LMFDB - Number field 28.0.20444429365013571404907915184985572799247638801.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675)

gp: K = bnfinit(y^28 - 6*y^27 + 18*y^26 + 6*y^25 - 69*y^24 + 47*y^23 + 257*y^22 - 674*y^21 + 321*y^20 + 545*y^19 - 15*y^18 - 692*y^17 + 1854*y^16 - 6800*y^15 + 5094*y^14 + 6261*y^13 - 5929*y^12 + 6262*y^11 - 14458*y^10 - 7839*y^9 + 27851*y^8 - 6775*y^7 + 6005*y^6 - 1487*y^5 - 35142*y^4 + 22011*y^3 + 18446*y^2 - 12525*y + 2675, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675)

$ x^{28} - 6 x^{27} + 18 x^{26} + 6 x^{25} - 69 x^{24} + 47 x^{23} + 257 x^{22} - 674 x^{21} + 321 x^{20} + \cdots + 2675 $ x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$28$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 14]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$20444429365013571404907915184985572799247638801$ 20444429365013571404907915184985572799247638801 $\medspace = 7^{14}\cdot 449^{13}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$45.08$	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:	$7^{1/2}449^{1/2}\approx 56.06246516163912$
Ramified primes:	$7$, $449$ 7, 449	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q(\sqrt{449}) $
$\card{ \Aut(K/\Q) }$:	$2$	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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{2}{7}a^{6}+\frac{1}{7}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{7}+\frac{1}{7}a$, $\frac{1}{7}a^{14}-\frac{2}{7}a^{8}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{15}-\frac{2}{7}a^{9}+\frac{1}{7}a^{3}$, $\frac{1}{35}a^{16}+\frac{1}{35}a^{13}+\frac{1}{7}a^{10}+\frac{1}{7}a^{7}+\frac{1}{35}a^{4}+\frac{1}{35}a$, $\frac{1}{35}a^{17}+\frac{1}{35}a^{14}+\frac{1}{7}a^{11}+\frac{1}{7}a^{8}+\frac{1}{35}a^{5}+\frac{1}{35}a^{2}$, $\frac{1}{35}a^{18}+\frac{1}{35}a^{15}+\frac{1}{7}a^{9}+\frac{11}{35}a^{6}+\frac{1}{35}a^{3}-\frac{1}{7}$, $\frac{1}{245}a^{19}+\frac{3}{245}a^{18}-\frac{1}{245}a^{17}+\frac{3}{245}a^{15}-\frac{16}{245}a^{14}-\frac{6}{245}a^{13}-\frac{3}{49}a^{12}+\frac{20}{49}a^{11}-\frac{3}{7}a^{10}+\frac{3}{49}a^{9}-\frac{2}{49}a^{8}-\frac{19}{245}a^{7}+\frac{4}{35}a^{6}-\frac{106}{245}a^{5}+\frac{3}{7}a^{4}+\frac{3}{245}a^{3}-\frac{86}{245}a^{2}-\frac{46}{245}a+\frac{8}{49}$, $\frac{1}{245}a^{20}-\frac{3}{245}a^{18}+\frac{3}{245}a^{17}+\frac{3}{245}a^{16}+\frac{17}{245}a^{15}+\frac{1}{35}a^{14}+\frac{3}{245}a^{13}+\frac{1}{49}a^{12}+\frac{17}{49}a^{11}+\frac{17}{49}a^{10}-\frac{18}{49}a^{9}+\frac{81}{245}a^{8}+\frac{17}{49}a^{7}-\frac{78}{245}a^{6}-\frac{67}{245}a^{5}-\frac{67}{245}a^{4}-\frac{53}{245}a^{3}-\frac{68}{245}a^{2}-\frac{67}{245}a-\frac{10}{49}$, $\frac{1}{1225}a^{21}+\frac{2}{1225}a^{20}+\frac{2}{1225}a^{19}-\frac{16}{1225}a^{18}-\frac{17}{1225}a^{17}+\frac{2}{1225}a^{16}+\frac{4}{175}a^{15}-\frac{12}{175}a^{14}-\frac{3}{49}a^{13}+\frac{11}{245}a^{12}-\frac{66}{245}a^{11}+\frac{86}{245}a^{10}-\frac{409}{1225}a^{9}+\frac{582}{1225}a^{8}-\frac{38}{1225}a^{7}+\frac{274}{1225}a^{6}-\frac{17}{1225}a^{5}+\frac{317}{1225}a^{4}+\frac{58}{1225}a^{3}-\frac{409}{1225}a^{2}+\frac{4}{245}a+\frac{11}{49}$, $\frac{1}{1225}a^{22}-\frac{2}{1225}a^{20}+\frac{1}{245}a^{18}+\frac{16}{1225}a^{17}-\frac{11}{1225}a^{16}+\frac{1}{49}a^{15}-\frac{52}{1225}a^{14}+\frac{2}{49}a^{13}-\frac{8}{245}a^{12}-\frac{117}{245}a^{11}+\frac{131}{1225}a^{10}-\frac{9}{49}a^{9}-\frac{527}{1225}a^{8}-\frac{41}{245}a^{7}+\frac{11}{49}a^{6}-\frac{544}{1225}a^{5}+\frac{264}{1225}a^{4}-\frac{72}{245}a^{3}+\frac{74}{175}a^{2}+\frac{101}{245}a+\frac{3}{49}$, $\frac{1}{1225}a^{23}-\frac{1}{1225}a^{20}-\frac{1}{1225}a^{19}+\frac{4}{1225}a^{18}-\frac{3}{245}a^{17}+\frac{2}{175}a^{16}-\frac{76}{1225}a^{15}+\frac{6}{175}a^{14}+\frac{6}{245}a^{13}-\frac{554}{1225}a^{11}+\frac{1}{35}a^{10}+\frac{71}{245}a^{9}-\frac{396}{1225}a^{8}-\frac{386}{1225}a^{7}-\frac{201}{1225}a^{6}+\frac{87}{245}a^{5}-\frac{9}{25}a^{4}-\frac{321}{1225}a^{3}-\frac{303}{1225}a^{2}-\frac{4}{35}a+\frac{23}{49}$, $\frac{1}{8575}a^{24}+\frac{2}{8575}a^{23}-\frac{2}{8575}a^{22}-\frac{2}{8575}a^{21}-\frac{1}{8575}a^{20}+\frac{1}{1715}a^{19}+\frac{1}{175}a^{18}+\frac{69}{8575}a^{17}+\frac{1}{1225}a^{16}-\frac{138}{8575}a^