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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
89.1-A1 89.1-A 3.1.23.1 \( 89 \) $0$ $\Z/10\Z$ $1$ $80.93816251$ 0.337535470 \( \frac{197037}{89} a^{2} - \frac{337201}{89} a + \frac{275391}{89} \) \( \bigl[a + 1\) , \( -a^{2} - a - 1\) , \( a^{2} + a\) , \( -a^{2}\) , \( -a^{2} + 1\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}-a^{2}{x}-a^{2}+1$
89.1-A2 89.1-A 3.1.23.1 \( 89 \) $0$ $\Z/10\Z$ $1$ $40.46908125$ 0.337535470 \( \frac{102722246265}{7921} a^{2} - \frac{179321519586}{7921} a + \frac{134701062490}{7921} \) \( \bigl[a + 1\) , \( -a^{2} - a - 1\) , \( a^{2} + a\) , \( -6 a^{2} + 5 a - 5\) , \( 5 a^{2} - 11 a + 9\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+\left(-6a^{2}+5a-5\right){x}+5a^{2}-11a+9$
89.1-A3 89.1-A 3.1.23.1 \( 89 \) $0$ $\Z/2\Z$ $1$ $3.237526500$ 0.337535470 \( \frac{10320094419026087722}{5584059449} a^{2} - \frac{18110627314505490995}{5584059449} a + \frac{13670942613281923954}{5584059449} \) \( \bigl[a^{2} + a + 1\) , \( -a^{2} - 1\) , \( a^{2} + 1\) , \( -133 a^{2} + 216 a - 155\) , \( -870 a^{2} + 1533 a - 1172\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}+\left(a^{2}+1\right){y}={x}^{3}+\left(-a^{2}-1\right){x}^{2}+\left(-133a^{2}+216a-155\right){x}-870a^{2}+1533a-1172$
89.1-A4 89.1-A 3.1.23.1 \( 89 \) $0$ $\Z/2\Z$ $1$ $1.618763250$ 0.337535470 \( \frac{743348644076538917997765}{31181719929966183601} a^{2} - \frac{1319483559157003453097049}{31181719929966183601} a + \frac{1008804885134049329340601}{31181719929966183601} \) \( \bigl[a + 1\) , \( -a^{2} - a - 1\) , \( a^{2} + a\) , \( -31 a^{2} + 80 a - 50\) , \( -115 a^{2} + 276 a - 245\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+\left(-31a^{2}+80a-50\right){x}-115a^{2}+276a-245$
107.1-A1 107.1-A 3.1.23.1 \( 107 \) $0$ $\Z/9\Z$ $1$ $72.11684248$ 0.371293855 \( -\frac{36864}{107} a^{2} + \frac{102400}{107} a + \frac{61440}{107} \) \( \bigl[0\) , \( a^{2} + a - 1\) , \( a^{2} + 1\) , \( a^{2} - a - 1\) , \( -a^{2}\bigr] \) ${y}^2+\left(a^{2}+1\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{2}-a-1\right){x}-a^{2}$
107.1-A2 107.1-A 3.1.23.1 \( 107 \) $0$ $\Z/3\Z$ $1$ $8.012982497$ 0.371293855 \( -\frac{13431610740736}{1225043} a^{2} - \frac{34043228901376}{1225043} a - \frac{18044231077888}{1225043} \) \( \bigl[0\) , \( a^{2} - a + 1\) , \( a^{2} + a\) , \( -30 a^{2} - 11 a + 11\) , \( -101 a^{2} + 29 a + 80\bigr] \) ${y}^2+\left(a^{2}+a\right){y}={x}^{3}+\left(a^{2}-a+1\right){x}^{2}+\left(-30a^{2}-11a+11\right){x}-101a^{2}+29a+80$
107.1-A3 107.1-A 3.1.23.1 \( 107 \) $0$ $\mathsf{trivial}$ $1$ $0.890331388$ 0.371293855 \( -\frac{270815754742792058998784}{1838459212420154507} a^{2} + \frac{475186041093450994348032}{1838459212420154507} a - \frac{358714265350210784702464}{1838459212420154507} \) \( \bigl[0\) , \( -a^{2} - a\) , \( 1\) , \( -59 a^{2} + 100 a - 71\) , \( -274 a^{2} + 484 a - 422\bigr] \) ${y}^2+{y}={x}^{3}+\left(-a^{2}-a\right){x}^{2}+\left(-59a^{2}+100a-71\right){x}-274a^{2}+484a-422$
