LMFDB - Number field 24.0.3785323726785214561740247642669056.1

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)

gp: K = bnfinit(y^24 - 6*y^23 + 18*y^22 - 32*y^21 + 30*y^20 + 4*y^19 - 62*y^18 + 110*y^17 - 118*y^16 + 44*y^15 + 206*y^14 - 696*y^13 + 1249*y^12 - 1392*y^11 + 824*y^10 + 352*y^9 - 1888*y^8 + 3520*y^7 - 3968*y^6 + 512*y^5 + 7680*y^4 - 16384*y^3 + 18432*y^2 - 12288*y + 4096, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)

$ x^{24} - 6 x^{23} + 18 x^{22} - 32 x^{21} + 30 x^{20} + 4 x^{19} - 62 x^{18} + 110 x^{17} - 118 x^{16} + \cdots + 4096 $ x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$24$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 12]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$3785323726785214561740247642669056$ 3785323726785214561740247642669056 $\medspace = 2^{24}\cdot 3^{12}\cdot 23^{4}\cdot 79^{8}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$25.07$	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:	$2\cdot 3^{1/2}23^{1/2}79^{1/2}\approx 147.66177569025777$
Ramified primes:	$2$, $3$, $23$, $79$ 2, 3, 23, 79	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Aut(K/\Q) }$:	$8$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:	unavailable$^{2048}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{13}-\frac{1}{2}a$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}+\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{8}a^{8}+\frac{3}{8}a^{6}+\frac{1}{4}a^{5}-\frac{7}{16}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{15}-\frac{1}{8}a^{14}-\frac{1}{16}a^{13}-\frac{1}{4}a^{12}-\frac{3}{16}a^{11}-\frac{3}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{4}a^{8}+\frac{3}{16}a^{7}-\frac{1}{8}a^{6}-\frac{15}{32}a^{5}-\frac{5}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{128}a^{18}-\frac{1}{64}a^{17}+\frac{1}{64}a^{16}-\frac{1}{16}a^{15}+\frac{7}{64}a^{14}-\frac{1}{32}a^{13}+\frac{1}{64}a^{12}+\frac{11}{64}a^{11}+\frac{9}{64}a^{10}+\frac{1}{32}a^{9}+\frac{7}{64}a^{8}-\frac{1}{4}a^{7}-\frac{15}{128}a^{6}-\frac{11}{32}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{256}a^{19}-\frac{1}{128}a^{17}-\frac{1}{64}a^{16}-\frac{1}{128}a^{15}+\frac{3}{32}a^{14}+\frac{29}{128}a^{13}+\frac{13}{128}a^{12}+\frac{31}{128}a^{11}-\frac{3}{32}a^{10}+\frac{11}{128}a^{9}+\frac{15}{64}a^{8}-\frac{15}{256}a^{7}+\frac{27}{128}a^{6}-\frac{3}{32}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{512}a^{20}-\frac{1}{256}a^{18}-\frac{1}{128}a^{17}-\frac{1}{256}a^{16}+\frac{3}{64}a^{15}+\frac{29}{256}a^{14}+\frac{13}{256}a^{13}+\frac{31}{256}a^{12}-\frac{3}{64}a^{11}-\frac{53}{256}a^{10}-\frac{17}{128}a^{9}+\frac{113}{512}a^{8}+\frac{27}{256}a^{7}+\frac{29}{64}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}+\frac{3}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{1024}a^{21}-\frac{1}{512}a^{19}-\frac{1}{256}a^{18}-\frac{1}{512}a^{17}+\frac{3}{128}a^{16}+\frac{29}{512}a^{15}+\frac{13}{512}a^{14}-\frac{97}{512}a^{13}-\frac{3}{128}a^{12}-\frac{53}{512}a^{11}+\frac{47}{256}a^{10}-\frac{143}{1024}a^{9}+\frac{27}{512}a^{8}-\frac{3}{128}a^{7}-\frac{1}{16}a^{6}-\frac{1}{16}a^{5}+\frac{3}{8}a^{4}+\frac{7}{16}a^{3}+\frac{3}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{698368}a^{22}-\frac{19}{87296}a^{21}-\frac{149}{349184}a^{20}+\frac{3}{15872}a^{19}-\frac{457}{349184}a^{18}+\frac{1303}{87296}a^{17}+\frac{9909}{349184}a^{16}-\frac{19427}{349184}a^{15}+\frac{41