LMFDB - Number field 26.0.105838418476275527898387851073846090273368671.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675)

gp: K = bnfinit(y^26 - 4*y^25 + 29*y^24 - 114*y^23 + 437*y^22 - 1233*y^21 + 3738*y^20 - 6943*y^19 + 17106*y^18 - 22984*y^17 + 59870*y^16 - 97147*y^15 + 263098*y^14 - 450753*y^13 + 813670*y^12 - 1048444*y^11 + 1103470*y^10 - 985605*y^9 + 369592*y^8 - 235197*y^7 - 276734*y^6 + 186178*y^5 - 88862*y^4 + 240404*y^3 + 178609*y^2 + 96160*y + 61675, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675)

$ x^{26} - 4 x^{25} + 29 x^{24} - 114 x^{23} + 437 x^{22} - 1233 x^{21} + 3738 x^{20} - 6943 x^{19} + \cdots + 61675 $ x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$26$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 13]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$-105838418476275527898387851073846090273368671$ -105838418476275527898387851073846090273368671 $\medspace = -\,31^{13}\cdot 113^{12}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$49.35$	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:	$31^{1/2}113^{1/2}\approx 59.18614702783076$
Ramified primes:	$31$, $113$ 31, 113	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{-31}) $
$\card{ \Aut(K/\Q) }$:	$2$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{12}+\frac{2}{5}a^{11}-\frac{2}{5}a^{10}-\frac{1}{5}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{15}+\frac{1}{5}a^{12}+\frac{1}{5}a^{11}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{11}+\frac{1}{5}a^{10}+\frac{2}{5}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{12}+\frac{1}{5}a^{11}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{5}a^{18}-\frac{2}{5}a^{12}-\frac{1}{5}a^{11}-\frac{2}{5}a^{9}+\frac{2}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{25}a^{19}-\frac{2}{25}a^{18}-\frac{1}{25}a^{16}-\frac{1}{25}a^{14}-\frac{6}{25}a^{12}+\frac{6}{25}a^{11}+\frac{7}{25}a^{10}-\frac{6}{25}a^{9}-\frac{12}{25}a^{8}-\frac{1}{25}a^{7}-\frac{2}{25}a^{6}+\frac{4}{25}a^{5}-\frac{12}{25}a^{4}+\frac{2}{25}a^{3}+\frac{7}{25}a^{2}+\frac{1}{5}a$, $\frac{1}{25}a^{20}+\frac{1}{25}a^{18}-\frac{1}{25}a^{17}-\frac{2}{25}a^{16}-\frac{1}{25}a^{15}-\frac{2}{25}a^{14}-\frac{1}{25}a^{13}-\frac{11}{25}a^{12}-\frac{1}{25}a^{11}+\frac{3}{25}a^{10}+\frac{1}{25}a^{9}+\frac{1}{5}a^{8}+\frac{11}{25}a^{7}+\frac{1}{5}a^{6}+\frac{6}{25}a^{5}-\frac{12}{25}a^{4}-\frac{4}{25}a^{3}-\frac{1}{25}a^{2}$, $\frac{1}{775}a^{21}-\frac{7}{775}a^{20}-\frac{4}{775}a^{19}-\frac{58}{775}a^{18}-\frac{6}{155}a^{17}+\frac{3}{775}a^{16}+\frac{38}{775}a^{14}+\frac{71}{775}a^{13}-\frac{164}{775}a^{12}-\frac{11}{155}a^{11}-\frac{57}{155}a^{10}+\frac{233}{775}a^{9}+\frac{221}{775}a^{8}+\frac{213}{775}a^{7}-\frac{144}{775}a^{6}+\frac{101}{775}a^{5}+\frac{21}{155}a^{4}+\frac{202}{775}a^{3}+\frac{282}{775}a^{2}+\frac{21}{155}a+\frac{14}{31}$, $\frac{1}{775}a^{22}+\frac{9}{775}a^{20}+\frac{7}{775}a^{19}+\frac{12}{155}a^{18}+\frac{41}{775}a^{17}-\frac{41}{775}a^{16}-\frac{24}{775}a^{15}-\frac{7}{155}a^{14}-\frac{39}{775}a^{13}-\frac{118}{775}a^{12}-\frac{174}{775}a^{11}-\frac{6}{31}a^{10}+\frac{271}{775}a^{9}+\frac{334}{775}a^{8}-\frac{234}{775}a^{7}-\frac{318}{775}a^{6}-\frac{304}{775}a^{5}+\frac{162}{775}a^{4}+\frac{239}{775}a^{3}+\frac{33}{775}a^{2}+\frac{1}{5}a+\frac{5}{31}$, $\frac{1}{275125}a^{23}-\frac{3}{8875}a^{22}+\frac{164}{275125}a^{21}-\frac{4116}{275125}a^{20}+\frac{2509}{275125}a^{19}-\frac{1269}{55025}a^{18}-\frac{5063}{275125}a^{17}-\frac{179}{275125}a^{16}+\frac{14938}{275125}a^{15}+\frac{8858}{275125}a^{14}-\frac{4159}{55025}a^{13}+\frac{13466}{275125}a^{12}+\frac{29858}{275125}a^{11}-\frac{25087}{275125}a^{10}-\frac{4112}{55025}a^{9}+\frac{65579}{275125}a^{8}-\frac{44462}{275125}a^{7}-\frac{9708}{55025}a^{6}+\frac{120473}{275125}a^{5}+\frac{642}{2201}a^{4}+\frac{14496}{55025}a^{3}-\frac{1503