115.1-A1 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/12\Z$ $1$ $67.03413881$ 0.388266227 \( -\frac{296772}{575} a^{2} + \frac{501473}{575} a - \frac{384109}{575} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( a\) , \( 0\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+a{x}$
115.1-A2 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $33.51706940$ 0.388266227 \( \frac{524530283647}{330625} a^{2} - \frac{936669243273}{330625} a + \frac{716408628984}{330625} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( 6 a - 5\) , \( 4 a^{2} - 7 a + 2\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-5\right){x}+4a^{2}-7a+2$
115.1-A3 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/4\Z$ $1$ $7.448237646$ 0.388266227 \( -\frac{258468891972347}{190109375} a^{2} - \frac{300129881653427}{190109375} a - \frac{79092285157884}{190109375} \) \( \bigl[a^{2}\) , \( -a^{2} - 1\) , \( 0\) , \( 6 a^{2} + 15 a + 10\) , \( 29 a^{2} - 28 a - 39\bigr] \) ${y}^2+a^{2}{x}{y}={x}^{3}+\left(-a^{2}-1\right){x}^{2}+\left(6a^{2}+15a+10\right){x}+29a^{2}-28a-39$
115.1-A4 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/6\Z$ $1$ $8.379267352$ 0.388266227 \( -\frac{5581242271849501}{6996025} a^{2} - \frac{4930264753084216}{6996025} a - \frac{541308225756447}{6996025} \) \( \bigl[a^{2} + 1\) , \( a + 1\) , \( a + 1\) , \( 80 a^{2} + 75 a + 10\) , \( -199 a^{2} + 536 a + 518\bigr] \) ${y}^2+\left(a^{2}+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(80a^{2}+75a+10\right){x}-199a^{2}+536a+518$
115.1-A5 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/6\Z$ $1$ $16.75853470$ 0.388266227 \( \frac{75716587374138607891}{8984375} a^{2} - \frac{132873348136199426744}{8984375} a + \frac{100303122964106562977}{8984375} \) \( \bigl[a + 1\) , \( a^{2} + a\) , \( 0\) , \( -1347 a^{2} + 2368 a - 1788\) , \( 34299 a^{2} - 60188 a + 45434\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}+a\right){x}^{2}+\left(-1347a^{2}+2368a-1788\right){x}+34299a^{2}-60188a+45434$
115.1-A6 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $3.724118823$ 0.388266227 \( -\frac{24639055700669505303}{36141574462890625} a^{2} - \frac{34086830170754179598}{36141574462890625} a + \frac{95255415238700540209}{36141574462890625} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -15 a^{2} + 6 a - 10\) , \( -31 a^{2} + 6 a - 8\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a^{2}+6a-10\right){x}-31a^{2}+6a-8$
115.1-A7 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $0.931029705$ 0.388266227 \( \frac{477008324446884679504423036}{342416006750317515625} a^{2} - \frac{836616303539449005035716299}{342416006750317515625} a + \frac{631813749335688590262064917}{342416006750317515625} \) \( \bigl[1\) , \( -a^{2} + 1\) , \( a + 1\) , \( -341 a^{2} + 550 a - 401\) , \( -3926 a^{2} + 6899 a - 5146\bigr] \) ${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-341a^{2}+550a-401\right){x}-3926a^{2}+6899a-5146$