407}{349184}a^{14}-\frac{21185}{87296}a^{13}+\frac{41291}{349184}a^{12}+\frac{31567}{174592}a^{11}-\frac{129119}{698368}a^{10}+\frac{41255}{349184}a^{9}-\frac{8379}{43648}a^{8}+\frac{16749}{87296}a^{7}-\frac{2307}{43648}a^{6}+\frac{5309}{10912}a^{5}-\frac{3533}{10912}a^{4}+\frac{179}{496}a^{3}-\frac{149}{682}a^{2}+\frac{37}{1364}a-\frac{339}{682}$, $\frac{1}{101961728}a^{23}-\frac{19}{50980864}a^{22}-\frac{20407}{50980864}a^{21}-\frac{8663}{12745216}a^{20}+\frac{5703}{50980864}a^{19}+\frac{14749}{25490432}a^{18}-\frac{49279}{50980864}a^{17}-\frac{311089}{50980864}a^{16}-\frac{2436523}{50980864}a^{15}+\frac{2629503}{25490432}a^{14}-\frac{8141857}{50980864}a^{13}+\frac{279379}{12745216}a^{12}-\frac{23858367}{101961728}a^{11}+\frac{168327}{3186304}a^{10}+\frac{29709}{1158656}a^{9}+\frac{1579081}{12745216}a^{8}-\frac{73361}{796576}a^{7}-\frac{813111}{3186304}a^{6}-\frac{451997}{1593152}a^{5}+\frac{41535}{199144}a^{4}+\frac{4910}{24893}a^{3}+\frac{2309}{18104}a^{2}+\frac{8456}{24893}a-\frac{9093}{49786}$ 1, a, a^2, a^3, a^4, a^5, a^6, 1/2*a^7 - 1/2*a, 1/2*a^8 - 1/2*a^2, 1/2*a^9 - 1/2*a^3, 1/2*a^10 - 1/2*a^4, 1/2*a^11 - 1/2*a^5, 1/2*a^12 - 1/2*a^6, 1/2*a^13 - 1/2*a, 1/4*a^14 - 1/4*a^2, 1/8*a^15 - 1/4*a^13 - 1/4*a^11 - 1/4*a^9 - 1/4*a^8 - 1/4*a^7 - 1/2*a^6 - 1/4*a^5 - 1/2*a^4 - 3/8*a^3 + 1/4*a^2, 1/16*a^16 - 1/8*a^14 - 1/4*a^13 - 1/8*a^12 + 1/8*a^10 + 1/8*a^9 - 1/8*a^8 + 3/8*a^6 + 1/4*a^5 - 7/16*a^4 - 1/8*a^3 - 1/2*a^2 - 1/2*a, 1/32*a^17 - 1/16*a^15 - 1/8*a^14 - 1/16*a^13 - 1/4*a^12 - 3/16*a^11 - 3/16*a^10 - 1/16*a^9 - 1/4*a^8 + 3/16*a^7 - 1/8*a^6 - 15/32*a^5 - 5/16*a^4 + 1/4*a^3 - 1/2*a^2 - 1/2*a, 1/128*a^18 - 1/64*a^17 + 1/64*a^16 - 1/16*a^15 + 7/64*a^14 - 1/32*a^13 + 1/64*a^12 + 11/64*a^11 + 9/64*a^10 + 1/32*a^9 + 7/64*a^8 - 1/4*a^7 - 15/128*a^6 - 11/32*a^5 - 1/2*a^4 - 1/2*a^2 - 1/2*a - 1/2, 1/256*a^19 - 1/128*a^17 - 1/64*a^16 - 1/128*a^15 + 3/32*a^14 + 29/128*a^13 + 13/128*a^12 + 31/128*a^11 - 3/32*a^10 + 11/128*a^9 + 15/64*a^8 - 15/256*a^7 + 27/128*a^6 - 3/32*a^5 - 1/4*a^4 + 1/4*a^3 + 1/4*a - 1/2, 1/512*a^20 - 1/256*a^18 - 1/128*a^17 - 1/256*a^16 + 3/64*a^15 + 29/256*a^14 + 13/256*a^13 + 31/256*a^12 - 3/64*a^11 - 53/256*a^10 - 17/128*a^9 + 113/512*a^8 + 27/256*a^7 + 29/64*a^6 - 1/8*a^5 - 1/8*a^4 + 1/4*a^3 + 3/8*a^2 - 1/4*a, 1/1024*a^21 - 1/512*a^19 - 1/256*a^18 - 1/512*a^17 + 3/128*a^16 + 29/512*a^15 + 13/512*a^14 - 97/512*a^13 - 3/128*a^12 - 53/512*a^11 + 47/256*a^10 - 143/1024*a^9 + 27/512*a^8 - 3/128*a^7 - 1/16*a^6 - 1/16*a^5 + 3/8*a^4 + 7/16*a^3 + 3/8*a^2 - 1/2*a, 1/698368*a^22 - 19/87296*a^21 - 149/349184*a^20 + 3/15872*a^19 - 457/349184*a^18 + 1303/87296*a^17 + 9909/349184*a^16 - 19427/349184*a^15 + 41407/349184*a^14 - 21185/87296*a^13 + 41291/349184*a^12 + 31567/174592*a^11 - 129119/698368*a^10 + 41255/349184*a^9 - 8379/43648*a^8 + 16749/87296*a^7 - 2307/43648*a^6 + 5309/10912*a^5 - 3533/10912*a^4 + 179/496*a^3 - 149/682*a^2 + 37/1364*a - 339/682, 1/101961728*a^23 - 19/50980864*a^22 - 20407/50980864*a^21 - 8663/12745216*a^20 + 5703/50980864*a^19 + 14749/25490432*a^18 - 49279/50980864*a^17 - 311089/50980864*a^16 - 2436523/50980864*a^15 + 2629503/25490432*a^14 - 8141857/50980864*a^13 + 279379/12745216*a^12 - 23858367/101961728*a^11 + 168327/3186304*a^10 + 29709/1158656*a^9 + 1579081/12745216*a^8 - 73361/796576*a^7 - 813111/3186304*a^6 - 451997/1593152*a^5 + 41535/199144*a^4 + 4910/24893*a^3 + 2309/18104*a^2 + 8456/24893*a - 9093/49786