{15}+\frac{327}{8575}a^{14}+\frac{78}{1715}a^{13}-\frac{254}{8575}a^{12}-\frac{2223}{8575}a^{11}-\frac{617}{8575}a^{10}-\frac{1027}{8575}a^{9}-\frac{208}{1225}a^{8}-\frac{229}{1715}a^{7}-\frac{138}{1225}a^{6}-\frac{2566}{8575}a^{5}+\frac{962}{8575}a^{4}+\frac{3442}{8575}a^{3}+\frac{3202}{8575}a^{2}+\frac{129}{1715}a-\frac{152}{343}$, $\frac{1}{317275}a^{25}+\frac{13}{317275}a^{24}+\frac{34}{317275}a^{23}+\frac{88}{317275}a^{22}-\frac{1}{8575}a^{21}+\frac{78}{317275}a^{20}+\frac{587}{317275}a^{19}-\frac{1212}{317275}a^{18}+\frac{4336}{317275}a^{17}+\frac{818}{63455}a^{16}+\frac{4493}{317275}a^{15}+\frac{3777}{317275}a^{14}-\frac{7654}{317275}a^{13}-\frac{18842}{317275}a^{12}-\frac{48226}{317275}a^{11}-\frac{157572}{317275}a^{10}+\frac{96118}{317275}a^{9}+\frac{153723}{317275}a^{8}-\frac{96343}{317275}a^{7}+\frac{105423}{317275}a^{6}+\frac{20336}{317275}a^{5}-\frac{20536}{63455}a^{4}-\frac{41872}{317275}a^{3}+\frac{19067}{317275}a^{2}-\frac{11619}{63455}a+\frac{3970}{12691}$, $\frac{1}{239902679140075}a^{26}-\frac{372079181}{239902679140075}a^{25}-\frac{1669085671}{47980535828015}a^{24}-\frac{13096528929}{239902679140075}a^{23}+\frac{4060398581}{47980535828015}a^{22}+\frac{27458434343}{239902679140075}a^{21}-\frac{17153736906}{47980535828015}a^{20}+\frac{37491778448}{239902679140075}a^{19}+\frac{19888439}{31880754703}a^{18}-\frac{3125399970408}{239902679140075}a^{17}-\frac{1009391376933}{239902679140075}a^{16}-\frac{224447750206}{5579132073025}a^{15}-\frac{8921329677032}{239902679140075}a^{14}+\frac{15368694854459}{239902679140075}a^{13}+\frac{2721791037421}{47980535828015}a^{12}+\frac{9808378115153}{34271811305725}a^{11}-\frac{3495246989928}{9596107165603}a^{10}-\frac{65723651702222}{239902679140075}a^{9}+\frac{89431643986}{195838921747}a^{8}+\frac{15412654789428}{239902679140075}a^{7}+\frac{21576088607649}{47980535828015}a^{6}-\frac{1952172573088}{5851284857075}a^{5}-\frac{4439796119548}{239902679140075}a^{4}-\frac{15902389951734}{34271811305725}a^{3}+\frac{107187720306407}{239902679140075}a^{2}-\frac{10224426888601}{47980535828015}a-\frac{3361885110408}{9596107165603}$, $\frac{1}{25\!\cdots\!25}a^{27}-\frac{314849447}{63\!\cdots\!25}a^{26}-\frac{47\!\cdots\!94}{12\!\cdots\!75}a^{25}+\frac{12\!\cdots\!86}{25\!\cdots\!25}a^{24}-\frac{15\!\cdots\!49}{51\!\cdots\!25}a^{23}+\frac{72\!\cdots\!27}{25\!\cdots\!25}a^{22}+\frac{12\!\cdots\!07}{51\!\cdots\!25}a^{21}-\frac{34\!\cdots\!04}{25\!\cdots\!25}a^{20}-\frac{18\!\cdots\!62}{15\!\cdots\!75}a^{19}+\frac{52\!\cdots\!64}{51\!\cdots\!25}a^{18}-\frac{21\!\cdots\!47}{51\!\cdots\!25}a^{17}+\frac{36\!\cdots\!28}{25\!\cdots\!25}a^{16}+\frac{36\!\cdots\!11}{25\!\cdots\!25}a^{15}-\frac{23\!\cdots\!06}{25\!\cdots\!25}a^{14}+\frac{36\!\cdots\!72}{10\!\cdots\!85}a^{13}-\frac{11\!\cdots\!69}{25\!\cdots\!25}a^{12}-\frac{44\!\cdots\!07}{10\!\cdots\!85}a^{11}-\frac{11\!\cdots\!53}{25\!\cdots\!25}a^{10}+\frac{11\!\cdots\!33}{10\!\cdots\!85}a^{9}-\frac{41\!\cdots\!59}{25\!\cdots\!25}a^{8}-\frac{12\!\cdots\!24}{51\!\cdots\!25}a^{7}-\frac{45\!\cdots\!31}{14\!\cdots\!25}a^{6}-\frac{14\!\cdots\!07}{51\!\cdots\!25}a^{5}-\frac{32\!\cdots\!31}{37\!\cdots\!75}a^{4}+\frac{45\!\cdots\!27}{51\!\cdots\!25}a^{3}-\frac{22\!\cdots\!49}{25\!\cdots\!25}a^{2}+\frac{42\!\cdots\!72}{10\!\cdots\!85}a+\frac{29\!\cdots\!82}{10\!\cdots\!85}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, 1/7*a^12 - 2/7*a^6 + 1/7, 1/7*a^13 - 2/7*a^7 + 1/7*a, 1/7*a^14 - 2/7*a^8 + 1/7*a^2, 1/7*a^15 - 2/7*a^9 + 1/7*a^3, 1/35*a^16 + 1/35*a^13 + 1/7*a^10 + 1/7*a^7 + 1/35*a^4 + 1/35*a, 1/35*a^17 + 1/35*a^14 + 1/7*a^11 + 1/7*a^8 + 1/35*a^5 + 1/35*a^2, 1/35*a^18 + 1/35*a^15 + 1/7*a^9 + 11/35*a^6 + 1/35*a^3 - 1/7, 1/245*a^19 + 3/245*a^18 - 1/245*a^17 + 3/245*a^15 - 16/245*a^14 - 6/245*a^13 - 3/49*a^12 + 20/49*a^11 - 3/7*a^10 + 3/49*a^9 - 2/49*a^8 - 19/245*a^7 + 4/35*a^6 - 106/245*a^5 + 3/7*a^4 + 3/245*a^3 - 86/245*a^2 - 46/245*a + 8/49, 1/245*a^20 - 3/245*a^18 + 3/245*a^17 + 3/245*a^16 + 17/245*a^15 + 1/35*a^14 + 3/245*a^13 + 1/49*a^12 + 17/49*a^11 + 17/49*a^10 - 18/49*a^9 + 81/245*a^8 + 17/49*a^7 - 78/245*a^6 - 67/245*a^5 - 67/245*a^4 - 53/245*a^3 - 68/245*a^2 - 67/245*a - 10/49, 1/1225*a^21 + 