115.1-A8 115.1-A 3.1.23.1 \( 5 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $1.862059411$ 0.388266227 \( \frac{20007944569370849968678844}{725209712982177734375} a^{2} - \frac{7918974152713453255537371}{725209712982177734375} a + \frac{4956482510946229795232693}{725209712982177734375} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -130 a^{2} + 61 a - 30\) , \( 520 a^{2} - 496 a - 4\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-130a^{2}+61a-30\right){x}+520a^{2}-496a-4$
136.1-A1 136.1-A 3.1.23.1 \( 2^{3} \cdot 17 \) $0$ $\Z/9\Z$ $1$ $79.17989318$ 0.407658000 \( -\frac{55665}{17} a^{2} + \frac{197635}{34} a - \frac{136767}{34} \) \( \bigl[a\) , \( a - 1\) , \( 1\) , \( -a\) , \( 0\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}-a{x}$
136.1-A2 136.1-A 3.1.23.1 \( 2^{3} \cdot 17 \) $0$ $\Z/3\Z$ $1$ $8.797765909$ 0.407658000 \( \frac{214693602205}{19652} a^{2} + \frac{1064389027655}{39304} a + \frac{69858641986}{4913} \) \( \bigl[a^{2}\) , \( -1\) , \( a^{2} + 1\) , \( -a^{2} + 26 a + 21\) , \( -88 a^{2} + 13 a + 59\bigr] \) ${y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}-{x}^{2}+\left(-a^{2}+26a+21\right){x}-88a^{2}+13a+59$
136.1-A3 136.1-A 3.1.23.1 \( 2^{3} \cdot 17 \) $0$ $\mathsf{trivial}$ $1$ $0.977529545$ 0.407658000 \( -\frac{287944742921902931}{60716992766464} a^{2} + \frac{247958719176608235}{30358496383232} a - \frac{265887199961636953}{60716992766464} \) \( \bigl[a\) , \( a - 1\) , \( 1\) , \( -20 a^{2} - a - 30\) , \( -38 a^{2} + 36 a - 40\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-20a^{2}-a-30\right){x}-38a^{2}+36a-40$
161.2-A1 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/8\Z$ $1$ $68.25637837$ 0.444763710 \( \frac{29913}{161} a^{2} - \frac{183184}{161} a + \frac{131092}{161} \) \( \bigl[a^{2}\) , \( a^{2} + a\) , \( a^{2} + 1\) , \( -1\) , \( -a^{2}\bigr] \) ${y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}+\left(a^{2}+a\right){x}^{2}-{x}-a^{2}$
161.2-A2 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $17.06409459$ 0.444763710 \( \frac{865540714}{1127} a^{2} + \frac{111075631}{1127} a - \frac{222741042}{1127} \) \( \bigl[a^{2}\) , \( a^{2} + a\) , \( a^{2} + 1\) , \( -6\) , \( -6 a^{2} + 4 a + 1\bigr] \) ${y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}+\left(a^{2}+a\right){x}^{2}-6{x}-6a^{2}+4a+1$
161.2-A3 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $4.266023648$ 0.444763710 \( \frac{10722329721395253}{1270129} a^{2} - \frac{18816375995075321}{1270129} a + \frac{14204063955828727}{1270129} \) \( \bigl[1\) , \( -a^{2}\) , \( a^{2} + a + 1\) , \( -262 a^{2} + 457 a - 348\) , \( -2894 a^{2} + 5076 a - 3832\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a+1\right){y}={x}^{3}-a^{2}{x}^{2}+\left(-262a^{2}+457a-348\right){x}-2894a^{2}+5076a-3832$
161.2-A4 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/4\Z$ $1$ $4.266023648$ 0.444763710 \( \frac{70480768576475}{161} a^{2} - \frac{98873472543831}{161} a - \frac{114800157887607}{161} \) \( \bigl[a + 1\) , \( -a^{2} + a\) , \( 1\) , \( -336 a^{2} + 85 a + 256\) , \( -1008 a^{2} + 2146 a + 2194\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a\right){x}^{2}+\left(-336a^{2}+85a+256\right){x}-1008a^{2}+2146a+2194$