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{4}$, which has order $4$ (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:	$11$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -\frac{290201}{50980864} a^{23} + \frac{163497}{4634624} a^{22} - \frac{2171221}{25490432} a^{21} + \frac{2715659}{25490432} a^{20} - \frac{571313}{25490432} a^{19} - \frac{3867623}{25490432} a^{18} + \frac{7251851}{25490432} a^{17} - \frac{4062931}{12745216} a^{16} + \frac{1472277}{6372608} a^{15} + \frac{4176357}{25490432} a^{14} - \frac{3346569}{2317312} a^{13} + \frac{7550321}{2317312} a^{12} - \frac{205942813}{50980864} a^{11} + \frac{128302637}{50980864} a^{10} + \frac{14494175}{25490432} a^{9} - \frac{23181691}{6372608} a^{8} + \frac{26950403}{3186304} a^{7} - \frac{3669117}{289664} a^{6} + \frac{5971125}{796576} a^{5} + \frac{12539553}{796576} a^{4} - \frac{17844963}{398288} a^{3} + \frac{5194591}{99572} a^{2} - \frac{789966}{24893} a + \frac{266757}{49786} $ -(290201)/(50980864)a^(23) + (163497)/(4634624)a^(22) - (2171221)/(25490432)a^(21) + (2715659)/(25490432)a^(20) - (571313)/(25490432)a^(19) - (3867623)/(25490432)a^(18) + (7251851)/(25490432)a^(17) - (4062931)/(12745216)a^(16) + (1472277)/(6372608)a^(15) + (4176357)/(25490432)a^(14) - (3346569)/(2317312)a^(13) + (7550321)/(2317312)a^(12) - (205942813)/(50980864)a^(11) + (128302637)/(50980864)a^(10) + (14494175)/(25490432)a^(9) - (23181691)/(6372608)a^(8) + (26950403)/(3186304)a^(7) - (3669117)/(289664)a^(6) + (5971125)/(796576)a^(5) + (12539553)/(796576)a^(4) - (17844963)/(398288)a^(3) + (5194591)/(99572)a^(2) - (789966)/(24893)*a + (266757)/(49786) (order $12$)	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:	$\frac{40555}{12745216}a^{23}-\frac{194089}{4634624}a^{22}+\frac{3607173}{25490432}a^{21}-\frac{6490327}{25490432}a^{20}+\frac{1331661}{6372608}a^{19}+\frac{3178139}{25490432}a^{18}-\frac{6992917}{12745216}a^{17}+\frac{20594913}{25490432}a^{16}-\frac{19040545}{25490432}a^{15}+\frac{7047529}{25490432}a^{14}+\frac{1854399}{1158656}a^{13}-\frac{13116367}{2317312}a^{12}+\frac{121459677}{12745216}a^{11}-\frac{475431371}{50980864}a^{10}+\frac{5046705}{1593152}a^{9}+\frac{15123755}{3186304}a^{8}-\frac{45993421}{3186304}a^{7}+\frac{61003}{2263}a^{6}-\frac{23637937}{796576}a^{5}-\frac{1282907}{796576}a^{4}+\frac{1822977}{24893}a^{3}-\frac{12850479}{99572}a^{2}+\frac{2920321}{24893}a-\frac{40477}{803}$, $\frac{1183995}{101961728}a^{23}-\frac{736389}{12745216}a^{22}+\frac{6502739}{50980864}a^{21}-\frac{3432733}{25490432}a^{20}-\frac{710583}{50980864}a^{19}+\frac{3380715}{12745216}a^{18}-\frac{21172733}{50980864}a^{17}+\frac{1827197}{4634624}a^{16}-\frac{12706615}{50980864}a^{15}-\frac{2756029}{6372608}a^{14}+\frac{120877685}{50980864}a^{13}-\frac{3969677}{822272}a^{12}+\frac{537366099}{101961728}a^{11}-\frac{118703131}{50980864}a^{10}-\frac{51054143}{25490432}a^{9}+\frac{38509689}{6372608}a^{8}-\frac{79204525}{6372608}a^{7}+\frac{27888699}{1593152}a^{6}-\frac{881355}{144832}a^{5}-\frac{24212777}{796576}a^{4}+\frac{26848109}{398288}a^