2/1225*a^20 + 2/1225*a^19 - 16/1225*a^18 - 17/1225*a^17 + 2/1225*a^16 + 4/175*a^15 - 12/175*a^14 - 3/49*a^13 + 11/245*a^12 - 66/245*a^11 + 86/245*a^10 - 409/1225*a^9 + 582/1225*a^8 - 38/1225*a^7 + 274/1225*a^6 - 17/1225*a^5 + 317/1225*a^4 + 58/1225*a^3 - 409/1225*a^2 + 4/245*a + 11/49, 1/1225*a^22 - 2/1225*a^20 + 1/245*a^18 + 16/1225*a^17 - 11/1225*a^16 + 1/49*a^15 - 52/1225*a^14 + 2/49*a^13 - 8/245*a^12 - 117/245*a^11 + 131/1225*a^10 - 9/49*a^9 - 527/1225*a^8 - 41/245*a^7 + 11/49*a^6 - 544/1225*a^5 + 264/1225*a^4 - 72/245*a^3 + 74/175*a^2 + 101/245*a + 3/49, 1/1225*a^23 - 1/1225*a^20 - 1/1225*a^19 + 4/1225*a^18 - 3/245*a^17 + 2/175*a^16 - 76/1225*a^15 + 6/175*a^14 + 6/245*a^13 - 554/1225*a^11 + 1/35*a^10 + 71/245*a^9 - 396/1225*a^8 - 386/1225*a^7 - 201/1225*a^6 + 87/245*a^5 - 9/25*a^4 - 321/1225*a^3 - 303/1225*a^2 - 4/35*a + 23/49, 1/8575*a^24 + 2/8575*a^23 - 2/8575*a^22 - 2/8575*a^21 - 1/8575*a^20 + 1/1715*a^19 + 1/175*a^18 + 69/8575*a^17 + 1/1225*a^16 - 138/8575*a^15 + 327/8575*a^14 + 78/1715*a^13 - 254/8575*a^12 - 2223/8575*a^11 - 617/8575*a^10 - 1027/8575*a^9 - 208/1225*a^8 - 229/1715*a^7 - 138/1225*a^6 - 2566/8575*a^5 + 962/8575*a^4 + 3442/8575*a^3 + 3202/8575*a^2 + 129/1715*a - 152/343, 1/317275*a^25 + 13/317275*a^24 + 34/317275*a^23 + 88/317275*a^22 - 1/8575*a^21 + 78/317275*a^20 + 587/317275*a^19 - 1212/317275*a^18 + 4336/317275*a^17 + 818/63455*a^16 + 4493/317275*a^15 + 3777/317275*a^14 - 7654/317275*a^13 - 18842/317275*a^12 - 48226/317275*a^11 - 157572/317275*a^10 + 96118/317275*a^9 + 153723/317275*a^8 - 96343/317275*a^7 + 105423/317275*a^6 + 20336/317275*a^5 - 20536/63455*a^4 - 41872/317275*a^3 + 19067/317275*a^2 - 11619/63455*a + 3970/12691, 1/239902679140075*a^26 - 372079181/239902679140075*a^25 - 1669085671/47980535828015*a^24 - 13096528929/239902679140075*a^23 + 4060398581/47980535828015*a^22 + 27458434343/239902679140075*a^21 - 17153736906/47980535828015*a^20 + 37491778448/239902679140075*a^19 + 19888439/31880754703*a^18 - 3125399970408/239902679140075*a^17 - 1009391376933/239902679140075*a^16 - 224447750206/5579132073025*a^15 - 8921329677032/239902679140075*a^14 + 15368694854459/239902679140075*a^13 + 2721791037421/47980535828015*a^12 + 9808378115153/34271811305725*a^11 - 3495246989928/9596107165603*a^10 - 65723651702222/239902679140075*a^9 + 89431643986/195838921747*a^8 + 15412654789428/239902679140075*a^7 + 21576088607649/47980535828015*a^6 - 1952172573088/5851284857075*a^5 - 4439796119548/239902679140075*a^4 - 15902389951734/34271811305725*a^3 + 107187720306407/239902679140075*a^2 - 10224426888601/47980535828015*a - 3361885110408/9596107165603, 1/25974637773290955737442125*a^27 - 314849447/633527750568072091157125*a^26 - 47391558836009694/120812268712981189476475*a^25 + 1260363234268205255186/25974637773290955737442125*a^24 - 153098375144052646149/5194927554658191147488425*a^23 + 7224169299395329353827/25974637773290955737442125*a^22 + 129211431934504875307/5194927554658191147488425*a^21 - 3414811189977577279004/25974637773290955737442125*a^20 - 18689322019588240262/15145561383843122878975*a^19 + 52533451903175870390964/5194927554658191147488425*a^18 - 21160207840310393390447/5194927554658191147488425*a^17 + 367336420210594628003828/25974637773290955737442125*a^16 + 36727865296166722183411/25974637773290955737442125*a^15 - 230460392031535652070606/25974637773290955737442125*a^14 + 36114056255241525781172/1038985510931638229497685*a^13 - 1120963118776252889276569/25974637773290955737442125*a^12 - 446591093297662333312507/1038985510931638229497685*a^11 - 11652570999457683460235953/25974637773290955737442125*a^10 + 113926294469806835449633/1038985510931638229497685*a^9 - 4143666150335868965602859/25974637773290955737442125*a^8 - 1299272957749327239248924/5194927554658191147488425*a^7 - 45645616744665658536731/140403447423194355337525*a^6 - 1454747202573961108124507/5194927554658191147488425*a^5 - 324277694356358538197931/3710662539041565105348875*a^4 + 455529277338756412579327/5194927554658191147488425*a^3 - 2278806181125286183753649/25974637773290955737442125*a^2 + 422813904623347664778272/1038985510931638229497685*a + 295641202301014397180982/1038985510931638229497685

