1. Introduction. Let N ≥ 1 be an integer and let X 0 (N ) be the modular curve over Q which corresponds to the modular group Γ 0 (N ). As a defining equation of X 0 (N ) we have the so-called modular equation of level N . It has many good properties, e.g. it reflects the defining property of X 0 (N ), it is the coarse moduli space of the isomorphism classes of the generalized elliptic curves with a cyclic subgroup of order N . But its degree and coefficients are too large to be applied to practical calculations on X 0 (N ). While it is an important problem to determine the algebraic points on X 0 (N ), we need a more manageable defining equation, which will also help to solve other related problems. In the case of a hyperelliptic modular curve, a kind of normal form of a defining equation is given by N. Murabayashi ([9]) and M. Shimura ([13]).In this paper, we give a relation between the modular equation of level N and the normal form in the case of a hyperelliptic modular curve X 0 (N ) except for N = 40, 48. First recall that the modular equation of level N is written in the following form:
lbr~mrl madim .~d ~0.sk.rk 1-inw6im-dth o b t m w ~ f i ~ Tho. prtiat, a JO d8y8 old 9/15,7 6-ti-d i m k l modim dell-, (w + CIa)/lOO m l OLB, 24.7/29.1 ml v m 19.8 m l l a o a t r o b. S o d i m at tho dllu-w-t, (WqQO + C11. 1 X 100, 60.6/ 56.5 n, 86.7 $ i. ~ t r o b. Reputed m t l a l o m at 17-0 oi H L-t i. I z d i a p m d m l m a d i m 9 d M ~ t e ~ j k tia, but a .od-to Lpzwmmont in diatal .e @ ~ 6 8 t i ~ 4 in modllm rodmorptioa at tb illlut*-t. Thim o u e demmmtrrtoa thrt hlaotioarS ab-tiem I n obmtrPatit. twopathy of W a n-oy .y 8-both pmxbal d d i n k 1 puk oi the n o-, d-.that ~ 8-of tPI h h r hraoaat .rj not obli&ewl4 f o l l s r domoktmatioa. 8 L. C l t f S S B ; A. L. * ola P. (I d s-: Bttdiem o ILr urd K ~ ~ E U O O in B a r t t e r ' ~ SJmdr. Thin report acmoerrm two ohildren, one male and one female, fifteen Md semn yeart old roe peotival, disgnosea aa Bsrtter's Symimme. A sodlam and potasaim balanoe wsrr done in e.ob one of thane two patients under the next oondi-tionmi) B ~ a l ~ituation;2 overload of C l q a t a dose of 200 m E q / d ; 3 spiranolaatone a t a dome of 3 rypg ; 4) ClK a t 10 mEdps/24h; 5) C ~ K at 10 24h(C-e 1)and 7 mWlk/24h , plum ClBa a t a dome a t ' 4 d!Q/pg/24h; 7 d K c c l 2 4 b PI-C-, 4 WW24h PI-He glummate, 6 W p g / 2 4 h r Dming eaoh on. of theme m o a n , a d a i l y om-fro1 af plama~, rPrinrry snd lntrssrithroo)lr+r m and K oauoentntion vsll done, srr well i~ am driv food intake .rd fasoal exomtian; mid-m e oontrol wcu, also dme. 1Zt ham beon po~sible t o ua, t o demonstrate the tuietenoe af a 19s loam nyndrame in these two patients. 2.-Uhder C1K overload, these two pat i e n t ~ reaohed the higher plaxm K oonoentrstia* with p i t i-pQtaaGI1111 balance, proving, a1-thou& iaoaopleteb. the nemtive rodira balan-oe d a t i n g previ&~. 3.-In o m experienoe, it 88-t h a t the m i t i p a t i o n of the b8balan-00 b e d i f 6 the p0t~8iUll baleme, h both patient ., 4.-W e oan't prove olearly the biogbnc t i 0 aiieot af Ilg aa 1BI utd K m e t a b o l ~ ; but we oan muppa~e it, a t loam*, i n a m af them two patient... DDAVP-test for estimation of renal concentration capacity in infante and children. A new method for estimation of renal concentrating perfonnance by intranasal administration of DDAVP (1-deamino-8-D-argi-nine-vasopressin) has been tested in 79 children and 25 infants. By comparative studies of different doses of intrqvenous and intranasal DDAVP it has been possible to elaborate a standard procedure using 20 pg DDAVP in children and 10 pg DD-ZVP in infants by the intranasal route. The DDAVP-test with none or only moderate short-term fluidrestriotion yields urine oemolality values equiva!.ent to those after 22 hours of prolonged dehydration and significantly higher Ohan those after combined pitressin and fluid deprivation test. In alinear study of the urine ooncen-trating performance postnatally in 28 infants as estimated by the DDAVP-test both lower maximum urine osmolality and shorter DDAVP-resgonse curves are found in preterm and asphyxiated babies. The lastme...
An elliptic curve E. defined over l is called a Q-curve if E and E are isogenous over Q for any ff in Gal(Q/Q). Many examples of Q-curves defined over quadratic fields have already been known. In this paper, we will give families of Q-curves defined over quartic and octic number fields.
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