}{8875}a^{2}+\frac{23926}{55025}a-\frac{154}{355}$, $\frac{1}{89415625}a^{24}+\frac{3}{17883125}a^{23}-\frac{6449}{17883125}a^{22}+\frac{13596}{89415625}a^{21}-\frac{1699784}{89415625}a^{20}-\frac{1520668}{89415625}a^{19}-\frac{3627948}{89415625}a^{18}-\frac{3323793}{89415625}a^{17}+\frac{53267}{6878125}a^{16}+\frac{1234502}{89415625}a^{15}-\frac{4389131}{89415625}a^{14}+\frac{2832746}{89415625}a^{13}-\frac{30003249}{89415625}a^{12}-\frac{3550748}{89415625}a^{11}+\frac{22590774}{89415625}a^{10}-\frac{36366606}{89415625}a^{9}-\frac{273134}{1375625}a^{8}+\frac{2359}{40625}a^{7}-\frac{11699997}{89415625}a^{6}+\frac{42857729}{89415625}a^{5}-\frac{8071262}{17883125}a^{4}+\frac{25304402}{89415625}a^{3}+\frac{2045707}{6878125}a^{2}+\frac{2670188}{17883125}a-\frac{946562}{3576625}$, $\frac{1}{78\!\cdots\!25}a^{25}-\frac{19\!\cdots\!82}{78\!\cdots\!25}a^{24}+\frac{19\!\cdots\!27}{31\!\cdots\!25}a^{23}+\frac{13\!\cdots\!01}{71\!\cdots\!75}a^{22}+\frac{40\!\cdots\!29}{78\!\cdots\!25}a^{21}+\frac{53\!\cdots\!16}{15\!\cdots\!25}a^{20}-\frac{55\!\cdots\!52}{78\!\cdots\!25}a^{19}+\frac{47\!\cdots\!88}{78\!\cdots\!25}a^{18}+\frac{11\!\cdots\!67}{78\!\cdots\!25}a^{17}+\frac{14\!\cdots\!53}{15\!\cdots\!25}a^{16}+\frac{26\!\cdots\!03}{12\!\cdots\!75}a^{15}+\frac{40\!\cdots\!06}{60\!\cdots\!25}a^{14}-\frac{65\!\cdots\!61}{78\!\cdots\!25}a^{13}-\frac{28\!\cdots\!79}{15\!\cdots\!25}a^{12}-\frac{35\!\cdots\!29}{15\!\cdots\!25}a^{11}-\frac{52\!\cdots\!34}{78\!\cdots\!25}a^{10}-\frac{33\!\cdots\!78}{78\!\cdots\!25}a^{9}+\frac{21\!\cdots\!79}{78\!\cdots\!25}a^{8}-\frac{11\!\cdots\!42}{33\!\cdots\!75}a^{7}-\frac{17\!\cdots\!12}{78\!\cdots\!25}a^{6}+\frac{18\!\cdots\!77}{78\!\cdots\!25}a^{5}+\frac{20\!\cdots\!29}{54\!\cdots\!75}a^{4}+\frac{16\!\cdots\!47}{78\!\cdots\!25}a^{3}-\frac{22\!\cdots\!67}{71\!\cdots\!75}a^{2}-\frac{50\!\cdots\!96}{15\!\cdots\!25}a+\frac{14\!\cdots\!89}{31\!\cdots\!25}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, 1/5*a^13 + 1/5*a^12 + 2/5*a^11 - 1/5*a^10 + 2/5*a^9 + 1/5*a^8 - 2/5*a^7 + 1/5*a^6 + 2/5*a^5 + 1/5*a^3 + 2/5*a^2 + 1/5*a, 1/5*a^14 + 1/5*a^12 + 2/5*a^11 - 2/5*a^10 - 1/5*a^9 + 2/5*a^8 - 2/5*a^7 + 1/5*a^6 - 2/5*a^5 + 1/5*a^4 + 1/5*a^3 - 1/5*a^2 - 1/5*a, 1/5*a^15 + 1/5*a^12 + 1/5*a^11 + 2/5*a^8 - 2/5*a^7 + 2/5*a^6 - 1/5*a^5 + 1/5*a^4 - 2/5*a^3 + 2/5*a^2 - 1/5*a, 1/5*a^16 - 2/5*a^11 + 1/5*a^10 + 2/5*a^8 - 1/5*a^7 - 2/5*a^6 - 1/5*a^5 - 2/5*a^4 + 1/5*a^3 + 2/5*a^2 - 1/5*a, 1/5*a^17 - 2/5*a^12 + 1/5*a^11 + 2/5*a^9 - 1/5*a^8 - 2/5*a^7 - 1/5*a^6 - 2/5*a^5 + 1/5*a^4 + 2/5*a^3 - 1/5*a^2, 1/5*a^18 - 2/5*a^12 - 1/5*a^11 - 2/5*a^9 + 2/5*a^4 + 1/5*a^3 - 1/5*a^2 + 2/5*a, 1/25*a^19 - 2/25*a^18 - 1/25*a^16 - 1/25*a^14 - 6/25*a^12 + 6/25*a^11 + 7/25*a^10 - 6/25*a^9 - 12/25*a^8 - 1/25*a^7 - 2/25*a^6 + 4/25*a^5 - 12/25*a^4 + 2/25*a^3 + 7/25*a^2 + 1/5*a, 1/25*a^20 + 1/25*a^18 - 1/25*a^17 - 2/25*a^16 - 1/25*a^15 - 2/25*a^14 - 1/25*a^13 - 11/25*a^12 - 1/25*a^11 + 3/25*a^10 + 1/25*a^9 + 1/5*a^8 + 11/25*a^7 + 1/5*a^6 + 6/25*a^5 - 12/25*a^4 - 4/25*a^3 - 1/25*a^2, 1/775*a^21 - 7/775*a^20 - 4/775*a^19 - 58/775*a^18 - 6/155*a^17 + 3/775*a^16 + 38/775*a^14 + 71/775*a^13 - 164/775*a^12 - 11/155*a^11 - 57/155*a^10 + 233/775*a^9 + 221/775*a^8 + 213/775*a^7 - 144/775*a^6 + 101/775*a^5 + 21/155*a^4 + 202/775*a^3 + 282/775*a^2 + 21/155*a + 14/31, 1/775*a^22 + 9/775*a^20 + 7/775*a^19 + 12/155*a^18 + 41/775*a^17 - 41/775*a^16 - 24/775*a^15 - 7/155*a^14 - 39/775*a^13 - 118/775*a^12 - 174/775*a^11 - 6/31*a^10 + 271/775*a^9 + 334/775*a^8 - 234/775*a^7 - 318/775*a^6 - 304/775*a^5 + 162/775*a^4 + 239/775*a^3 + 33/775*a^2 + 1/5*a + 5/31, 1/275125*a^23 - 3/8875*a^22 + 164/275125*a^21 - 4116/275125*a^20 + 2509/275125*a^19 - 1269/55025*a^18 - 5063/275125*a^17 - 179/275125*a^16 + 14938/275125*a^15 + 8858/275125*a^14 - 4159/55025*a^13 + 13466/275125*a^12 + 29858/275125*a^11 - 25087/275125*a^10 - 4112/55025*a^9 + 65579/275125*a^8 - 44462/275125*a^7 - 9708/55025*a^6 + 120473/275125*a^5 + 642/2201*a^4 + 14496/55025*a^3 - 1503/8875*a^2 + 23926/55025*a - 154/355, 1/89415625*a^24 + 3/17883125*a^23 - 6449/17883125*a^22 + 13596/89415625*a^21 - 1699784/89415625*a^20 - 1520668/89415625*a^19 - 3627948/89415625*a^18 - 3323793/89415625*a^17 + 53267/6878125*a^16 + 1234502/89415625*a^15 - 4389131/89415625*a^14 + 2832746/89415625*a^13 - 30003249/89415625*a^12 - 3550748/89415625*a^11 + 22590774/89415625*a^10 - 36366606/89415625*a^9 - 273134/1375625*a^8 + 2359/40625*a^7 - 11699997/89415625*a^6 + 42857729/89415625*a^5 - 8071262/17883125*a^4 + 25304402/89415625*a^3 + 2045707/6878125*a^2 + 2670188/17883125*a - 946562/3576625, 1/784803989526277132786021644580620722639830203197216328125*a^25 - 1911911312047144992295631852576040423383802275882/784803989526277132786021644580620722639830203197216328125*a^24 + 1929019010572000732413235578865627817065408621927/31392159581051085311440865783224828905593208127888653125*a^23 + 13207212660022549983008151462516478175435236024305601/71345817229661557526001967689147338421802745745201484375*a^22 + 409600020839943461908499992025992783225049167746062229/784803989526277132786021644580620722639830203197216328125*a^21 + 537998017938366206814935128476675633405607435163700216/156960797905255426557204328916124144527966040639443265625*a^20 - 5508541992230012527326221822343359664904213935282858252/784803989526277132786021644580620722639830203197216328125*a^19 + 47464683718048354185407806626218488539520945045238320188/784803989526277132786021644580620722639830203197216328125*a^18 + 11629471588819314147221701428272628271801730408222367667/784803989526277132786021644580620722639830203197216328125*a^17 + 14700821535716782153600492734720277559568431231924101953/156960797905255426557204328916124144527966040639443265625*a^16 + 268425601927097551721194040979962996209836280194803/12602231867142145849635032430038068609230513098309375*a^15 + 4069326205596102970421309979845004944331122744312357106/60369537655867471752770895736970824818448477169016640625*a^14 - 65353893339408606893419174849902199049378779653047409161/784803989526277132786021644580620722639830203197216328125*a^13 - 28304737764432365219540515200030681633826084552891487979/156960797905255426557204328916124144527966040639443265625*a^12 - 35553757831050772194767060401241444056373470265326178829/156960797905255426557204328916124144527966040639443265625*a^11 - 52696841442702763943069985113940380878982569493991704134/784803989526277132786021644580620722639830203197216328125*a^10 - 331838192680291201375569140586598254507167098869904784378/784803989526277132786021644580620722639830203197216328125*a^9 + 211733458476571929778996661366737543355306310432483818279/784803989526277132786021644580620722639830203197216328125*a^8 - 1186144081297168640994429679467757530621440528508683742/3339591444792668650153283593960088181446085971051984375*a^7 - 177526463346251054096029912069689836238519896409940937712/784803989526277132786021644580620722639830203197216328125*a^6 + 18895037870411059097110472139404923648727560830308608777/784803989526277132786021644580620722639830203197216328125*a^5 + 2052129287367135637235133347418720219476917403873336429/5488139786897042886615535976088256801677134288092421875*a^4 + 168034406916441353007903694997153811211365207350403088847/784803989526277132786021644580620722639830203197216328125*a^3 - 22467195984712454168751648014369880751620356188411921967/71345817229661557526001967689147338421802745745201484375*a^2 - 50208697795280718145822003294281017408146219635552556196/156960797905255426557204328916124144527966040639443265625*a + 14925255832939004311646284809493023243617975482303495489/31392159581051085311440865783224828905593208127888653125