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$, $7$

Class group and class number

Trivial group, which has order $1$ (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:	$13$	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:	$\frac{40\!\cdots\!91}{25\!\cdots\!25}a^{27}-\frac{52\!\cdots\!37}{63\!\cdots\!25}a^{26}+\frac{11\!\cdots\!51}{51\!\cdots\!25}a^{25}+\frac{70\!\cdots\!11}{25\!\cdots\!25}a^{24}-\frac{83\!\cdots\!67}{85\!\cdots\!25}a^{23}+\frac{20\!\cdots\!97}{25\!\cdots\!25}a^{22}+\frac{22\!\cdots\!64}{51\!\cdots\!25}a^{21}-\frac{19\!\cdots\!04}{25\!\cdots\!25}a^{20}-\frac{32\!\cdots\!34}{20\!\cdots\!75}a^{19}+\frac{53\!\cdots\!68}{51\!\cdots\!25}a^{18}+\frac{34\!\cdots\!83}{51\!\cdots\!25}a^{17}-\frac{25\!\cdots\!77}{25\!\cdots\!25}a^{16}+\frac{54\!\cdots\!71}{25\!\cdots\!25}a^{15}-\frac{22\!\cdots\!46}{25\!\cdots\!25}a^{14}+\frac{65\!\cdots\!98}{51\!\cdots\!25}a^{13}+\frac{34\!\cdots\!81}{25\!\cdots\!25}a^{12}-\frac{76\!\cdots\!78}{51\!\cdots\!25}a^{11}+\frac{13\!\cdots\!17}{25\!\cdots\!25}a^{10}-\frac{88\!\cdots\!18}{51\!\cdots\!25}a^{9}-\frac{68\!\cdots\!59}{25\!\cdots\!25}a^{8}+\frac{16\!\cdots\!46}{51\!\cdots\!25}a^{7}+\frac{79\!\cdots\!97}{51\!\cdots\!25}a^{6}+\frac{50\!\cdots\!53}{51\!\cdots\!25}a^{5}+\frac{15\!\cdots\!54}{37\!\cdots\!75}a^{4}-\frac{28\!\cdots\!54}{51\!\cdots\!25}a^{3}-\frac{94\!\cdots\!99}{25\!\cdots\!25}a^{2}+\frac{40\!\cdots\!41}{10\!\cdots\!85}a+\frac{47\!\cdots\!67}{10\!\cdots\!85}$, $\frac{90\!\cdots\!11}{70\!\cdots\!25}a^{27}-\frac{25\!\cdots\!04}{25\!\cdots\!25}a^{26}+\frac{19\!\cdots\!31}{51\!\cdots\!25}a^{25}-\frac{11\!\cdots\!13}{25\!\cdots\!25}a^{24}-\frac{75\!\cdots\!33}{11\!\cdots\!65}a^{23}+\frac{53\!\cdots\!19}{25\!\cdots\!25}a^{22}+\frac{63\!\cdots\!76}{51\!\cdots\!25}a^{21}-\frac{32\!\cdots\!08}{25\!\cdots\!25}a^{20}+\frac{16\!\cdots\!53}{74\!\cdots\!75}a^{19}-\frac{14\!\cdots\!73}{10\!\cdots\!85}a^{18}+\frac{78\!\cdots\!39}{51\!\cdots\!25}a^{17}-\frac{17\!\cdots\!94}{25\!\cdots\!25}a^{16}+\frac{83\!\cdots\!37}{25\!\cdots\!25}a^{15}-\frac{35\!\cdots\!07}{25\!\cdots\!25}a^{14}+\frac{52\!\cdots\!58}{20\!\cdots\!37}a^{13}-\frac{32\!\cdots\!73}{25\!\cdots\!25}a^{12}-\frac{32\!\cdots\!92}{51\!\cdots\!25}a^{11}+\frac{53\!\cdots\!09}{25\!\cdots\!25}a^{10}-\frac{29\!\cdots\!53}{51\!\cdots\!25}a^{9}+\frac{22\!\cdots\!12}{42\!\cdots\!25}a^{8}+\frac{12\!\cdots\!48}{10\!\cdots\!85}a^{7}-\frac{14\!\cdots\!57}{51\!\cdots\!25}a^{6}+\frac{24\!\cdots\!64}{51\!\cdots\!25}a^{5}-\frac{39\!\cdots\!16}{53\!\cdots\!25}a^{4}-\frac{10\!\cdots\!89}{51\!\cdots\!25}a^{3}+\frac{28\!\cdots\!12}{42\!\cdots\!25}a^{2}-\frac{13\!\cdots\!77}{10\!\cdots\!85}a-\frac{13\!\cdots\!91}{10\!\cdots\!85}$, $\frac{79\!\cdots\!71}{37\!\cdots\!75}a^{27}-\frac{48\!\cdots\!67}{37\!\cdots\!75}a^{26}+\frac{35\!\cdots\!93}{74\!\cdots\!75}a^{25}-\frac{13\!\cdots\!64}{37\!\cdots\!75}a^{24}-\frac{53\!\cdots\!37}{74\!\cdots\!75}a^{23}+\frac{25\!\cdots\!12}{37\!\cdots\!75}a^{22}+\frac{23\!\cdots\!06}{14\!\cdots\!55}a^{21}-\frac{43\!\cdots\!66}{41\!\cdots\!75}a^{20}+\frac{16\!\cdots\!19}{74\!\cdots\!75}a^{19}-\frac{80\!\cdots\!66}{20\!\cdots\!75}a^{18}+\frac{37\!\cdots\!13}{74\!\cdots\!75}a^{17}-\frac{93\!\cdots\!82}{37\!\cdots\!75}a^{16}+\frac{27\!\cdots\!91}{37\!\cdots\!75}a^{15}-\frac{85\!\cdots\!26}{37\!\cdots\!75}a^{14}+\frac{93\!\cdots\!89}{29\!\cdots\!91}a^{13}-\frac{24\!\cdots\!92}{53\!\cdots\!25}a^{12}+\frac{41\!\cdots\!47}{74\!\cdots\!75}a^{11}-\frac{24\!\cdots\!93}{37\!\cdots\!75}a^{10}-\frac{21\!\cdots\!02}{74\!\cdots\!75}a^{9}+\frac{60\!\cdots\!71}{37\!\cdots\!75}a^{8}-\frac{30\!\cdots\!29}{74\!\cdots\!75}a^{7}-\frac{41\!\cdots\!43}{74\!\cdots\!75}a^{6}+\frac{92\!\cdots\!33}{74\!\cdots\!75}a^{5}-\frac{50\!\cdots\!61}{53\!\cdots\!25}a^{4}+\frac{38\!\cdots\!49}{74\!\cdots\!75}a^{3}-\frac{18\!\cdots\!29}{37\!\cdots\!75}a^{2}-\frac{13\!\cdots\!96}{26\!\cdots\!35}a+\frac{18\!\cdots\!16}{21\!\cdots\!65}$, $\frac{46\!\cdots\!42}{25\!\cdots\!25}a^{27}-\frac{30\!\cdots\!69}{25\!\cdots\!25}a^{26}-\frac{16\!\cdots\!06}{51\!\cdots\!25}a^{25}+\frac{49\!\cdots\!17}{25\!\cdots\!25}a^{24}-\frac{35\!\cdots\!48}{51\!\cdots\!25}a^{23}-\frac{23\!\cdots\!41}{25\!\cdots\!25}a^{22}+\frac{23\!\cdots\!94}{51\!\cdots\!25}a^{21}+\frac{44\!\cdots\!32}{25\!\cdots\!25}a^{20}-\frac{10\!\cdots\!51}{14\!\cdots\!55}a^{19}+\frac{77\!\cdots\!72}{12\!\cdots\!25}a^{18}+\frac{20\!\cdots\!78}{20\!\cdots\!37}a^{17}+\frac{50\!\cdots\!23}{70\!\cdots\!25}a^{16}-\frac{33\!\cdots\!63}{25\!\cdots\!25}a^{15}-\frac{11\!\cdots\!32}{25\!\cdots\!25}a^{14}-\frac{31\!\cdots\!42}{51\!\cdots\!25}a^{13}+\frac{13\!\cdots\!57}{25\!\cdots\!25}a^{12}+\frac{89\!\cdots\!49}{10\!\cdots\!85}a^{11}+\frac{44\!\cdots\!99}{25\!\cdots\!25}a^{10}-\frac{55\!\cdots\!23}{10\!