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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
4.1-a1 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.844013029$ $15.59078888$ 1.948013704 \( -\frac{1680914269}{32768} \) \( \bigl[a^{3} - 4 a\) , \( a^{2} - 2\) , \( a^{3} - 3 a\) , \( a^{2} - 28\) , \( -31 a^{2} + 64\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{2}-28\right){x}-31a^{2}+64$
4.1-a2 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.168802605$ $389.7697222$ 1.948013704 \( \frac{1331}{8} \) \( \bigl[a^{3} - 4 a\) , \( a^{2} - 2\) , \( a^{3} - 3 a\) , \( a^{2} - 3\) , \( -1\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{2}-3\right){x}-1$
4.1-a3 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.506407817$ $43.30774691$ 1.948013704 \( \frac{461373}{2} a^{2} - 992771 \) \( \bigl[a^{3} - 3 a\) , \( a^{2} - 2\) , \( a^{3} - 3 a\) , \( a^{2} - 3\) , \( -1\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{2}-3\right){x}-1$
4.1-a4 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.281337676$ $140.3171000$ 1.948013704 \( \frac{1250637664527933}{32} a^{2} - \frac{2690606637259811}{16} \) \( \bigl[a^{3} - 3 a\) , \( a^{2} - 2\) , \( a\) , \( 119 a^{2} - 88\) , \( -380 a^{2} + 271\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(119a^{2}-88\right){x}-380a^{2}+271$
4.1-a5 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $2.532039089$ $1.732309876$ 1.948013704 \( -\frac{1250637664527933}{32} a^{2} + \frac{871975048120043}{32} \) \( \bigl[a\) , \( a^{2} - 2\) , \( a^{3} - 4 a\) , \( -118 a^{2} + 505\) , \( -498 a^{2} + 2132\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(-118a^{2}+505\right){x}-498a^{2}+2132$
4.1-a6 4.1-a 4.4.8112.1 \( 2^{2} \) $1$ $\Z/3\Z$ $\mathrm{SU}(2)$ $0.056267535$ $3507.927500$ 1.948013704 \( -\frac{461373}{2} a^{2} + \frac{321323}{2} \) \( \bigl[a\) , \( a^{2} - 2\) , \( 0\) , \( a^{2} - 1\) , \( 1\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a^{2}-2\right){x}^{2}+\left(a^{2}-1\right){x}+1$
4.1-b1 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\Z/5\Z$ $\mathrm{SU}(2)$ $1$ $815.4744629$ 0.724329847 \( \frac{1331}{8} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 4\) , \( 1\) , \( -a^{2} + 6\) , \( a^{2} + 1\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-a^{2}+6\right){x}+a^{2}+1$
4.1-b2 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.304759140$ 0.724329847 \( -\frac{1680914269}{32768} \) \( \bigl[a^{2} - 2\) , \( -a^{2} + 4\) , \( 1\) , \( -76 a^{2} + 56\) , \( -507 a^{2} + 351\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-76a^{2}+56\right){x}-507a^{2}+351$
4.1-b3 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.304759140$ 0.724329847 \( \frac{1250637664527933}{32} a^{2} - \frac{2690606637259811}{16} \) \( \bigl[1\) , \( 1\) , \( a^{2} - 3\) , \( 28 a^{2} - 82\) , \( 52 a^{2} - 262\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(28a^{2}-82\right){x}+52a^{2}-262$
4.1-b4 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\Z/5\Z$ $\mathrm{SU}(2)$ $1$ $815.4744629$ 0.724329847 \( -\frac{461373}{2} a^{2} + \frac{321323}{2} \) \( \bigl[1\) , \( 1\) , \( a^{2} - 3\) , \( -2 a^{2} + 3\) , \( a^{2} - 2\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(-2a^{2}+3\right){x}+a^{2}-2$
4.1-b5 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\Z/5\Z$ $\mathrm{SU}(2)$ $1$ $815.4744629$ 0.724329847 \( \frac{461373}{2} a^{2} - 992771 \) \( \bigl[1\) , \( 1\) , \( a^{2} - 2\) , \( a^{2} - 5\) , \( -a^{2} + 3\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}+\left(a^{2}-5\right){x}-a^{2}+3$
4.1-b6 4.1-b 4.4.8112.1 \( 2^{2} \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.304759140$ 0.724329847 \( -\frac{1250637664527933}{32} a^{2} + \frac{871975048120043}{32} \) \( \bigl[1\) , \( 1\) , \( a^{2} - 2\) , \( -29 a^{2} + 60\) , \( -52 a^{2} - 2\bigr] \) ${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+{x}^{2}+\left(-29a^{2}+60\right){x}-52a^{2}-2$