161.2-A5 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $1.066505912$ 0.444763710 \( \frac{258562058370012941913191}{1127} a^{2} - \frac{453744781572309571525470}{1127} a + \frac{342521801784920454613398}{1127} \) \( \bigl[a^{2} + a + 1\) , \( a^{2} + a\) , \( a\) , \( -122205 a^{2} + 214461 a - 161897\) , \( -29503518 a^{2} + 51775066 a - 39083845\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}+a{y}={x}^{3}+\left(a^{2}+a\right){x}^{2}+\left(-122205a^{2}+214461a-161897\right){x}-29503518a^{2}+51775066a-39083845$
161.2-A6 161.2-A 3.1.23.1 \( 7 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $1.066505912$ 0.444763710 \( -\frac{83698751031253985}{1613227676641} a^{2} + \frac{145641648220203186}{1613227676641} a - \frac{107112349272246506}{1613227676641} \) \( \bigl[a^{2}\) , \( a^{2} + a\) , \( a^{2} + 1\) , \( -40 a^{2} + 60 a - 21\) , \( -93 a^{2} + 198 a - 118\bigr] \) ${y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}+\left(a^{2}+a\right){x}^{2}+\left(-40a^{2}+60a-21\right){x}-93a^{2}+198a-118$
167.1-A1 167.1-A 3.1.23.1 \( 167 \) $0$ $\Z/7\Z$ $1$ $53.73833242$ 0.457355791 \( -\frac{303468}{167} a^{2} + \frac{159670}{167} a + \frac{352325}{167} \) \( \bigl[1\) , \( -a^{2} + 1\) , \( a\) , \( -a\) , \( 0\bigr] \) ${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}-a{x}$
167.1-A2 167.1-A 3.1.23.1 \( 167 \) $0$ $\mathsf{trivial}$ $1$ $1.096700661$ 0.457355791 \( -\frac{10662760246308487894808}{3622557586593623} a^{2} + \frac{18721837331562719138346}{3622557586593623} a - \frac{14121786168686506510523}{3622557586593623} \) \( \bigl[a^{2} + a\) , \( -a^{2} - a + 1\) , \( a + 1\) , \( -86 a^{2} + 129 a - 104\) , \( -517 a^{2} + 886 a - 666\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}-a+1\right){x}^{2}+\left(-86a^{2}+129a-104\right){x}-517a^{2}+886a-666$
185.1-A1 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/12\Z$ $1$ $55.81894521$ 0.484960610 \( \frac{2899619}{4625} a^{2} - \frac{6835221}{4625} a + \frac{8754868}{4625} \) \( \bigl[a^{2} + a\) , \( -a^{2}\) , \( a^{2} + a\) , \( -2 a^{2} + a + 1\) , \( -a^{2} + 1\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}-a^{2}{x}^{2}+\left(-2a^{2}+a+1\right){x}-a^{2}+1$
185.1-A2 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $13.95473630$ 0.484960610 \( \frac{1489529038113}{21390625} a^{2} - \frac{1401973775442}{21390625} a - \frac{1385370674639}{21390625} \) \( \bigl[a^{2} + a\) , \( -a^{2}\) , \( a^{2} + a\) , \( -2 a^{2} + a - 4\) , \( -a^{2} + a - 3\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}-a^{2}{x}^{2}+\left(-2a^{2}+a-4\right){x}-a^{2}+a-3$
185.1-A3 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/4\Z$ $1$ $6.202105023$ 0.484960610 \( \frac{41106476476641}{253265} a^{2} + \frac{3298915904486}{253265} a - \frac{20013184081503}{253265} \) \( \bigl[1\) , \( -a^{2}\) , \( a^{2}\) , \( -46 a^{2} + 5 a + 20\) , \( 91 a^{2} - 84 a - 126\bigr] \) ${y}^2+{x}{y}+a^{2}{y}={x}^{3}-a^{2}{x}^{2}+\left(-46a^{2}+5a+20\right){x}+91a^{2}-84a-126$
185.1-A4 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/6\Z$ $1$ $3.488684075$ 0.484960610 \( -\frac{42118422527744}{4625} a^{2} + \frac{17289052159471}{4625} a + \frac{39636307424257}{4625} \) \( \bigl[1\) , \( -a^{2}\) , \( a^{2}\) , \( -91 a^{2} + 85 a + 85\) , \( 240 a^{2} + 309 a + 34\bigr] \) ${y}^2+{x}{y}+a^{2}{y}={x}^{3}-a^{2}{x}^{2}+\left(-91a^{2}+85a+85\right){x}+240a^{2}+309a+34$