\cdots\!85}a^{9}-\frac{43\!\cdots\!53}{25\!\cdots\!25}a^{8}-\frac{45\!\cdots\!81}{51\!\cdots\!25}a^{7}+\frac{43\!\cdots\!05}{20\!\cdots\!37}a^{6}+\frac{65\!\cdots\!09}{10\!\cdots\!85}a^{5}+\frac{14\!\cdots\!73}{37\!\cdots\!75}a^{4}-\frac{24\!\cdots\!51}{51\!\cdots\!25}a^{3}-\frac{57\!\cdots\!78}{25\!\cdots\!25}a^{2}+\frac{11\!\cdots\!59}{25\!\cdots\!85}a+\frac{30\!\cdots\!57}{28\!\cdots\!05}$, $\frac{74\!\cdots\!46}{25\!\cdots\!25}a^{27}-\frac{31\!\cdots\!92}{25\!\cdots\!25}a^{26}+\frac{11\!\cdots\!67}{51\!\cdots\!25}a^{25}+\frac{41\!\cdots\!66}{42\!\cdots\!25}a^{24}-\frac{61\!\cdots\!53}{51\!\cdots\!25}a^{23}-\frac{77\!\cdots\!23}{25\!\cdots\!25}a^{22}+\frac{45\!\cdots\!19}{51\!\cdots\!25}a^{21}-\frac{35\!\cdots\!64}{25\!\cdots\!25}a^{20}-\frac{19\!\cdots\!48}{74\!\cdots\!75}a^{19}+\frac{33\!\cdots\!19}{51\!\cdots\!25}a^{18}+\frac{38\!\cdots\!76}{51\!\cdots\!25}a^{17}-\frac{55\!\cdots\!52}{25\!\cdots\!25}a^{16}-\frac{17\!\cdots\!09}{25\!\cdots\!25}a^{15}-\frac{21\!\cdots\!51}{25\!\cdots\!25}a^{14}+\frac{80\!\cdots\!89}{51\!\cdots\!25}a^{13}+\frac{53\!\cdots\!21}{25\!\cdots\!25}a^{12}+\frac{11\!\cdots\!36}{51\!\cdots\!25}a^{11}-\frac{81\!\cdots\!53}{25\!\cdots\!25}a^{10}-\frac{15\!\cdots\!53}{51\!\cdots\!25}a^{9}-\frac{90\!\cdots\!44}{25\!\cdots\!25}a^{8}+\frac{21\!\cdots\!61}{51\!\cdots\!25}a^{7}+\frac{25\!\cdots\!33}{51\!\cdots\!25}a^{6}-\frac{22\!\cdots\!19}{51\!\cdots\!25}a^{5}-\frac{28\!\cdots\!96}{37\!\cdots\!75}a^{4}+\frac{30\!\cdots\!79}{51\!\cdots\!25}a^{3}+\frac{71\!\cdots\!21}{25\!\cdots\!25}a^{2}-\frac{21\!\cdots\!56}{10\!\cdots\!85}a+\frac{22\!\cdots\!42}{10\!\cdots\!85}$, $\frac{76\!\cdots\!93}{25\!\cdots\!25}a^{27}-\frac{12\!\cdots\!13}{70\!\cdots\!25}a^{26}+\frac{27\!\cdots\!38}{51\!\cdots\!25}a^{25}+\frac{45\!\cdots\!58}{25\!\cdots\!25}a^{24}-\frac{10\!\cdots\!73}{51\!\cdots\!25}a^{23}+\frac{10\!\cdots\!61}{63\!\cdots\!25}a^{22}+\frac{38\!\cdots\!97}{51\!\cdots\!25}a^{21}-\frac{54\!\cdots\!27}{25\!\cdots\!25}a^{20}+\frac{15\!\cdots\!93}{14\!\cdots\!55}a^{19}+\frac{96\!\cdots\!74}{51\!\cdots\!25}a^{18}-\frac{60\!\cdots\!58}{51\!\cdots\!25}a^{17}-\frac{56\!\cdots\!51}{25\!\cdots\!25}a^{16}+\frac{19\!\cdots\!63}{25\!\cdots\!25}a^{15}-\frac{46\!\cdots\!43}{25\!\cdots\!25}a^{14}+\frac{64\!\cdots\!19}{51\!\cdots\!25}a^{13}+\frac{11\!\cdots\!48}{63\!\cdots\!25}a^{12}-\frac{12\!\cdots\!46}{51\!\cdots\!25}a^{11}+\frac{60\!\cdots\!36}{25\!\cdots\!25}a^{10}-\frac{18\!\cdots\!24}{51\!\cdots\!25}a^{9}-\frac{13\!\cdots\!58}{71\!\cdots\!25}a^{8}+\frac{38\!\cdots\!11}{51\!\cdots\!25}a^{7}-\frac{13\!\cdots\!34}{51\!\cdots\!25}a^{6}+\frac{67\!\cdots\!72}{51\!\cdots\!25}a^{5}+\frac{76\!\cdots\!61}{53\!\cdots\!25}a^{4}-\frac{57\!\cdots\!16}{51\!\cdots\!25}a^{3}+\frac{12\!\cdots\!53}{25\!\cdots\!25}a^{2}+\frac{62\!\cdots\!57}{10\!\cdots\!85}a-\frac{16\!\cdots\!84}{10\!\cdots\!85}$, $\frac{15\!\cdots\!62}{25\!\cdots\!25}a^{27}-\frac{88\!\cdots\!19}{25\!\cdots\!25}a^{26}+\frac{59\!\cdots\!01}{58\!\cdots\!25}a^{25}+\frac{94\!\cdots\!92}{25\!\cdots\!25}a^{24}-\frac{17\!\cdots\!46}{51\!\cdots\!25}a^{23}+\frac{50\!\cdots\!39}{25\!\cdots\!25}a^{22}+\frac{71\!\cdots\!96}{51\!\cdots\!25}a^{21}-\frac{91\!\cdots\!53}{25\!\cdots\!25}a^{20}+\frac{13\!\cdots\!34}{74\!\cdots\!75}a^{19}+\frac{77\!\cdots\!63}{51\!\cdots\!25}a^{18}+\frac{11\!\cdots\!19}{51\!\cdots\!25}a^{17}-\frac{92\!\cdots\!64}{25\!\cdots\!25}a^{16}+\frac{27\!\cdots\!32}{25\!\cdots\!25}a^{15}-\frac{10\!\cdots\!67}{25\!\cdots\!25}a^{14}+\frac{15\!\cdots\!38}{51\!\cdots\!25}a^{13}+\frac{54\!\cdots\!57}{25\!\cdots\!25}a^{12}-\frac{58\!\cdots\!08}{85\!\cdots\!25}a^{11}+\frac{96\!\cdots\!29}{25\!\cdots\!25}a^{10}-\frac{10\!\cdots\!86}{12\!\cdots\!75}a^{9}-\frac{10\!\cdots\!38}{25\!\cdots\!25}a^{8}+\frac{54\!\cdots\!42}{51\!\cdots\!25}a^{7}+\frac{32\!\cdots\!41}{51\!\cdots\!25}a^{6}+\frac{37\!\cdots\!89}{51\!\cdots\!25}a^{5}-\frac{75\!\cdots\!72}{37\!\cdots\!75}a^{4}-\frac{88\!\cdots\!71}{51\!\cdots\!25}a^{3}+\frac{12\!\cdots\!37}{25\!\cdots\!25}a^{2}+\frac{83\!\cdots\!32}{10\!\cdots\!85}a+\frac{51\!\cdots\!17}{28\!\cdots\!05}$, $\frac{221990904}{8191798799905}a^{27}-\frac{1208401713}{8191798799905}a^{26}+\frac{18275756473}{40958993999525}a^{25}+\frac{17761221759}{40958993999525}a^{24}-\frac{19315303783}{8191798799905}a^{23}+\frac{187035903807}{40958993999525}a^{22}+\frac{356226522178}{40958993999525}a^{21}-\frac{956588436127}{40958993999525}a^{20}+\frac{12272334029}{835897836725}a^{19}+\frac{57814406036}{1106999837825}a^{18}-\frac{4138606663662}{40958993999525}a^{17}+\frac{1283964283797}{40958993999525}a^{16}+\frac{3472246558853}{40958993999525}a^{15}-\frac{116285983677}{1638359759981}a^{14}+\frac{1672510144548}{40958993999525}a^{13}+\frac{23669994177569}{40958993999525}a^{12}-\frac{10546307213447}{8191798799905}a^{11}+\frac{634768046294}{952534744175}a^{10}-\frac{11117321294157}{40958993999525}a^{9}-\frac{65691949882}{40958993999525}a^{8}+\frac{99168016498526}{40958993999525}a^{7}-\frac{53127637364583}{40958993999525}a^{6}-\frac{32321403162992}{40958993999525}a^{5}+\frac{59967143716}{95922702575}a^{4}-\frac{84971670737407}{40958993999525}a^{3}+\frac{753567309081}{221399967565}a^{2}+\frac{23314847649623}{8191798799905}a-\frac{2435517117904}{1638359759981}$, $\frac{17\!