862689019510844790783220425081855692811680985548}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{15} - \frac{9883759314756985047431475023699281955430092577726426799299256565033696292873359092778290203561926427226110658192704017608696019462289547183250417603052658}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{14} - \frac{824255750476646084637335613000202884874305594107193032869999703331747821827405136265330973626322835844235801843846701809838245062378898465842412301462431}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{13} - \frac{2681777840147269759563921232777606469965341714992243210008353685177581348863522009033082998543014117516225733520256411348537953455354424369268873660014540}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{12} - \frac{27832189046377841189686589325585675325087263247516097545739743917317080800213545955888284280379981571883023598225489517763220108640339036414333143562183}{520354133280761747345422490972556192081100564199811509218017992323865165527701020950033595719040268488157828845920651243478947829641819904773112819646109} a^{11} - \frac{85431950285661046869464562250279099457523751306010343860927895036957925468186187034643797940561183823832119559584559406668543749040363006101210010395450}{576608634175979233544927625131751456089868192761953293998344261764283021260425455647334524985963540757147864396830991918449644892305800435018854746094337} a^{10} + \frac{7539551033374353740713413553907834943746956521447225080466310097953225906823398456607840099841389128683014832888149511757567248693029691942306591613857760}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{9} - \frac{2405058892533336703852920468138537562276029864119052300789374041554728883714443018053173113520595830906186389495483576662414071366996304447013680574377433}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{8} - \frac{1927726144323821687624733312252933170434753766663159796174336422568377487631793495284605873977462692950321200894976243523086361653552060623556227442915996}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{7} - \frac{4262125829243012327967673994042701758708209371512212771296604271700991756413039856850921764677760388521110120894504883032383511242812216216597415748154997}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{6} + \frac{1825665329135442521217600591000021180234209421807191102505038897777791518730118251628204303865673339091986254665334430426362633849657123523567054672804681}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{5} - \frac{3327244993873892516142526442173190757306968105225665480249398724498624768585016104136669788714971804489715659276481234143567667875927217864255633441802082}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{4} - \frac{7009827435392626205894533262027196407868186679144751068829963049497420904364730055809267111600492155660700299641639570400343480914016597695544647591221886}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{3} - \frac{10592554220562583528032984139903204948348552950601538245856839355040451210508182968770310585081835412642992998323894602190329185205127738138658333597042025}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a^{2} - \frac{9349174116383778730908411213728600621646484022651997114532756975103821134965757659424440637041938635451856912034048904592291510645947359891662880447428776}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469} a - \frac{3419509352014390621571360492256226066308742576220786155494471145603280673097881885806561277322609811519353430383359095420958924652706648129728238049201079}{21334519464511231641162322129874803875325123132192271877938737685278471786635741858951377424480651008014470982682746700982636861015314616095697625605490469}$

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:  $33$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 17.17.15400296222263289476715621650663041.1

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 $17^{2}$ $34$ $34$ $34$ $34$ $17^{2}$ R $17^{2}$ $34$ $34$ $34$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{17}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{17}$ $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$

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
17Data not computed
137Data not computed