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

Class group and class number

$C_{3}$, which has order $3$ (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:	$12$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -1 (order $2$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	$\frac{25\!\cdots\!39}{78\!\cdots\!25}a^{25}-\frac{14\!\cdots\!73}{78\!\cdots\!25}a^{24}+\frac{34\!\cdots\!48}{31\!\cdots\!25}a^{23}-\frac{35\!\cdots\!11}{71\!\cdots\!75}a^{22}+\frac{14\!\cdots\!06}{78\!\cdots\!25}a^{21}-\frac{28\!\cdots\!31}{50\!\cdots\!75}a^{20}+\frac{12\!\cdots\!22}{78\!\cdots\!25}a^{19}-\frac{27\!\cdots\!93}{78\!\cdots\!25}a^{18}+\frac{52\!\cdots\!63}{78\!\cdots\!25}a^{17}-\frac{19\!\cdots\!18}{15\!\cdots\!25}a^{16}+\frac{26\!\cdots\!93}{12\!\cdots\!75}a^{15}-\frac{39\!\cdots\!83}{78\!\cdots\!25}a^{14}+\frac{63\!\cdots\!17}{60\!\cdots\!25}a^{13}-\frac{34\!\cdots\!16}{15\!\cdots\!25}a^{12}+\frac{54\!\cdots\!49}{15\!\cdots\!25}a^{11}-\frac{36\!\cdots\!76}{78\!\cdots\!25}a^{10}+\frac{36\!\cdots\!33}{78\!\cdots\!25}a^{9}-\frac{37\!\cdots\!64}{11\!\cdots\!75}a^{8}+\frac{78\!\cdots\!17}{33\!\cdots\!75}a^{7}-\frac{20\!\cdots\!68}{78\!\cdots\!25}a^{6}+\frac{32\!\cdots\!03}{78\!\cdots\!25}a^{5}+\frac{19\!\cdots\!28}{71\!\cdots\!75}a^{4}-\frac{62\!\cdots\!17}{78\!\cdots\!25}a^{3}-\frac{25\!\cdots\!13}{71\!\cdots\!75}a^{2}-\frac{55\!\cdots\!94}{15\!\cdots\!25}a-\frac{25\!\cdots\!54}{31\!\cdots\!25}$, $\frac{54\!\cdots\!96}{78\!\cdots\!25}a^{25}-\frac{15\!\cdots\!72}{78\!\cdots\!25}a^{24}+\frac{51\!\cdots\!82}{31\!\cdots\!25}a^{23}-\frac{12\!\cdots\!09}{23\!\cdots\!25}a^{22}+\frac{15\!\cdots\!84}{78\!\cdots\!25}a^{21}-\frac{67\!\cdots\!09}{15\!\cdots\!25}a^{20}+\frac{10\!\cdots\!08}{78\!\cdots\!25}a^{19}-\frac{85\!\cdots\!02}{78\!\cdots\!25}a^{18}+\frac{33\!\cdots\!57}{78\!\cdots\!25}a^{17}+\frac{23\!\cdots\!43}{15\!\cdots\!25}a^{16}+\frac{74\!\cdots\!18}{50\!\cdots\!75}a^{15}-\frac{63\!\cdots\!37}{78\!\cdots\!25}a^{14}+\frac{45\!\cdots\!13}{60\!\cdots\!25}a^{13}-\frac{80\!\cdots\!04}{15\!\cdots\!25}a^{12}+\frac{10\!\cdots\!51}{15\!\cdots\!25}a^{11}+\frac{11\!\cdots\!86}{78\!\cdots\!25}a^{10}-\frac{35\!\cdots\!88}{78\!\cdots\!25}a^{9}+\frac{47\!\cdots\!84}{78\!\cdots\!25}a^{8}-\frac{28\!\cdots\!22}{33\!\cdots\!75}a^{7}+\frac{23\!\cdots\!73}{78\!\cdots\!25}a^{6}-\frac{18\!\cdots\!83}{78\!\cdots\!25}a^{5}-\frac{13\!\cdots\!58}{71\!\cdots\!75}a^{4}+\frac{21\!\cdots\!87}{78\!\cdots\!25}a^{3}+\frac{56\!\cdots\!18}{71\!\cdots\!75}a^{2}+\frac{41\!\cdots\!84}{15\!\cdots\!25}a+\frac{48\!\cdots\!19}{31\!\cdots\!25}$, $\frac{55\!\cdots\!17}{25\!\cdots\!25}a^{25}-\frac{35\!\cdots\!24}{25\!\cdots\!25}a^{24}+\frac{79\!\cdots\!41}{10\!\cdots\!25}a^{23}-\frac{85\!\cdots\!83}{23\!\cdots\!75}a^{22}+\frac{34\!\cdots\!13}{25\!\cdots\!25}a^{21}-\frac{21\!\cdots\!64}{51\!\cdots\!25}a^{20}+\frac{30\!\cdots\!56}{25\!\cdots\!25}a^{19}-\frac{68\!\cdots\!39}{25\!\cdots\!25}a^{18}+\frac{13\!\cdots\!79}{25\!\cdots\!25}a^{17}-\frac{31\!\cdots\!78}{33\!\cdots\!75}a^{16}+\frac{33\!\cdots\!93}{20\!\cdots\!75}a^{15}-\frac{13\!\cdots\!39}{36\!\cdots\!75}a^{14}+\frac{20\!\cdots\!08}{25\!\cdots\!25}a^{13}-\frac{88\!\cdots\!14}{51\!\cdots\!25}a^{12}+\frac{27\!\cdots\!83}{10\!\cdots\!25}a^{11}-\frac{96\!\cdots\!23}{25\!\cdots\!25}a^{10}+\frac{97\!\cdots\!29}{25\!\cdots\!25}a^{9}-\frac{72\!\cdots\!07}{25\!\cdots\!25}a^{8}+\frac{15\!\cdots\!73}{84\!\cdots\!75}a^{7}-\frac{25\!\cdots\!44}{25\!\cdots\!25}a^{6}+\frac{42\!\cdots\!89}{25\!\cdots\!25}a^{5}+\frac{13\!\cdots\!59}{23\!\cdots\!75}a^{4}-\frac{18\!\cdots\!11}{25\!\cdots\!25}a^{3}-\frac{75\!\cdots\!94}{23\!\cdots\!75}a^{2}-\frac{18\!\cdots\!72}{51\!\cdots\!25}a-\frac{21\!\cdots\!27}{10\!\cdots\!25}$, $\frac{27\!\cdots\!51}{71\!\cdots\!75}a^{25}-\frac{12\!\cdots\!32}{71\!\cdots\!75}a^{24}+\frac{33\!\cdots\!17}{28\!