\cdots\!38}{13\!\cdots\!15}a^{27}-\frac{49\!\cdots\!09}{51\!\cdots\!25}a^{26}+\frac{16\!\cdots\!17}{51\!\cdots\!25}a^{25}-\frac{13\!\cdots\!28}{10\!\cdots\!85}a^{24}-\frac{52\!\cdots\!53}{51\!\cdots\!25}a^{23}+\frac{42\!\cdots\!56}{51\!\cdots\!25}a^{22}+\frac{21\!\cdots\!22}{51\!\cdots\!25}a^{21}-\frac{53\!\cdots\!93}{51\!\cdots\!25}a^{20}+\frac{47\!\cdots\!63}{74\!\cdots\!75}a^{19}+\frac{18\!\cdots\!41}{58\!\cdots\!25}a^{18}+\frac{11\!\cdots\!19}{51\!\cdots\!25}a^{17}-\frac{24\!\cdots\!08}{51\!\cdots\!25}a^{16}+\frac{13\!\cdots\!87}{51\!\cdots\!25}a^{15}-\frac{35\!\cdots\!31}{51\!\cdots\!25}a^{14}+\frac{11\!\cdots\!67}{51\!\cdots\!25}a^{13}-\frac{99\!\cdots\!42}{20\!\cdots\!37}a^{12}-\frac{24\!\cdots\!43}{51\!\cdots\!25}a^{11}-\frac{10\!\cdots\!19}{51\!\cdots\!25}a^{10}-\frac{11\!\cdots\!13}{51\!\cdots\!25}a^{9}+\frac{26\!\cdots\!62}{51\!\cdots\!25}a^{8}+\frac{20\!\cdots\!61}{51\!\cdots\!25}a^{7}-\frac{15\!\cdots\!91}{51\!\cdots\!25}a^{6}-\frac{33\!\cdots\!42}{12\!\cdots\!75}a^{5}-\frac{41\!\cdots\!69}{74\!\cdots\!75}a^{4}+\frac{29\!\cdots\!87}{51\!\cdots\!25}a^{3}+\frac{30\!\cdots\!03}{51\!\cdots\!25}a^{2}-\frac{40\!\cdots\!78}{10\!\cdots\!85}a+\frac{18\!\cdots\!30}{20\!\cdots\!37}$, $\frac{12\!\cdots\!04}{25\!\cdots\!25}a^{27}-\frac{69\!\cdots\!13}{25\!\cdots\!25}a^{26}+\frac{37\!\cdots\!36}{51\!\cdots\!25}a^{25}+\frac{20\!\cdots\!24}{25\!\cdots\!25}a^{24}-\frac{16\!\cdots\!63}{51\!\cdots\!25}a^{23}+\frac{10\!\cdots\!03}{25\!\cdots\!25}a^{22}+\frac{74\!\cdots\!44}{51\!\cdots\!25}a^{21}-\frac{69\!\cdots\!46}{25\!\cdots\!25}a^{20}-\frac{35\!\cdots\!97}{15\!\cdots\!75}a^{19}+\frac{46\!\cdots\!36}{12\!\cdots\!25}a^{18}+\frac{19\!\cdots\!49}{12\!\cdots\!75}a^{17}-\frac{12\!\cdots\!98}{25\!\cdots\!25}a^{16}+\frac{25\!\cdots\!94}{25\!\cdots\!25}a^{15}-\frac{80\!\cdots\!04}{25\!\cdots\!25}a^{14}+\frac{50\!\cdots\!49}{51\!\cdots\!25}a^{13}+\frac{10\!\cdots\!79}{25\!\cdots\!25}a^{12}-\frac{54\!\cdots\!17}{85\!\cdots\!25}a^{11}-\frac{10\!\cdots\!67}{25\!\cdots\!25}a^{10}-\frac{13\!\cdots\!44}{51\!\cdots\!25}a^{9}-\frac{27\!\cdots\!16}{25\!\cdots\!25}a^{8}+\frac{67\!\cdots\!42}{51\!\cdots\!25}a^{7}+\frac{29\!\cdots\!12}{51\!\cdots\!25}a^{6}-\frac{18\!\cdots\!18}{51\!\cdots\!25}a^{5}+\frac{20\!\cdots\!21}{37\!\cdots\!75}a^{4}-\frac{10\!\cdots\!02}{51\!\cdots\!25}a^{3}+\frac{30\!\cdots\!79}{25\!\cdots\!25}a^{2}+\frac{43\!\cdots\!39}{25\!\cdots\!85}a-\frac{43\!\cdots\!87}{10\!\cdots\!85}$, $\frac{49\!\cdots\!48}{29\!\cdots\!25}a^{27}-\frac{45\!\cdots\!49}{63\!\cdots\!25}a^{26}+\frac{35\!\cdots\!91}{51\!\cdots\!25}a^{25}+\frac{22\!\cdots\!12}{25\!\cdots\!25}a^{24}-\frac{66\!\cdots\!22}{51\!\cdots\!25}a^{23}-\frac{96\!\cdots\!96}{25\!\cdots\!25}a^{22}+\frac{43\!\cdots\!36}{51\!\cdots\!25}a^{21}+\frac{75\!\cdots\!42}{25\!\cdots\!25}a^{20}-\frac{51\!\cdots\!77}{14\!\cdots\!55}a^{19}+\frac{19\!\cdots\!99}{10\!\cdots\!85}a^{18}+\frac{44\!\cdots\!49}{51\!\cdots\!25}a^{17}-\frac{13\!\cdots\!09}{25\!\cdots\!25}a^{16}-\frac{31\!\cdots\!63}{25\!\cdots\!25}a^{15}-\frac{11\!\cdots\!67}{25\!\cdots\!25}a^{14}+\frac{17\!\cdots\!02}{20\!\cdots\!37}a^{13}+\frac{74\!\cdots\!27}{25\!\cdots\!25}a^{12}+\frac{15\!\cdots\!66}{51\!\cdots\!25}a^{11}-\frac{15\!\cdots\!31}{25\!\cdots\!25}a^{10}-\frac{18\!\cdots\!18}{51\!\cdots\!25}a^{9}-\frac{29\!\cdots\!18}{25\!\cdots\!25}a^{8}+\frac{26\!\cdots\!54}{51\!\cdots\!25}a^{7}+\frac{62\!\cdots\!48}{51\!\cdots\!25}a^{6}-\frac{46\!\cdots\!96}{51\!\cdots\!25}a^{5}-\frac{33\!\cdots\!07}{37\!\cdots\!75}a^{4}+\frac{29\!\cdots\!43}{51\!\cdots\!25}a^{3}+\frac{64\!\cdots\!96}{70\!\cdots\!25}a^{2}-\frac{67\!\cdots\!03}{10\!\cdots\!85}a-\frac{10\!\cdots\!01}{10\!\cdots\!85}$, $\frac{84\!\cdots\!83}{25\!\cdots\!25}a^{27}-\frac{49\!\cdots\!66}{25\!\cdots\!25}a^{26}+\frac{25\!\cdots\!72}{51\!\cdots\!25}a^{25}+\frac{12\!\cdots\!58}{25\!\cdots\!25}a^{24}-\frac{13\!\cdots\!82}{51\!\cdots\!25}a^{23}-\frac{25\!\cdots\!84}{25\!\cdots\!25}a^{22}+\frac{87\!\cdots\!18}{10\!\cdots\!85}a^{21}-\frac{44\!\cdots\!62}{25\!\cdots\!25}a^{20}-\frac{80\!\cdots\!68}{29\!\cdots\!91}a^{19}+\frac{99\!\cdots\!13}{51\!\cdots\!25}a^{18}+\frac{16\!\cdots\!83}{51\!\cdots\!25}a^{17}+\frac{54\!\cdots\!29}{25\!\cdots\!25}a^{16}+\frac{29\!\cdots\!93}{25\!\cdots\!25}a^{15}-\frac{78\!