{3}-\frac{6629661}{99572}a^{2}+\frac{2896475}{99572}a+\frac{49987}{24893}$, $\frac{1348585}{101961728}a^{23}-\frac{2446053}{50980864}a^{22}+\frac{328157}{4634624}a^{21}-\frac{60867}{6372608}a^{20}-\frac{7506013}{50980864}a^{19}+\frac{6005075}{25490432}a^{18}-\frac{8664083}{50980864}a^{17}+\frac{499859}{50980864}a^{16}+\frac{7208565}{50980864}a^{15}-\frac{1685371}{2317312}a^{14}+\frac{102714715}{50980864}a^{13}-\frac{7805121}{3186304}a^{12}+\frac{71380865}{101961728}a^{11}+\frac{67525839}{25490432}a^{10}-\frac{102588561}{25490432}a^{9}+\frac{29325991}{6372608}a^{8}-\frac{44216279}{6372608}a^{7}+\frac{237007}{51392}a^{6}+\frac{17589027}{1593152}a^{5}-\frac{1798383}{49786}a^{4}+\frac{14538257}{398288}a^{3}-\frac{104279}{24893}a^{2}-\frac{3165559}{99572}a+\frac{739364}{24893}$, $\frac{232585}{101961728}a^{23}-\frac{1292087}{50980864}a^{22}+\frac{4067353}{50980864}a^{21}-\frac{1702265}{12745216}a^{20}+\frac{4257423}{50980864}a^{19}+\frac{2735185}{25490432}a^{18}-\frac{1427637}{4634624}a^{17}+\frac{20088399}{50980864}a^{16}-\frac{17115083}{50980864}a^{15}+\frac{2180811}{25490432}a^{14}+\frac{53476295}{50980864}a^{13}-\frac{40795775}{12745216}a^{12}+\frac{45822587}{9269248}a^{11}-\frac{4787149}{1158656}a^{10}+\frac{8701519}{12745216}a^{9}+\frac{37999765}{12745216}a^{8}-\frac{25547041}{3186304}a^{7}+\frac{45418589}{3186304}a^{6}-\frac{22703305}{1593152}a^{5}-\frac{2618027}{398288}a^{4}+\frac{1089147}{24893}a^{3}-\frac{13044697}{199144}a^{2}+\frac{116945}{2263}a-\frac{851467}{49786}$, $\frac{19365}{4634624}a^{23}-\frac{76957}{2317312}a^{22}+\frac{117057}{1158656}a^{21}-\frac{24043}{144832}a^{20}+\frac{263637}{2317312}a^{19}+\frac{124817}{1158656}a^{18}-\frac{879949}{2317312}a^{17}+\frac{1196407}{2317312}a^{16}-\frac{1081097}{2317312}a^{15}+\frac{47953}{579328}a^{14}+\frac{3015721}{2317312}a^{13}-\frac{1139815}{289664}a^{12}+\frac{28950785}{4634624}a^{11}-\frac{6472553}{1158656}a^{10}+\frac{3734453}{2317312}a^{9}+\frac{4267497}{1158656}a^{8}-\frac{5878797}{579328}a^{7}+\frac{40654}{2263}a^{6}-\frac{1286975}{72416}a^{5}-\frac{56615}{9052}a^{4}+\frac{1852855}{36208}a^{3}-\frac{1509913}{18104}a^{2}+\frac{640699}{9052}a-\frac{2094}{73}$, $\frac{280105}{50980864}a^{23}-\frac{26529}{1158656}a^{22}+\frac{39707}{796576}a^{21}-\frac{607095}{12745216}a^{20}-\frac{391463}{25490432}a^{19}+\frac{617075}{6372608}a^{18}-\frac{4016549}{25490432}a^{17}+\frac{3647009}{25490432}a^{16}-\frac{2370099}{25490432}a^{15}-\frac{2648779}{12745216}a^{14}+\frac{2260955}{2317312}a^{13}-\frac{2069073}{1158656}a^{12}+\frac{97468229}{50980864}a^{11}-\frac{17545159}{25490432}a^{10}-\frac{17302447}{25490432}a^{9}+\frac{31680903}{12745216}a^{8}-\frac{31497419}{6372608}a^{7}+\frac{914577}{144832}a^{6}-\frac{1135293}{796576}a^{5}-\frac{5173177}{398288}a^{4}+\frac{9610619}{398288}a^{3}-\frac{4769203}{199144}a^{2}+\frac{1007673}{99572}a+\frac{1596}{803}$, $\frac{27641}{822272}a^{23}-\frac{141289}{822272}a^{22}+\frac{32183}{74752}a^{21}-\frac{117829}{205568}a^{20}+\frac{46019}{205568}a^{19}+\frac{266221}{411136}a^{18}-\frac{310417}{205