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Label Class Base field Conductor norm Rank Torsion CM Sato-Tate Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
79.3-a1 79.3-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.019263609$ $53060.19998$ 2.51617 \( -\frac{3250240894}{79} a^{5} + \frac{9006299501}{79} a^{4} + \frac{301818019}{79} a^{3} - \frac{13536315178}{79} a^{2} + \frac{4470082885}{79} a + \frac{1835599092}{79} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + a + 1\) , \( a^{4} - 4 a^{2} + a + 3\) , \( -3 a^{5} - a^{4} + 15 a^{3} + 6 a^{2} - 11 a - 2\) , \( 4 a^{5} + 3 a^{4} - 16 a^{3} - 11 a^{2} + 10 a + 4\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){x}{y}+\left(a^{4}-4a^{2}+a+3\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+a+1\right){x}^{2}+\left(-3a^{5}-a^{4}+15a^{3}+6a^{2}-11a-2\right){x}+4a^{5}+3a^{4}-16a^{3}-11a^{2}+10a+4$
79.3-a2 79.3-a \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.038527218$ $13265.04999$ 2.51617 \( -\frac{308721461939500395175}{6241} a^{5} + \frac{855440019384751139506}{6241} a^{4} + \frac{28698270198869399627}{6241} a^{3} - \frac{1285707960303197449691}{6241} a^{2} + \frac{424546949726100418011}{6241} a + \frac{174329076201954485036}{6241} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + a + 1\) , \( a^{4} - 4 a^{2} + a + 3\) , \( 2 a^{5} - 11 a^{4} + 26 a^{2} + 4 a - 7\) , \( -5 a^{5} + 16 a^{4} - 12 a^{3} - 22 a^{2} + 23 a\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-2a+3\right){x}{y}+\left(a^{4}-4a^{2}+a+3\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+a+1\right){x}^{2}+\left(2a^{5}-11a^{4}+26a^{2}+4a-7\right){x}-5a^{5}+16a^{4}-12a^{3}-22a^{2}+23a$
79.3-b1 79.3-b \(\Q(\zeta_{13})^+\) \( 79 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.001171369$ $107610.2235$ 2.48239 \( -\frac{21702535350}{6241} a^{5} - \frac{22817295041}{6241} a^{4} + \frac{67159076059}{6241} a^{3} + \frac{54772511851}{6241} a^{2} - \frac{36084307562}{6241} a - \frac{16274230797}{6241} \) \( \bigl[a^{5} - 5 a^{3} + 5 a\) , \( a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 9 a - 4\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -8 a^{5} + 3 a^{4} + 40 a^{3} - 8 a^{2} - 45 a\) , \( 9 a^{5} - 4 a^{4} - 48 a^{3} + 9 a^{2} + 59 a + 7\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+5a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+9a-4\right){x}^{2}+\left(-8a^{5}+3a^{4}+40a^{3}-8a^{2}-45a\right){x}+9a^{5}-4a^{4}-48a^{3}+9a^{2}+59a+7$
79.3-c1 79.3-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $45606.09482$ 1.49691 \( \frac{312138828714566}{6241} a^{5} - \frac{666874973719595}{6241} a^{4} - \frac{803095442738995}{6241} a^{3} + \frac{2161421201156764}{6241} a^{2} - \frac{582552361525647}{6241} a - \frac{274796597365423}{6241} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( a^{4} - 3 a^{2} - a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( -57 a^{5} + 16 a^{4} + 293 a^{3} - 17 a^{2} - 350 a - 80\) , \( 390 a^{5} - 113 a^{4} - 2030 a^{3} + 120 a^{2} + 2423 a + 548\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}-a-1\right){x}^{2}+\left(-57a^{5}+16a^{4}+293a^{3}-17a^{2}-350a-80\right){x}+390a^{5}-113a^{4}-2030a^{3}+120a^{2}+2423a+548$
79.3-c2 79.3-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $91212.18964$ 1.49691 \( \frac{6187974}{79} a^{5} - \frac{12923287}{79} a^{4} - \frac{16188247}{79} a^{3} + \frac{41817819}{79} a^{2} - \frac{10836900}{79} a - \frac{5220894}{79} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( -a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a - 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( 15 a^{5} + 9 a^{4} - 64 a^{3} - 43 a^{2} + 37 a + 19\) , \( -56 a^{5} - 29 a^{4} + 239 a^{3} + 138 a^{2} - 138 a - 42\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-3a^{2}-5a-1\right){x}^{2}+\left(15a^{5}+9a^{4}-64a^{3}-43a^{2}+37a+19\right){x}-56a^{5}-29a^{4}+239a^{3}+138a^{2}-138a-42$