\cdots\!73}{25\!\cdots\!25}a^{14}+\frac{19\!\cdots\!33}{51\!\cdots\!25}a^{13}+\frac{10\!\cdots\!43}{25\!\cdots\!25}a^{12}+\frac{55\!\cdots\!34}{20\!\cdots\!37}a^{11}+\frac{56\!\cdots\!76}{25\!\cdots\!25}a^{10}-\frac{44\!\cdots\!91}{51\!\cdots\!25}a^{9}-\frac{30\!\cdots\!27}{25\!\cdots\!25}a^{8}+\frac{22\!\cdots\!86}{51\!\cdots\!25}a^{7}+\frac{93\!\cdots\!77}{10\!\cdots\!85}a^{6}+\frac{64\!\cdots\!18}{51\!\cdots\!25}a^{5}+\frac{20\!\cdots\!17}{12\!\cdots\!75}a^{4}-\frac{97\!\cdots\!98}{51\!\cdots\!25}a^{3}-\frac{16\!\cdots\!17}{25\!\cdots\!25}a^{2}+\frac{95\!\cdots\!31}{23\!\cdots\!33}a-\frac{10\!\cdots\!64}{10\!\cdots\!85}$, $\frac{90\!\cdots\!76}{25\!\cdots\!25}a^{27}-\frac{26\!\cdots\!42}{25\!\cdots\!25}a^{26}+\frac{36\!\cdots\!36}{51\!\cdots\!25}a^{25}+\frac{46\!\cdots\!31}{25\!\cdots\!25}a^{24}-\frac{10\!\cdots\!29}{10\!\cdots\!85}a^{23}-\frac{73\!\cdots\!53}{25\!\cdots\!25}a^{22}+\frac{52\!\cdots\!11}{51\!\cdots\!25}a^{21}+\frac{17\!\cdots\!21}{25\!\cdots\!25}a^{20}-\frac{26\!\cdots\!86}{74\!\cdots\!75}a^{19}+\frac{32\!\cdots\!22}{10\!\cdots\!85}a^{18}+\frac{54\!\cdots\!78}{51\!\cdots\!25}a^{17}+\frac{97\!\cdots\!68}{25\!\cdots\!25}a^{16}+\frac{80\!\cdots\!16}{25\!\cdots\!25}a^{15}-\frac{10\!\cdots\!61}{25\!\cdots\!25}a^{14}-\frac{21\!\cdots\!62}{51\!\cdots\!25}a^{13}+\frac{10\!\cdots\!26}{25\!\cdots\!25}a^{12}-\frac{79\!\cdots\!01}{51\!\cdots\!25}a^{11}+\frac{14\!\cdots\!47}{42\!\cdots\!25}a^{10}+\frac{14\!\cdots\!24}{51\!\cdots\!25}a^{9}-\frac{35\!\cdots\!34}{25\!\cdots\!25}a^{8}+\frac{28\!\cdots\!37}{10\!\cdots\!85}a^{7}+\frac{40\!\cdots\!64}{51\!\cdots\!25}a^{6}+\frac{26\!\cdots\!53}{85\!\cdots\!25}a^{5}+\frac{10\!\cdots\!23}{86\!\cdots\!25}a^{4}-\frac{87\!\cdots\!02}{85\!\cdots\!25}a^{3}-\frac{19\!\cdots\!14}{25\!\cdots\!25}a^{2}+\frac{27\!\cdots\!47}{11\!\cdots\!65}a+\frac{26\!\cdots\!52}{10\!\cdots\!85}$ 40603611874649439518491/25974637773290955737442125a^27 - 5269634338290712596937/633527750568072091157125a^26 + 114159211189962590467951/5194927554658191147488425a^25 + 706451731775966739989011/25974637773290955737442125a^24 - 8306878006945618860367/85162746797675264712925a^23 + 20096068037725773156797/25974637773290955737442125a^22 + 2267240751366317082028864/5194927554658191147488425a^21 - 19962429187171367572038604/25974637773290955737442125a^20 - 3267603298016871360334/20057635346170622191075a^19 + 5395404337246912863838968/5194927554658191147488425a^18 + 3408102744991121427243383/5194927554658191147488425a^17 - 25528448550219241284716777/25974637773290955737442125a^16 + 54340909133448712128964571/25974637773290955737442125a^15 - 227329488050952565011623446/25974637773290955737442125a^14 + 6518474076240224478124498/5194927554658191147488425a^13 + 348476376986478165096751181/25974637773290955737442125a^12 - 7610409515365787578713878/5194927554658191147488425a^11 + 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3634528336039935530536/5194927554658191147488425a^25 + 460076140795264679200231/25974637773290955737442125a^24 - 10211633590491645053229/1038985510931638229497685a^23 - 730570168777332865726453/25974637773290955737442125a^22 + 529669051079418538100411/5194927554658191147488425a^21 + 174689048653811094805921/25974637773290955737442125a^20 - 265191837510475787954386/742132507808313021069775a^19 + 328703293790369419011422/1038985510931638229497685a^18 + 543750693964061277579378/5194927554658191147488425a^17 + 9731350020897723191884768/25974637773290955737442125a^16 + 8026552737956723118358216/25974637773290955737442125a^15 - 10035278972826136529764461/25974637773290955737442125a^14 - 21792185827740069999711262/5194927554658191147488425a^13 + 109990530925999472774379726/25974637773290955737442125a^12 - 798558268561699099482701/5194927554658191147488425a^11 + 1464430610512218218374247/425813733988376323564625a^10 + 14914209589523876792620724/5194927554658191147488425a^9 - 358764358110445135520226734/25974637773290955737442125a^8 + 2853764210696602965568237/1038985510931638229497685a^7 + 4026224206367251740123864/5194927554658191147488425a^6 + 265996642286902645890253/85162746797675264712925a^5 + 1040159363332481825398323/86294477652129421054625a^4 - 875196094590439457232902/85162746797675264712925a^3 - 191520413141165174337431414/25974637773290955737442125a^2 + 27067159350859319068547/11673994504849867747165*a + 269549958255252205129352/1038985510931638229497685 (assuming GRH)	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	$ 4093870424302.4214 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 4093870424302.4214 \cdot 1}{2\cdot\sqrt{20444429365013571404907915184985572799247638801}}\cr\approx \mathstrut & 2.13961068156221 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675)