185.1-A5 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/12\Z$ $1$ $3.488684075$ 0.484960610 \( -\frac{10622336863650048}{457558837890625} a^{2} + \frac{109862649535073857}{457558837890625} a + \frac{95300849679068319}{457558837890625} \) \( \bigl[a^{2} + a\) , \( -a^{2}\) , \( a^{2} + a\) , \( -2 a^{2} - 4 a + 1\) , \( 10 a - 13\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}-a^{2}{x}^{2}+\left(-2a^{2}-4a+1\right){x}+10a-13$
185.1-A6 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $1.550526255$ 0.484960610 \( \frac{263188403436724755665988}{64143160225} a^{2} - \frac{461863452181436668806417}{64143160225} a + \frac{348650404446288146701261}{64143160225} \) \( \bigl[a^{2} + a + 1\) , \( a^{2} + a\) , \( a^{2} + a + 1\) , \( -1407 a^{2} + 2472 a - 1882\) , \( -36680 a^{2} + 64355 a - 48601\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}+\left(a^{2}+a+1\right){y}={x}^{3}+\left(a^{2}+a\right){x}^{2}+\left(-1407a^{2}+2472a-1882\right){x}-36680a^{2}+64355a-48601$
185.1-A7 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/2\Z$ $1$ $0.387631563$ 0.484960610 \( \frac{13726702146510635059614676998359}{253265} a^{2} - \frac{24088683028132051336493850205641}{253265} a + \frac{18184008827232638954852241783208}{253265} \) \( \bigl[a + 1\) , \( -a^{2}\) , \( 1\) , \( -19287979 a^{2} + 33848042 a - 25551131\) , \( -58496625438 a^{2} + 102654421532 a - 77491530156\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a^{2}{x}^{2}+\left(-19287979a^{2}+33848042a-25551131\right){x}-58496625438a^{2}+102654421532a-77491530156$
185.1-A8 185.1-A 3.1.23.1 \( 5 \cdot 37 \) $0$ $\Z/4\Z$ $1$ $0.387631563$ 0.484960610 \( -\frac{4643971392557468963566571623}{4114345003650022050625} a^{2} + \frac{8148410084872811532229913657}{4114345003650022050625} a - \frac{6152898026705077651587034456}{4114345003650022050625} \) \( \bigl[a^{2} + a\) , \( -a^{2}\) , \( a^{2} + a\) , \( -142 a^{2} + 291 a - 179\) , \( -1459 a^{2} + 2242 a - 1431\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+a\right){y}={x}^{3}-a^{2}{x}^{2}+\left(-142a^{2}+291a-179\right){x}-1459a^{2}+2242a-1431$
223.3-A1 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $38.57473619$ 0.502711782 \( \frac{25211780051350}{49729} a^{2} - \frac{44243413922775}{49729} a + \frac{33398510059362}{49729} \) \( \bigl[1\) , \( a^{2}\) , \( a^{2} + a + 1\) , \( -16 a^{2} + 26 a - 22\) , \( 37 a^{2} - 68 a + 52\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a+1\right){y}={x}^{3}+a^{2}{x}^{2}+\left(-16a^{2}+26a-22\right){x}+37a^{2}-68a+52$
223.3-A2 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/8\Z$ $1$ $77.14947239$ 0.502711782 \( -\frac{2856217}{223} a^{2} + \frac{4856528}{223} a - \frac{3550580}{223} \) \( \bigl[a\) , \( a^{2} - a - 1\) , \( 1\) , \( -a^{2}\) , \( 0\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}-a^{2}{x}$