\cdots\!75}a^{23}-\frac{34\!\cdots\!39}{71\!\cdots\!75}a^{22}+\frac{13\!\cdots\!79}{71\!\cdots\!75}a^{21}-\frac{75\!\cdots\!79}{14\!\cdots\!75}a^{20}+\frac{11\!\cdots\!73}{71\!\cdots\!75}a^{19}-\frac{21\!\cdots\!62}{71\!\cdots\!75}a^{18}+\frac{48\!\cdots\!67}{71\!\cdots\!75}a^{17}-\frac{12\!\cdots\!17}{14\!\cdots\!75}a^{16}+\frac{95\!\cdots\!93}{45\!\cdots\!25}a^{15}-\frac{25\!\cdots\!22}{71\!\cdots\!75}a^{14}+\frac{22\!\cdots\!44}{23\!\cdots\!25}a^{13}-\frac{26\!\cdots\!49}{14\!\cdots\!75}a^{12}+\frac{44\!\cdots\!06}{14\!\cdots\!75}a^{11}-\frac{25\!\cdots\!59}{71\!\cdots\!75}a^{10}+\frac{14\!\cdots\!22}{71\!\cdots\!75}a^{9}+\frac{56\!\cdots\!34}{23\!\cdots\!25}a^{8}-\frac{13\!\cdots\!07}{30\!\cdots\!25}a^{7}+\frac{30\!\cdots\!63}{71\!\cdots\!75}a^{6}-\frac{27\!\cdots\!98}{71\!\cdots\!75}a^{5}+\frac{11\!\cdots\!72}{71\!\cdots\!75}a^{4}+\frac{39\!\cdots\!22}{71\!\cdots\!75}a^{3}+\frac{98\!\cdots\!13}{71\!\cdots\!75}a^{2}+\frac{14\!\cdots\!04}{14\!\cdots\!75}a+\frac{64\!\cdots\!64}{28\!\cdots\!75}$, $\frac{18\!\cdots\!31}{25\!\cdots\!75}a^{25}-\frac{27\!\cdots\!02}{78\!\cdots\!25}a^{24}+\frac{73\!\cdots\!77}{31\!\cdots\!25}a^{23}-\frac{70\!\cdots\!39}{71\!\cdots\!75}a^{22}+\frac{29\!\cdots\!44}{78\!\cdots\!25}a^{21}-\frac{17\!\cdots\!14}{15\!\cdots\!25}a^{20}+\frac{25\!\cdots\!78}{78\!\cdots\!25}a^{19}-\frac{52\!\cdots\!32}{78\!\cdots\!25}a^{18}+\frac{11\!\cdots\!12}{78\!\cdots\!25}a^{17}-\frac{37\!\cdots\!82}{15\!\cdots\!25}a^{16}+\frac{64\!\cdots\!12}{12\!\cdots\!75}a^{15}-\frac{74\!\cdots\!42}{78\!\cdots\!25}a^{14}+\frac{18\!\cdots\!54}{78\!\cdots\!25}a^{13}-\frac{69\!\cdots\!59}{15\!\cdots\!25}a^{12}+\frac{92\!\cdots\!02}{12\!\cdots\!25}a^{11}-\frac{82\!\cdots\!74}{78\!\cdots\!25}a^{10}+\frac{86\!\cdots\!17}{78\!\cdots\!25}a^{9}-\frac{73\!\cdots\!31}{78\!\cdots\!25}a^{8}+\frac{13\!\cdots\!33}{33\!\cdots\!75}a^{7}-\frac{51\!\cdots\!07}{78\!\cdots\!25}a^{6}-\frac{13\!\cdots\!06}{60\!\cdots\!25}a^{5}+\frac{18\!\cdots\!97}{71\!\cdots\!75}a^{4}-\frac{73\!\cdots\!08}{78\!\cdots\!25}a^{3}+\frac{27\!\cdots\!98}{23\!\cdots\!25}a^{2}+\frac{60\!\cdots\!63}{12\!\cdots\!25}a-\frac{41\!\cdots\!96}{31\!\cdots\!25}$, $\frac{66\!\cdots\!33}{78\!\cdots\!25}a^{25}-\frac{32\!\cdots\!81}{78\!\cdots\!25}a^{24}+\frac{79\!\cdots\!66}{31\!\cdots\!25}a^{23}-\frac{78\!\cdots\!42}{71\!\cdots\!75}a^{22}+\frac{10\!\cdots\!72}{25\!\cdots\!75}a^{21}-\frac{18\!\cdots\!22}{15\!\cdots\!25}a^{20}+\frac{25\!\cdots\!84}{78\!\cdots\!25}a^{19}-\frac{39\!\cdots\!67}{60\!\cdots\!25}a^{18}+\frac{10\!\cdots\!11}{78\!\cdots\!25}a^{17}-\frac{33\!\cdots\!51}{15\!\cdots\!25}a^{16}+\frac{41\!\cdots\!58}{96\!\cdots\!75}a^{15}-\frac{73\!\cdots\!51}{78\!\cdots\!25}a^{14}+\frac{16\!\cdots\!12}{78\!\cdots\!25}a^{13}-\frac{65\!\cdots\!57}{15\!\cdots\!25}a^{12}+\frac{97\!\cdots\!68}{15\!\cdots\!25}a^{11}-\frac{15\!\cdots\!99}{19\!\cdots\!75}a^{10}+\frac{51\!\cdots\!01}{78\!\cdots\!25}a^{9}-\frac{33\!\cdots\!68}{78\!\cdots\!25}a^{8}+\frac{55\!\cdots\!64}{33\!\cdots\!75}a^{7}+\frac{84\!\cdots\!29}{78\!\cdots\!25}a^{6}-\frac{20\!\cdots\!09}{78\!\cdots\!25}a^{5}+\frac{12\!\cdots\!41}{71\!\cdots\!75}a^{4}-\frac{69\!\cdots\!99}{78\!\cdots\!25}a^{3}-\frac{55\!\cdots\!86}{71\!\cdots\!75}a^{2}-\frac{10\!\cdots\!43}{15\!\cdots\!25}a-\frac{14\!\cdots\!88}{31\!\cdots\!25}$, $\frac{33\!\cdots\!76}{78\!\cdots\!25}a^{25}-\frac{69\!\cdots\!07}{78\!\cdots\!25}a^{24}+\frac{27\!\cdots\!17}{31\!\cdots\!25}a^{23}-\frac{17\!\cdots\!24}{71\!\cdots\!75}a^{22}+\frac{68\!\cdots\!04}{78\!\cdots\!25}a^{21}-\frac{24\!\cdots\!54}{15\!\cdots\!25}a^{20}+\frac{13\!\cdots\!58}{25\!\cdots\!75}a^{19}+\frac{12\!\cdots\!88}{78\!\cdots\!25}a^{18}+\frac{10\!\cdots\!92}{78\!\cdots\!25}a^{17}+\frac{61\!\cdots\!58}{15\!\cdots\!25}a^{16}+\frac{13\!\cdots\!48}{25\!\cdots\!75}a^{15}+\frac{41\!\cdots\!53}{78\!