568}a^{17}+\frac{371527}{205568}a^{16}-\frac{596139}{411136}a^{15}-\frac{13235}{18688}a^{14}+\frac{1416911}{205568}a^{13}-\frac{6795015}{411136}a^{12}+\frac{18355527}{822272}a^{11}-\frac{13583083}{822272}a^{10}+\frac{793049}{822272}a^{9}+\frac{958475}{51392}a^{8}-\frac{8910423}{205568}a^{7}+\frac{6918189}{102784}a^{6}-\frac{1196509}{25696}a^{5}-\frac{202471}{3212}a^{4}+\frac{2828743}{12848}a^{3}-\frac{235871}{803}a^{2}+\frac{683527}{3212}a-\frac{113777}{1606}$, $\frac{2378969}{101961728}a^{23}-\frac{5732669}{50980864}a^{22}+\frac{13710447}{50980864}a^{21}-\frac{2280053}{6372608}a^{20}+\frac{7182291}{50980864}a^{19}+\frac{10003179}{25490432}a^{18}-\frac{47868835}{50980864}a^{17}+\frac{60130451}{50980864}a^{16}-\frac{4310625}{4634624}a^{15}-\frac{12614801}{25490432}a^{14}+\frac{225838571}{50980864}a^{13}-\frac{8253467}{796576}a^{12}+\frac{1433502129}{101961728}a^{11}-\frac{274917653}{25490432}a^{10}+\frac{29386871}{25490432}a^{9}+\frac{74729483}{6372608}a^{8}-\frac{179139903}{6372608}a^{7}+\frac{67002473}{1593152}a^{6}-\frac{45806791}{1593152}a^{5}-\frac{15325435}{398288}a^{4}+\frac{54302393}{398288}a^{3}-\frac{18347093}{99572}a^{2}+\frac{14071291}{99572}a-\frac{1343283}{24893}$, $\frac{378493}{50980864}a^{23}-\frac{708155}{50980864}a^{22}-\frac{390581}{25490432}a^{21}+\frac{2653055}{25490432}a^{20}-\frac{4455371}{25490432}a^{19}+\frac{2123571}{25490432}a^{18}+\frac{3615957}{25490432}a^{17}-\frac{4073257}{12745216}a^{16}+\frac{209265}{579328}a^{15}-\frac{14886389}{25490432}a^{14}+\frac{14822391}{25490432}a^{13}+\frac{22364553}{25490432}a^{12}-\frac{186572791}{50980864}a^{11}+\frac{269448027}{50980864}a^{10}-\frac{83685341}{25490432}a^{9}+\frac{8982729}{12745216}a^{8}+\frac{1207689}{796576}a^{7}-\frac{12793463}{1593152}a^{6}+\frac{15992247}{796576}a^{5}-\frac{16437583}{796576}a^{4}-\frac{4159901}{398288}a^{3}+\frac{10579645}{199144}a^{2}-\frac{1640168}{24893}a+\frac{774657}{24893}$, $\frac{106389}{25490432}a^{23}-\frac{14793}{6372608}a^{22}-\frac{708369}{25490432}a^{21}+\frac{524097}{6372608}a^{20}-\frac{762797}{6372608}a^{19}+\frac{182803}{6372608}a^{18}+\frac{9715}{72416}a^{17}-\frac{3131175}{12745216}a^{16}+\frac{1047901}{3186304}a^{15}-\frac{134061}{411136}a^{14}+\frac{864099}{6372608}a^{13}+\frac{7679099}{6372608}a^{12}-\frac{6985247}{2317312}a^{11}+\frac{4462553}{1158656}a^{10}-\frac{62497527}{25490432}a^{9}-\frac{1997381}{12745216}a^{8}+\frac{1768435}{796576}a^{7}-\frac{6795073}{796576}a^{6}+\frac{667031}{49786}a^{5}-\frac{4257589}{398288}a^{4}-\frac{5283199}{398288}a^{3}+\frac{8720531}{199144}a^{2}-\frac{103384}{2263}a+\frac{741428}{24893}$, $\frac{70913}{6372608}a^{23}-\frac{55139}{579328}a^{22}+\frac{971233}{3186304}a^{21}-\frac{6501467}{12745216}a^{20}+\frac{284147}{796576}a^{19}+\frac{2194959}{6372608}a^{18}-\frac{938445}{796576}a^{17}+\frac{9912043}{6372608}a^{16}-\frac{2186161}{1593152}a^{15}+\frac{1935499}{6372608}a^{14}+\frac{2210717}{579328}a^{13}-\frac{6984273}{579328}a^{12}+\frac{122442217}{6372608}a^{11}-\frac{27049067}{1593152}a^{10}+\frac{828093}{199144}a^{9}+\frac{144850641}{12745216}a^{8}-\frac{192034719}{6372608}a^{7}+\frac{15801233}{289664}a^{6}-\frac{22248047}{398288}a^{5}-\frac{452720}{24893}a^{4}+\frac{31889207}{199144}a^{3}-\frac{51292273}{199144}a^{2}+\frac{21227917}{99572}a-\frac{125519}{1606}$ 