79.3-c3 79.3-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $5.837580137$ 1.49691 \( -\frac{15509334442908689571428}{3077056399} a^{5} + \frac{44198347225679886465327}{3077056399} a^{4} - \frac{1469794761354986220538}{3077056399} a^{3} - \frac{65596401146446137115029}{3077056399} a^{2} + \frac{25462955612740497650596}{3077056399} a + \frac{8445950409748499043000}{3077056399} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 4\) , \( a^{5} + a^{4} - 4 a^{3} - 4 a^{2} + 2 a + 3\) , \( 11 a^{5} - 45 a^{4} - 15 a^{3} + 102 a^{2} + 9 a - 54\) , \( 59 a^{5} - 254 a^{4} - 57 a^{3} + 539 a^{2} + 2 a - 247\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+4\right){x}^{2}+\left(11a^{5}-45a^{4}-15a^{3}+102a^{2}+9a-54\right){x}+59a^{5}-254a^{4}-57a^{3}+539a^{2}+2a-247$
79.3-c4 79.3-c \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $2.918790068$ 1.49691 \( -\frac{354107829458843827802411767457358588717502}{9468276082626847201} a^{5} + \frac{439473793889618077364787680142691941182070}{9468276082626847201} a^{4} + \frac{1664593722944885880969335611051937863705766}{9468276082626847201} a^{3} - \frac{1817720520510626376452406301038917963206945}{9468276082626847201} a^{2} - \frac{1686442982404405482780445086453016221746437}{9468276082626847201} a + \frac{1468879965454357964032378550143526285469230}{9468276082626847201} \) \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( -a^{5} + 5 a^{3} - a^{2} - 6 a + 2\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 3 a\) , \( -435 a^{5} - 121 a^{4} + 1512 a^{3} + 155 a^{2} - 866 a - 246\) , \( -8307 a^{5} - 5881 a^{4} + 27378 a^{3} + 12730 a^{2} - 16010 a - 4422\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+3a\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-435a^{5}-121a^{4}+1512a^{3}+155a^{2}-866a-246\right){x}-8307a^{5}-5881a^{4}+27378a^{3}+12730a^{2}-16010a-4422$
79.3-d1 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $25.31742482$ 1.32957 \( -\frac{2872965026316702096470899522320}{38950081} a^{5} - \frac{1427925376305808382926176094042}{38950081} a^{4} + \frac{12227190148093368600436919005452}{38950081} a^{3} + \frac{6812506386251548298987875696395}{38950081} a^{2} - \frac{7039321653741016566532971180276}{38950081} a - \frac{1919120756343925478761954791834}{38950081} \) \( \bigl[a^{4} - 4 a^{2} + 2\) , \( a^{3} - a^{2} - 2 a + 3\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( 329 a^{5} + 161 a^{4} - 1399 a^{3} - 774 a^{2} + 813 a + 208\) , \( 4482 a^{5} + 3022 a^{4} - 18157 a^{3} - 13334 a^{2} + 7316 a + 2284\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+2\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-2a+3\right){x}^{2}+\left(329a^{5}+161a^{4}-1399a^{3}-774a^{2}+813a+208\right){x}+4482a^{5}+3022a^{4}-18157a^{3}-13334a^{2}+7316a+2284$
79.3-d2 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $25.31742482$ 1.32957 \( -\frac{1806310987189829549208270954886}{59091511031674153381441} a^{5} + \frac{1867280061295172198918453052615}{59091511031674153381441} a^{4} + \frac{4523353594532011391296947710931}{59091511031674153381441} a^{3} - \frac{1449083696217322076004722167442}{59091511031674153381441} a^{2} - \frac{2048522112959738678989772027522}{59091511031674153381441} a - \frac{353715935476625976028359928413}{59091511031674153381441} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{5} - a^{4} + 4 a^{3} + 5 a^{2} - 3 a - 4\) , \( a^{5} - 5 a^{3} + 6 a + 1\) , \( -50 a^{5} - 22 a^{4} + 304 a^{3} + 108 a^{2} - 451 a - 112\) , \( -360 a^{5} + 56 a^{4} + 1956 a^{3} + 50 a^{2} - 2509 a - 581\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){x}{y}+\left(a^{5}-5a^{3}+6a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+5a^{2}-3a-4\right){x}^{2}+\left(-50a^{5}-22a^{4}+304a^{3}+108a^{2}-451a-112\right){x}-360a^{5}+56a^{4}+1956a^{3}+50a^{2}-2509a-581$
79.3-d3 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $12962.52150$ 1.32957 \( \frac{1576208327952}{493039} a^{5} - \frac{546637380727}{493039} a^{4} - \frac{8080139654401}{493039} a^{3} + \frac{801868607748}{493039} a^{2} + \frac{9414033518410}{493039} a + \frac{2112843364295}{493039} \) \( \bigl[a^{5} - 5 a^{3} + 5 a + 1\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - a + 3\) , \( a\) , \( -4 a^{5} - 2 a^{4} + 17 a^{3} + 9 a^{2} - 10 a + 1\) , \( 12 a^{5} + 7 a^{4} - 51 a^{3} - 33 a^{2} + 29 a + 11\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+3\right){x}^{2}+\left(-4a^{5}-2a^{4}+17a^{3}+9a^{2}-10a+1\right){x}+12a^{5}+7a^{4}-51a^{3}-33a^{2}+29a+11$