223.3-A3 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $9.643684049$ 0.502711782 \( \frac{23571524496388989}{2472973441} a^{2} + \frac{4230234187020679}{2472973441} a - \frac{10136936292236441}{2472973441} \) \( \bigl[a^{2}\) , \( -a^{2} - a\) , \( a^{2} + 1\) , \( 15 a^{2} - 34 a - 40\) , \( 66 a^{2} - 98 a - 108\bigr] \) ${y}^2+a^{2}{x}{y}+\left(a^{2}+1\right){y}={x}^{3}+\left(-a^{2}-a\right){x}^{2}+\left(15a^{2}-34a-40\right){x}+66a^{2}-98a-108$
223.3-A4 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/4\Z$ $1$ $19.28736809$ 0.502711782 \( \frac{184522353444590388125}{223} a^{2} - \frac{323814156983193431065}{223} a + \frac{244440075121115638263}{223} \) \( \bigl[a^{2} + 1\) , \( -a^{2} - a\) , \( a^{2} + a + 1\) , \( -22604 a^{2} + 39664 a - 29941\) , \( 2340639 a^{2} - 4107538 a + 3100689\bigr] \) ${y}^2+\left(a^{2}+1\right){x}{y}+\left(a^{2}+a+1\right){y}={x}^{3}+\left(-a^{2}-a\right){x}^{2}+\left(-22604a^{2}+39664a-29941\right){x}+2340639a^{2}-4107538a+3100689$
223.3-A5 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/2\Z$ $1$ $2.410921012$ 0.502711782 \( -\frac{2545411286988224748223}{6115597639891380481} a^{2} - \frac{17180319046885161879922}{6115597639891380481} a + \frac{4238248567079522477622}{6115597639891380481} \) \( \bigl[a\) , \( a^{2} - a - 1\) , \( 1\) , \( -11 a^{2} + 40 a\) , \( -84 a^{2} + 34 a + 18\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-11a^{2}+40a\right){x}-84a^{2}+34a+18$
223.3-A6 223.3-A 3.1.23.1 \( 223 \) $0$ $\Z/2\Z$ $1$ $2.410921012$ 0.502711782 \( \frac{2398483247031296383}{49729} a^{2} - \frac{5215465637566830606}{49729} a - \frac{5303790920318513126}{49729} \) \( \bigl[a^{2} + a + 1\) , \( -a^{2} - a - 1\) , \( 0\) , \( 2058 a^{2} - 1279 a - 2138\) , \( -53646 a^{2} + 19860 a + 45562\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+\left(2058a^{2}-1279a-2138\right){x}-53646a^{2}+19860a+45562$
253.1-A1 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/8\Z$ $1$ $40.38798431$ 0.526342305 \( \frac{417863}{2783} a^{2} - \frac{34767}{2783} a + \frac{3149985}{2783} \) \( \bigl[a^{2} + a + 1\) , \( -a^{2} - a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+{x}$
253.1-A2 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $20.19399215$ 0.526342305 \( \frac{2062387370025}{7745089} a^{2} - \frac{476450732000}{7745089} a - \frac{1516352690708}{7745089} \) \( \bigl[a + 1\) , \( a^{2} - a\) , \( 1\) , \( -5 a^{2} + 4 a + 5\) , \( -3 a^{2} - 3 a - 1\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a\right){x}^{2}+\left(-5a^{2}+4a+5\right){x}-3a^{2}-3a-1$
253.1-A3 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $5.048498038$ 0.526342305 \( \frac{898671238793091074}{59986403617921} a^{2} - \frac{1600222393063032451}{59986403617921} a + \frac{1227014935973120994}{59986403617921} \) \( \bigl[a^{2} + a + 1\) , \( -a^{2} - a - 1\) , \( 0\) , \( -10 a^{2} + 15 a - 14\) , \( -23 a^{2} + 37 a - 18\bigr] \) ${y}^2+\left(a^{2}+a+1\right){x}{y}={x}^{3}+\left(-a^{2}-a-1\right){x}^{2}+\left(-10a^{2}+15a-14\right){x}-23a^{2}+37a-18$