40555/12745216a^23 - 194089/4634624a^22 + 3607173/25490432a^21 - 6490327/25490432a^20 + 1331661/6372608a^19 + 3178139/25490432a^18 - 6992917/12745216a^17 + 20594913/25490432a^16 - 19040545/25490432a^15 + 7047529/25490432a^14 + 1854399/1158656a^13 - 13116367/2317312a^12 + 121459677/12745216a^11 - 475431371/50980864a^10 + 5046705/1593152a^9 + 15123755/3186304a^8 - 45993421/3186304a^7 + 61003/2263a^6 - 23637937/796576a^5 - 1282907/796576a^4 + 1822977/24893a^3 - 12850479/99572a^2 + 2920321/24893a - 40477/803, 1183995/101961728a^23 - 736389/12745216a^22 + 6502739/50980864a^21 - 3432733/25490432a^20 - 710583/50980864a^19 + 3380715/12745216a^18 - 21172733/50980864a^17 + 1827197/4634624a^16 - 12706615/50980864a^15 - 2756029/6372608a^14 + 120877685/50980864a^13 - 3969677/822272a^12 + 537366099/101961728a^11 - 118703131/50980864a^10 - 51054143/25490432a^9 + 38509689/6372608a^8 - 79204525/6372608a^7 + 27888699/1593152a^6 - 881355/144832a^5 - 24212777/796576a^4 + 26848109/398288a^3 - 6629661/99572a^2 + 2896475/99572a + 49987/24893, 1348585/101961728a^23 - 2446053/50980864a^22 + 328157/4634624a^21 - 60867/6372608a^20 - 7506013/50980864a^19 + 6005075/25490432a^18 - 8664083/50980864a^17 + 499859/50980864a^16 + 7208565/50980864a^15 - 1685371/2317312a^14 + 102714715/50980864a^13 - 7805121/3186304a^12 + 71380865/101961728a^11 + 67525839/25490432a^10 - 102588561/25490432a^9 + 29325991/6372608a^8 - 44216279/6372608a^7 + 237007/51392a^6 + 17589027/1593152a^5 - 1798383/49786a^4 + 14538257/398288a^3 - 104279/24893a^2 - 3165559/99572a + 739364/24893, 232585/101961728a^23 - 1292087/50980864a^22 + 4067353/50980864a^21 - 1702265/12745216a^20 + 4257423/50980864a^19 + 2735185/25490432a^18 - 1427637/4634624a^17 + 20088399/50980864a^16 - 17115083/50980864a^15 + 2180811/25490432a^14 + 53476295/50980864a^13 - 40795775/12745216a^12 + 45822587/9269248a^11 - 4787149/1158656a^10 + 8701519/12745216a^9 + 37999765/12745216a^8 - 25547041/3186304a^7 + 45418589/3186304a^6 - 22703305/1593152a^5 - 2618027/398288a^4 + 1089147/24893a^3 - 13044697/199144a^2 + 116945/2263a - 851467/49786, 19365/4634624a^23 - 76957/2317312a^22 + 117057/1158656a^21 - 24043/144832a^20 + 263637/2317312a^19 + 124817/1158656a^18 - 879949/2317312a^17 + 1196407/2317312a^16 - 1081097/2317312a^15 + 47953/579328a^14 + 3015721/2317312a^13 - 1139815/289664a^12 + 28950785/4634624a^11 - 6472553/1158656a^10 + 3734453/2317312a^9 + 4267497/1158656a^8 - 5878797/579328a^7 + 40654/2263a^6 - 1286975/72416a^5 - 56615/9052a^4 + 1852855/36208a^3 - 1509913/18104a^2 + 640699/9052a - 2094/73, 280105/50980864a^23 - 26529/1158656a^22 + 39707/796576a^21 - 607095/12745216a^20 - 391463/25490432a^19 + 617075/6372608a^18 - 4016549/25490432a^17 + 3647009/25490432a^16 - 2370099/25490432a^15 - 2648779/12745216a^14 + 2260955/2317312a^13 - 2069073/1158656a^12 + 97468229/50980864a^11 - 17545159/25490432a^10 - 17302447/25490432a^9 + 31680903/12745216a^8 - 31497419/6372608a^7 + 914577/144832a^6 - 1135293/796576a^5 - 5173177/398288a^4 + 9610619/398288a^3 - 4769203/199144a^2 + 1007673/99572a + 1596/803, 27641/822272a^23 - 141289/822272a^22 + 32183/74752a^21 - 117829/205568a^20 + 46019/205568a^19 + 266221/411136a^18 - 310417/205568a^17 + 371527/205568a^16 - 596139/411136a^15 - 13235/18688a^14 + 1416911/205568a^13 - 6795015/411136a^12 + 18355527/822272a^11 - 13583083/822272a^10 + 793049/822272a^9 + 958475/51392a^8 - 8910423/205568a^7 + 6918189/102784a^6 - 1196509/25696a^5 - 202471/3212a^4 + 2828743/12848a^3 - 235871/803a^2 + 683527/3212a - 113777/1606, 2378969/101961728a^23 - 5732669/50980864a^22 + 13710447/50980864a^21 - 2280053/6372608a^20 + 7182291/50980864a^19 + 10003179/25490432a^18 - 47868835/50980864a^17 + 60130451/50980864a^16 - 4310625/4634624a^15 - 12614801/25490432a^14 + 225838571/50980864a^13 - 8253467/796576a^12 + 1433502129/101961728a^11 - 274917653/25490432a^10 + 29386871/25490432a^9 + 74729483/6372608a^8 - 179139903/6372608a^7 + 67002473/1593152a^6 - 45806791/1593152a^5 - 15325435/398288a^4 + 54302393/398288a^3 - 18347093/99572a^2 + 14071291/99572a - 1343283/24893, 378493/50980864a^23 - 708155/50980864a^22 - 390581/25490432a^21 + 2653055/25490432a^20 - 4455371/25490432a^19 + 2123571/25490432a^18 + 3615957/25490432a^17 - 4073257/12745216a^16 + 209265/579328a^15 - 14886389/25490432a^14 + 14822391/25490432a^13 + 22364553/25490432a^12 - 186572791/50980864a^11 + 269448027/50980864a^10 - 83685341/25490432a^9 + 8982729/12745216a^8 + 1207689/796576a^7 - 12793463/1593152a^6 + 15992247/796576a^5 - 16437583/796576a^4 - 4159901/398288a^3 + 10579645/199144a^2 - 1640168/24893a + 774657/24893, 106389/25490432a^23 - 14793/6372608a^22 - 708369/25490432a^21 + 524097/6372608a^20 - 762797/6372608a^19 + 182803/6372608a^18 + 9715/72416a^17 - 3131175/12745216a^16 + 1047901/3186304a^15 - 134061/411136a^14 + 864099/6372608a^13 + 7679099/6372608a^12 - 6985247/2317312a^11 + 4462553/1158656a^10 - 62497527/25490432a^9 - 1997381/12745216a^8 + 1768435/796576a^7 - 6795073/796576a^6 + 667031/49786a^5 - 4257589/398288a^4 - 5283199/398288a^3 + 8720531/199144a^2 - 103384/2263a + 741428/24893, 70913/6372608a^23 - 55139/579328a^22 + 971233/3186304a^21 - 6501467/12745216a^20 + 284147/796576a^19 + 2194959/6372608a^18 - 938445/796576a^17 + 9912043/6372608a^16 - 2186161/1593152a^15 + 1935499/6372608a^14 + 2210717/579328a^13 - 6984273/579328a^12 + 122442217/6372608a^11 - 27049067/1593152a^10 + 828093/199144a^9 + 144850641/12745216a^8 - 192034719/6372608a^7 + 15801233/289664a^6 - 22248047/398288a^5 - 452720/24893a^4 + 31889207/199144a^3 - 51292273/199144a^2 + 21227917/99572*a - 125519/1606 (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:	$ 42401055.974884115 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 42401055.974884115 \cdot 4}{12\cdot\sqrt{3785323726785214561740247642669056}}\cr\approx \mathstrut & 0.869685691062020 \end{aligned}\] (assuming GRH)