\cdots\!25}a^{14}+\frac{19\!\cdots\!64}{78\!\cdots\!25}a^{13}+\frac{29\!\cdots\!01}{15\!\cdots\!25}a^{12}-\frac{69\!\cdots\!94}{15\!\cdots\!25}a^{11}+\frac{16\!\cdots\!91}{78\!\cdots\!25}a^{10}-\frac{31\!\cdots\!03}{78\!\cdots\!25}a^{9}+\frac{35\!\cdots\!04}{78\!\cdots\!25}a^{8}-\frac{19\!\cdots\!07}{33\!\cdots\!75}a^{7}+\frac{17\!\cdots\!13}{78\!\cdots\!25}a^{6}-\frac{14\!\cdots\!48}{78\!\cdots\!25}a^{5}-\frac{19\!\cdots\!73}{71\!\cdots\!75}a^{4}+\frac{10\!\cdots\!47}{78\!\cdots\!25}a^{3}+\frac{66\!\cdots\!83}{71\!\cdots\!75}a^{2}+\frac{17\!\cdots\!79}{15\!\cdots\!25}a+\frac{24\!\cdots\!64}{31\!\cdots\!25}$, $\frac{14\!\cdots\!06}{78\!\cdots\!25}a^{25}-\frac{47\!\cdots\!92}{78\!\cdots\!25}a^{24}+\frac{11\!\cdots\!64}{24\!\cdots\!25}a^{23}-\frac{12\!\cdots\!94}{71\!\cdots\!75}a^{22}+\frac{38\!\cdots\!48}{60\!\cdots\!25}a^{21}-\frac{19\!\cdots\!78}{12\!\cdots\!25}a^{20}+\frac{39\!\cdots\!88}{78\!\cdots\!25}a^{19}-\frac{56\!\cdots\!72}{78\!\cdots\!25}a^{18}+\frac{15\!\cdots\!52}{78\!\cdots\!25}a^{17}-\frac{25\!\cdots\!42}{15\!\cdots\!25}a^{16}+\frac{88\!\cdots\!69}{12\!\cdots\!75}a^{15}-\frac{70\!\cdots\!32}{78\!\cdots\!25}a^{14}+\frac{25\!\cdots\!09}{78\!\cdots\!25}a^{13}-\frac{68\!\cdots\!09}{15\!\cdots\!25}a^{12}+\frac{11\!\cdots\!31}{15\!\cdots\!25}a^{11}-\frac{49\!\cdots\!54}{78\!\cdots\!25}a^{10}+\frac{15\!\cdots\!32}{78\!\cdots\!25}a^{9}+\frac{11\!\cdots\!49}{78\!\cdots\!25}a^{8}-\frac{36\!\cdots\!22}{33\!\cdots\!75}a^{7}+\frac{30\!\cdots\!03}{78\!\cdots\!25}a^{6}-\frac{49\!\cdots\!13}{78\!\cdots\!25}a^{5}-\frac{40\!\cdots\!13}{71\!\cdots\!75}a^{4}+\frac{33\!\cdots\!82}{78\!\cdots\!25}a^{3}+\frac{20\!\cdots\!23}{71\!\cdots\!75}a^{2}+\frac{80\!\cdots\!74}{15\!\cdots\!25}a+\frac{71\!\cdots\!59}{31\!\cdots\!25}$, $\frac{89\!\cdots\!57}{15\!\cdots\!25}a^{25}-\frac{13\!\cdots\!29}{50\!\cdots\!75}a^{24}+\frac{11\!\cdots\!99}{62\!\cdots\!25}a^{23}-\frac{11\!\cdots\!68}{14\!\cdots\!75}a^{22}+\frac{46\!\cdots\!03}{15\!\cdots\!25}a^{21}-\frac{27\!\cdots\!43}{31\!\cdots\!25}a^{20}+\frac{41\!\cdots\!36}{15\!\cdots\!25}a^{19}-\frac{84\!\cdots\!84}{15\!\cdots\!25}a^{18}+\frac{19\!\cdots\!94}{15\!\cdots\!25}a^{17}-\frac{59\!\cdots\!84}{31\!\cdots\!25}a^{16}+\frac{10\!\cdots\!99}{25\!\cdots\!75}a^{15}-\frac{37\!\cdots\!34}{50\!\cdots\!75}a^{14}+\frac{28\!\cdots\!98}{15\!\cdots\!25}a^{13}-\frac{35\!\cdots\!68}{10\!\cdots\!75}a^{12}+\frac{19\!\cdots\!62}{31\!\cdots\!25}a^{11}-\frac{13\!\cdots\!13}{15\!\cdots\!25}a^{10}+\frac{13\!\cdots\!54}{15\!\cdots\!25}a^{9}-\frac{10\!\cdots\!22}{15\!\cdots\!25}a^{8}+\frac{17\!\cdots\!96}{66\!\cdots\!75}a^{7}+\frac{48\!\cdots\!91}{15\!\cdots\!25}a^{6}-\frac{11\!\cdots\!06}{50\!\cdots\!75}a^{5}+\frac{28\!\cdots\!89}{14\!\cdots\!75}a^{4}-\frac{10\!\cdots\!21}{15\!\cdots\!25}a^{3}+\frac{10\!\cdots\!31}{14\!\cdots\!75}a^{2}+\frac{10\!\cdots\!78}{31\!\cdots\!25}a-\frac{26\!\cdots\!77}{62\!\cdots\!25}$, $\frac{41\!\cdots\!43}{60\!\cdots\!25}a^{25}-\frac{23\!\cdots\!13}{78\!\cdots\!25}a^{24}+\frac{62\!\cdots\!28}{31\!\cdots\!25}a^{23}-\frac{57\!\cdots\!41}{71\!\cdots\!75}a^{22}+\frac{23\!\cdots\!86}{78\!\cdots\!25}a^{21}-\frac{13\!\cdots\!11}{15\!\cdots\!25}a^{20}+\frac{19\!\cdots\!32}{78\!\cdots\!25}a^{19}-\frac{36\!\cdots\!08}{78\!\cdots\!25}a^{18}+\frac{83\!\cdots\!78}{78\!\cdots\!25}a^{17}-\frac{17\!\cdots\!06}{12\!\cdots\!25}a^{16}+\frac{88\!\cdots\!64}{25\!\cdots\!75}a^{15}-\frac{49\!\cdots\!23}{78\!\cdots\!25}a^{14}+\frac{12\!\cdots\!26}{78\!\cdots\!25}a^{13}-\frac{45\!\cdots\!41}{15\!\cdots\!25}a^{12}+\frac{76\!\cdots\!29}{15\!\cdots\!25}a^{11}-\frac{44\!\cdots\!81}{78\!\cdots\!25}a^{10}+\frac{37\!\cdots\!48}{78\!\cdots\!25}a^{9}-\frac{17\!\cdots\!53}{60\!\cdots\!25}a^{8}-\frac{30\!\cdots\!13}{33\!\cdots\!75}a^{7}+\frac{10\!\cdots\!42}{78\!\cdots\!25}a^{6}-\frac{17\!\cdots\!07}{78\!\cdots\!25}a^{5}+\frac{58\!