253.1-A4 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/4\Z$ $1$ $10.09699607$ 0.526342305 \( -\frac{174588745152190}{2783} a^{2} - \frac{130042620481635}{2783} a + \frac{1327777229138}{2783} \) \( \bigl[a^{2} + a\) , \( -a\) , \( a^{2} + 1\) , \( 174 a^{2} - 85 a - 169\) , \( 1133 a^{2} - 241 a - 824\bigr] \) ${y}^2+\left(a^{2}+a\right){x}{y}+\left(a^{2}+1\right){y}={x}^{3}-a{x}^{2}+\left(174a^{2}-85a-169\right){x}+1133a^{2}-241a-824$
253.1-A5 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $1.262124509$ 0.526342305 \( \frac{833654495257217661469537}{1146551135499121} a^{2} - \frac{1463012109643929405934449}{1146551135499121} a + \frac{1104329696417202846546921}{1146551135499121} \) \( \bigl[a^{2} + 1\) , \( a - 1\) , \( 0\) , \( -449 a^{2} + 838 a - 639\) , \( -6934 a^{2} + 12248 a - 9210\bigr] \) ${y}^2+\left(a^{2}+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-449a^{2}+838a-639\right){x}-6934a^{2}+12248a-9210$
253.1-A6 253.1-A 3.1.23.1 \( 11 \cdot 23 \) $0$ $\Z/2\Z$ $1$ $1.262124509$ 0.526342305 \( \frac{12096844958864335269183}{24307407097829673169} a^{2} + \frac{81129911893400377438417}{24307407097829673169} a + \frac{54350390176978478717063}{24307407097829673169} \) \( \bigl[a + 1\) , \( a^{2} - a\) , \( 1\) , \( -41 a - 30\) , \( -82 a^{2} - 80 a - 50\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a^{2}-a\right){x}^{2}+\left(-41a-30\right){x}-82a^{2}-80a-50$
259.1-A1 259.1-A 3.1.23.1 \( 7 \cdot 37 \) $0$ $\Z/9\Z$ $1$ $33.65942021$ 0.519886984 \( -\frac{8925184}{12691} a^{2} + \frac{23601152}{12691} a + \frac{39010304}{12691} \) \( \bigl[0\) , \( a^{2}\) , \( 1\) , \( -a\) , \( 0\bigr] \) ${y}^2+{y}={x}^{3}+a^{2}{x}^{2}-a{x}$
259.1-A2 259.1-A 3.1.23.1 \( 7 \cdot 37 \) $0$ $\Z/9\Z$ $1$ $3.739935579$ 0.519886984 \( \frac{533423413596160}{2044031255371} a^{2} + \frac{14942973020037120}{2044031255371} a + \frac{5291457412308992}{2044031255371} \) \( \bigl[0\) , \( a^{2}\) , \( 1\) , \( 9 a\) , \( 8 a^{2} - 15\bigr] \) ${y}^2+{y}={x}^{3}+a^{2}{x}^{2}+9a{x}+8a^{2}-15$
259.1-A3 259.1-A 3.1.23.1 \( 7 \cdot 37 \) $0$ $\Z/3\Z$ $1$ $0.415548397$ 0.519886984 \( -\frac{578622056111156353812664320}{44576876749711411} a^{2} + \frac{1015410575032007747136962560}{44576876749711411} a - \frac{766510266683364441673433088}{44576876749711411} \) \( \bigl[0\) , \( -a + 1\) , \( a\) , \( -1768 a^{2} + 3086 a - 2357\) , \( -52408 a^{2} + 92024 a - 69423\bigr] \) ${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1768a^{2}+3086a-2357\right){x}-52408a^{2}+92024a-69423$
259.1-A4 259.1-A 3.1.23.1 \( 7 \cdot 37 \) $0$ $\mathsf{trivial}$ $1$ $0.046172044$ 0.519886984 \( -\frac{8036649736036295472025012606273884160}{354571} a^{2} + \frac{14103337133217473594167573368251924480}{354571} a - \frac{10646294221413528700483995824789659648}{354571} \) \( \bigl[0\) , \( a^{2} + a - 1\) , \( a + 1\) , \( -1159242728 a^{2} + 2034329174 a - 1535669655\) , \( -27257216724773 a^{2} + 47833080874361 a - 36108124459830\bigr] \) ${y}^2+\left(a+1\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-1159242728a^{2}+2034329174a-1535669655\right){x}-27257216724773a^{2}+47833080874361a-36108124459830$
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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.