79.3-d4 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $12962.52150$ 1.32957 \( -\frac{66621778871}{79} a^{5} + \frac{184673057900}{79} a^{4} + \frac{6049766083}{79} a^{3} - \frac{277512370424}{79} a^{2} + \frac{91787049677}{79} a + \frac{37562009690}{79} \) \( \bigl[a^{2} + a - 1\) , \( -a^{2} - a + 3\) , \( a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( -a^{5} + 3 a^{4} + 8 a^{3} - 12 a^{2} - 20 a - 1\) , \( 12 a^{5} + 6 a^{4} - 53 a^{3} - 30 a^{2} + 34 a + 10\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-a^{5}+3a^{4}+8a^{3}-12a^{2}-20a-1\right){x}+12a^{5}+6a^{4}-53a^{3}-30a^{2}+34a+10$
79.3-d5 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1620.315188$ 1.32957 \( -\frac{2775957273063928}{6241} a^{5} + \frac{2158369015824389}{6241} a^{4} + \frac{12719957740377180}{6241} a^{3} - \frac{8693763912465636}{6241} a^{2} - \frac{11508811251959304}{6241} a + \frac{7949534780236512}{6241} \) \( \bigl[a^{3} - 2 a + 1\) , \( -a^{4} + a^{3} + 4 a^{2} - 4 a - 3\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( 4 a^{5} - a^{4} - 15 a^{3} - 16 a^{2} + 26 a - 2\) , \( -30 a^{5} + 79 a^{4} + 46 a^{3} - 216 a^{2} + 53 a + 38\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a-3\right){x}^{2}+\left(4a^{5}-a^{4}-15a^{3}-16a^{2}+26a-2\right){x}-30a^{5}+79a^{4}+46a^{3}-216a^{2}+53a+38$
79.3-d6 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1620.315188$ 1.32957 \( \frac{9864017269525778498272880}{243087455521} a^{5} - \frac{2868359809586230613384259}{243087455521} a^{4} - \frac{51354355161968675541349721}{243087455521} a^{3} + \frac{3035058454072124864303972}{243087455521} a^{2} + \frac{61336596742421539337407014}{243087455521} a + \frac{13908462111951939654564430}{243087455521} \) \( \bigl[a^{5} - 5 a^{3} + 5 a + 1\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - a + 3\) , \( a\) , \( 41 a^{5} + 18 a^{4} - 173 a^{3} - 86 a^{2} + 95 a + 16\) , \( 259 a^{5} + 137 a^{4} - 1094 a^{3} - 644 a^{2} + 601 a + 171\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+5a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+3\right){x}^{2}+\left(41a^{5}+18a^{4}-173a^{3}-86a^{2}+95a+16\right){x}+259a^{5}+137a^{4}-1094a^{3}-644a^{2}+601a+171$
79.3-d7 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $202.5393985$ 1.32957 \( -\frac{41494566893660714555288080}{79} a^{5} + \frac{51497801567134241086634054}{79} a^{4} + \frac{195058086408065742590919756}{79} a^{3} - \frac{213001575932755291713651877}{79} a^{2} - \frac{197618395730591263439326644}{79} a + \frac{172124231458098209062968630}{79} \) \( \bigl[a^{3} - 2 a + 1\) , \( -a^{4} + a^{3} + 4 a^{2} - 4 a - 3\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( -56 a^{5} + 144 a^{4} + 90 a^{3} - 446 a^{2} + 276 a - 42\) , \( -1161 a^{5} + 2386 a^{4} + 3200 a^{3} - 7763 a^{2} + 1278 a + 1453\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-4a-3\right){x}^{2}+\left(-56a^{5}+144a^{4}+90a^{3}-446a^{2}+276a-42\right){x}-1161a^{5}+2386a^{4}+3200a^{3}-7763a^{2}+1278a+1453$
79.3-d8 79.3-d \(\Q(\zeta_{13})^+\) \( 79 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $202.5393985$ 1.32957 \( \frac{3428757028066412689128269890262938}{493039} a^{5} - \frac{997049030798831835894322909079689}{493039} a^{4} - \frac{17850902058216642220750712826565565}{493039} a^{3} + \frac{1054993896343092497695425618836686}{493039} a^{2} + \frac{21320754151285089843137628848091614}{493039} a + \frac{4834616150757005557838566061341763}{493039} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( 85 a^{5} - 76 a^{4} - 434 a^{3} + 346 a^{2} + 492 a - 365\) , \( -795 a^{5} + 975 a^{4} + 3695 a^{3} - 4054 a^{2} - 3484 a + 3086\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(85a^{5}-76a^{4}-434a^{3}+346a^{2}+492a-365\right){x}-795a^{5}+975a^{4}+3695a^{3}-4054a^{2}-3484a+3086$
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  *The rank, regulator and analytic order of Ш are not known for all curves in the database; curves for which these are unknown will not appear in searches specifying one of these quantities.