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^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675, 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^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675);

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^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675);

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

$D_{28}$ (as 28T10):

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 56

The 17 conjugacy class representatives for $D_{28}$

Character table for $D_{28}$

Intermediate fields

$\Q(\sqrt{-7}) $, 4.0.22001.1, 7.1.31047965207.1, 14.0.6747833004465577869943.1

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 28 sibling:	deg 28
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	${\href{/padicField/2.7.0.1}{7} }^{4}$	$28$	${\href{/padicField/5.2.0.1}{2} }^{14}$	R	${\href{/padicField/11.14.0.1}{14} }^{2}$	$28$	$28$	$28$	${\href{/padicField/23.14.0.1}{14} }^{2}$	${\href{/padicField/29.2.0.1}{2} }^{13}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	${\href{/padicField/31.4.0.1}{4} }^{7}$	${\href{/padicField/37.2.0.1}{2} }^{13}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{14}$	${\href{/padicField/43.2.0.1}{2} }^{13}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.4.0.1}{4} }^{7}$	${\href{/padicField/53.14.0.1}{14} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{14}$

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
$7$ 7	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$449$ 449	$\Q_{449}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{449}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.449.2t1.a.a	$1$	$ 449 $	$\Q(\sqrt{449}) $	$C_2$ (as 2T1)	$1$	$1$
	1.3143.2t1.a.a	$1$	$ 7 \cdot 449 $	$\Q(\sqrt{-3143}) $	$C_2$ (as 2T1)	$1$	$-1$
*	1.7.2t1.a.a	$1$	$ 7 $	$\Q(\sqrt{-7}) $	$C_2$ (as 2T1)	$1$	$-1$
*	2.3143.4t3.c.a	$2$	$ 7 \cdot 449 $	4.2.1411207.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.3143.7t2.a.b	$2$	$ 7 \cdot 449 $	7.1.31047965207.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3143.14t3.a.a	$2$	$ 7 \cdot 449 $	14.2.432825288429292066229201.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3143.14t3.a.c	$2$	$ 7 \cdot 449 $	14.2.432825288429292066229201.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3143.7t2.a.a	$2$	$ 7 \cdot 449 $	7.1.31047965207.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3143.7t2.a.c	$2$	$ 7 \cdot 449 $	7.1.31047965207.1	$D_{7}$ (as 7T2)	$1$	$0$
*	2.3143.14t3.a.b	$2$	$ 7 \cdot 449 $	14.2.432825288429292066229201.1	$D_{14}$ (as 14T3)	$1$	$0$
*	2.3143.28t10.a.b	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3143.28t10.a.c	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3143.28t10.a.a	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3143.28t10.a.d	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3143.28t10.a.e	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$
*	2.3143.28t10.a.f	$2$	$ 7 \cdot 449 $	28.0.20444429365013571404907915184985572799247638801.1	$D_{28}$ (as 28T10)	$1$	$0$

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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