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

x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096)

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^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096, 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^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);

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^24 - 6*x^23 + 18*x^22 - 32*x^21 + 30*x^20 + 4*x^19 - 62*x^18 + 110*x^17 - 118*x^16 + 44*x^15 + 206*x^14 - 696*x^13 + 1249*x^12 - 1392*x^11 + 824*x^10 + 352*x^9 - 1888*x^8 + 3520*x^7 - 3968*x^6 + 512*x^5 + 7680*x^4 - 16384*x^3 + 18432*x^2 - 12288*x + 4096);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^3\times S_4$ (as 24T400):

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 192

The 40 conjugacy class representatives for $C_2^3\times S_4$

Character table for $C_2^3\times S_4$

Intermediate fields

$\Q(\sqrt{-1}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{3}) $, 3.3.316.1, $\Q(\zeta_{12})$, 6.0.2296688.1, 6.0.399424.1, 6.0.2696112.1, 6.0.248042304.1, 6.6.9186752.1, 6.6.62010576.1, 6.6.10784448.1, 12.12.61524984573628416.1, 12.0.84396412309504.1, 12.0.61524984573628416.1, 12.0.3845311535851776.1, 12.0.61524984573628416.3, 12.0.116304318664704.2, 12.0.61524984573628416.2

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 24 siblings:	data not computed
Degree 32 siblings:	data not computed
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	R	R	${\href{/padicField/5.6.0.1}{6} }^{4}$	${\href{/padicField/7.6.0.1}{6} }^{4}$	${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$	${\href{/padicField/13.3.0.1}{3} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}$	${\href{/padicField/19.2.0.1}{2} }^{12}$	R	${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$	${\href{/padicField/31.2.0.1}{2} }^{12}$	${\href{/padicField/37.2.0.1}{2} }^{12}$	${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{12}$	${\href{/padicField/47.6.0.1}{6} }^{4}$	${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{4}$

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
$2$ 2	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$3$ 3	3.12.6.2	$x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$
$23$ 23	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	23.4.2.1	$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	23.4.0.1	$x^{4} + 3 x^{2} + 19 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$79$ 79	79.2.0.1	$x^{2} + 78 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	79.2.0.1	$x^{2} + 78 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	79.2.0.1	$x^{2} + 78 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	79.2.0.1	$x^{2} + 78 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	79.4.2.1	$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	79.4.2.1	$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	79.4.2.1	$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	79.4.2.1	$x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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