\cdots\!93}{71\!\cdots\!75}a^{4}+\frac{20\!\cdots\!23}{78\!\cdots\!25}a^{3}+\frac{26\!\cdots\!44}{54\!\cdots\!75}a^{2}+\frac{16\!\cdots\!61}{15\!\cdots\!25}a+\frac{97\!\cdots\!51}{31\!\cdots\!25}$, $\frac{48\!\cdots\!67}{71\!\cdots\!75}a^{25}-\frac{20\!\cdots\!69}{71\!\cdots\!75}a^{24}+\frac{59\!\cdots\!54}{28\!\cdots\!75}a^{23}-\frac{60\!\cdots\!38}{71\!\cdots\!75}a^{22}+\frac{23\!\cdots\!43}{71\!\cdots\!75}a^{21}-\frac{14\!\cdots\!63}{14\!\cdots\!75}a^{20}+\frac{21\!\cdots\!16}{71\!\cdots\!75}a^{19}-\frac{44\!\cdots\!04}{71\!\cdots\!75}a^{18}+\frac{11\!\cdots\!64}{71\!\cdots\!75}a^{17}-\frac{47\!\cdots\!04}{20\!\cdots\!25}a^{16}+\frac{64\!\cdots\!57}{11\!\cdots\!25}a^{15}-\frac{66\!\cdots\!24}{71\!\cdots\!75}a^{14}+\frac{12\!\cdots\!01}{54\!\cdots\!75}a^{13}-\frac{60\!\cdots\!03}{14\!\cdots\!75}a^{12}+\frac{11\!\cdots\!62}{14\!\cdots\!75}a^{11}-\frac{85\!\cdots\!03}{71\!\cdots\!75}a^{10}+\frac{10\!\cdots\!24}{71\!\cdots\!75}a^{9}-\frac{11\!\cdots\!07}{71\!\cdots\!75}a^{8}+\frac{35\!\cdots\!16}{30\!\cdots\!25}a^{7}-\frac{63\!\cdots\!54}{71\!\cdots\!75}a^{6}+\frac{18\!\cdots\!34}{71\!\cdots\!75}a^{5}-\frac{32\!\cdots\!51}{71\!\cdots\!75}a^{4}-\frac{58\!\cdots\!76}{71\!\cdots\!75}a^{3}+\frac{17\!\cdots\!46}{71\!\cdots\!75}a^{2}+\frac{21\!\cdots\!43}{14\!\cdots\!75}a+\frac{24\!\cdots\!88}{28\!\cdots\!75}$, $\frac{41\!\cdots\!69}{15\!\cdots\!25}a^{25}-\frac{90\!\cdots\!08}{15\!\cdots\!25}a^{24}+\frac{25\!\cdots\!91}{48\!\cdots\!25}a^{23}-\frac{21\!\cdots\!56}{14\!\cdots\!75}a^{22}+\frac{62\!\cdots\!02}{12\!\cdots\!25}a^{21}-\frac{20\!\cdots\!77}{24\!\cdots\!25}a^{20}+\frac{43\!\cdots\!37}{15\!\cdots\!25}a^{19}+\frac{40\!\cdots\!22}{15\!\cdots\!25}a^{18}+\frac{38\!\cdots\!23}{15\!\cdots\!25}a^{17}+\frac{10\!\cdots\!02}{31\!\cdots\!25}a^{16}+\frac{54\!\cdots\!98}{50\!\cdots\!75}a^{15}+\frac{88\!\cdots\!82}{15\!\cdots\!25}a^{14}+\frac{16\!\cdots\!91}{15\!\cdots\!25}a^{13}+\frac{73\!\cdots\!69}{31\!\cdots\!25}a^{12}-\frac{18\!\cdots\!86}{31\!\cdots\!25}a^{11}+\frac{30\!\cdots\!79}{15\!\cdots\!25}a^{10}-\frac{53\!\cdots\!07}{15\!\cdots\!25}a^{9}+\frac{58\!\cdots\!51}{15\!\cdots\!25}a^{8}-\frac{25\!\cdots\!83}{66\!\cdots\!75}a^{7}+\frac{10\!\cdots\!72}{15\!\cdots\!25}a^{6}+\frac{19\!\cdots\!38}{15\!\cdots\!25}a^{5}-\frac{22\!\cdots\!12}{14\!\cdots\!75}a^{4}+\frac{21\!\cdots\!93}{15\!\cdots\!25}a^{3}+\frac{23\!\cdots\!77}{14\!\cdots\!75}a^{2}+\frac{25\!\cdots\!26}{31\!\cdots\!25}a+\frac{50\!\cdots\!41}{62\!\cdots\!25}$ 25260318885990440618795962656207038222056866841224039/784803989526277132786021644580620722639830203197216328125a^25 - 148832578267512130670761171531382367211678952799974073/784803989526277132786021644580620722639830203197216328125a^24 + 34168090668815607464773005430220629867019487648890648/31392159581051085311440865783224828905593208127888653125a^23 - 359665504221775656154835526878566226147630583107422811/71345817229661557526001967689147338421802745745201484375a^22 + 14509939724693947628100267747548172882715339409261264006/784803989526277132786021644580620722639830203197216328125a^21 - 282487310684652418809251363499567515779929429533900631/5063251545330820211522720287616907887998904536756234375a^20 + 123639450379164719833959313807657299750137793565796890322/784803989526277132786021644580620722639830203197216328125a^19 - 270687443603961165070014312603732628826155786122330918393/784803989526277132786021644580620722639830203197216328125a^18 + 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K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 195886918009.72314 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 195886918009.72314 \cdot 3}{2\cdot\sqrt{105838418476275527898387851073846090273368671}}\cr\approx \mathstrut & 0.679380864292949 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675)

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^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675, 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^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675);

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^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675);

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

$D_{26}$ (as 26T3):

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 solvable group of order 52

The 16 conjugacy class representatives for $D_{26}$

Character table for $D_{26}$

Intermediate fields

$\Q(\sqrt{-31}) $, 13.1.1847739844104888853729.1

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]

Sibling fields

Degree 26 sibling:	deg 26
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/5.2.0.1}{2} }^{12}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$	${\href{/padicField/7.13.0.1}{13} }^{2}$	${\href{/padicField/11.2.0.1}{2} }^{13}$	${\href{/padicField/13.2.0.1}{2} }^{13}$	$26$	${\href{/padicField/19.2.0.1}{2} }^{12}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$	$26$	$26$	R	$26$	${\href{/padicField/41.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/47.2.0.1}{2} }^{12}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.2.0.1}{2} }^{13}$	${\href{/padicField/59.2.0.1}{2} }^{12}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$31$ 31	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$113$ 113	$\Q_{113}$	$x + 110$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{113}$	$x + 110$	$1$	$1$	$0$	Trivial	$[\ ]$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	113.2.1.1	$x^{2} + 113$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.113.2t1.a.a	$1$	$ 113 $	$\Q(\sqrt{113}) $	$C_2$ (as 2T1)	$1$	$1$
	1.3503.2t1.a.a	$1$	$ 31 \cdot 113 $	$\Q(\sqrt{-3503}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.31.2t1.a.a	$1$	$ 31 $	$\Q(\sqrt{-31}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.3503.26t3.a.a	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3503.13t2.a.d	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.13t2.a.a	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.26t3.a.d	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3503.13t2.a.e	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.26t3.a.b	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3503.13t2.a.c	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.13t2.a.f	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.13t2.a.b	$2$	$ 31 \cdot 113 $	13.1.1847739844104888853729.1	$D_{13}$ (as 13T2)	$1$	$0$
*	2.3503.26t3.a.f	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3503.26t3.a.e	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$
*	2.3503.26t3.a.c	$2$	$ 31 \cdot 113 $	26.0.105838418476275527898387851073846090273368671.1	$D_{26}$ (as 26T3)	$1$	$0$

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 *.

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