9.1-a1 9.1-a 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $183.9710567$ 1.021305183 \( 1286156672 a^{3} - 1073723264 a^{2} - 5534602752 a + 4621152576 \) \( \bigl[a^{2} + a - 3\) , \( a^{2} - 3\) , \( a^{3} + a^{2} - 3 a - 3\) , \( -3 a^{3} - 4 a^{2} + 6 a - 3\) , \( -3 a^{3} - 14 a^{2} - 6 a + 12\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-3a^{3}-4a^{2}+6a-3\right){x}-3a^{3}-14a^{2}-6a+12$
9.1-a2 9.1-a 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $183.9710567$ 1.021305183 \( -1286156672 a^{3} - 1073723264 a^{2} + 5534602752 a + 4621152576 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 16 a^{3} + 15 a^{2} - 69 a - 60\) , \( 68 a^{3} + 57 a^{2} - 291 a - 243\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(16a^{3}+15a^{2}-69a-60\right){x}+68a^{3}+57a^{2}-291a-243$
9.1-b1 9.1-b 4.4.8112.1 \( 3^{2} \) 0 $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $1741.163848$ 1.073997239 \( 75904 a^{3} - 59264 a^{2} - 336384 a + 277824 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 4\) , \( a^{3} - 4 a\) , \( 2 a^{3} + 4 a^{2} - 4 a - 6\) , \( -a^{3} + 6 a + 3\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-4\right){x}^{2}+\left(2a^{3}+4a^{2}-4a-6\right){x}-a^{3}+6a+3$
9.1-b2 9.1-b 4.4.8112.1 \( 3^{2} \) 0 $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $1741.163848$ 1.073997239 \( -75904 a^{3} - 59264 a^{2} + 336384 a + 277824 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + a^{2} + 3 a - 4\) , \( 1\) , \( a^{3} - a^{2} - 3 a + 4\) , \( 2 a^{3} - 2 a^{2} - 6 a + 5\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-4\right){x}^{2}+\left(a^{3}-a^{2}-3a+4\right){x}+2a^{3}-2a^{2}-6a+5$
9.1-b3 9.1-b 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $193.4626497$ 1.073997239 \( -302347904 a^{3} + 627178624 a^{2} + 210809856 a - 437279424 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -1\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 27 a^{3} + 22 a^{2} - 116 a - 96\) , \( 127 a^{3} + 106 a^{2} - 546 a - 456\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}-{x}^{2}+\left(27a^{3}+22a^{2}-116a-96\right){x}+127a^{3}+106a^{2}-546a-456$
9.1-b4 9.1-b 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $193.4626497$ 1.073997239 \( 302347904 a^{3} + 627178624 a^{2} - 210809856 a - 437279424 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -1\) , \( a^{2} + a - 2\) , \( -26 a^{3} + 21 a^{2} + 111 a - 92\) , \( -105 a^{3} + 87 a^{2} + 451 a - 376\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}-{x}^{2}+\left(-26a^{3}+21a^{2}+111a-92\right){x}-105a^{3}+87a^{2}+451a-376$
9.1-c1 9.1-c 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.6566600$ 2.235326260 \( 302347904 a^{3} + 627178624 a^{2} - 210809856 a - 437279424 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 4\) , \( a^{2} + a - 2\) , \( -2 a^{3} - a^{2} + 9 a + 7\) , \( -8 a^{3} - 10 a^{2} + 25 a + 23\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-4\right){x}^{2}+\left(-2a^{3}-a^{2}+9a+7\right){x}-8a^{3}-10a^{2}+25a+23$
9.1-c2 9.1-c 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.6566600$ 2.235326260 \( -302347904 a^{3} + 627178624 a^{2} + 210809856 a - 437279424 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + a^{2} + 3 a - 4\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 5 a^{3} - 4 a^{2} - 20 a + 9\) , \( 6 a^{3} - 7 a^{2} - 18 a + 12\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-4\right){x}^{2}+\left(5a^{3}-4a^{2}-20a+9\right){x}+6a^{3}-7a^{2}-18a+12$
9.1-c3 9.1-c 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.6566600$ 2.235326260 \( 75904 a^{3} - 59264 a^{2} - 336384 a + 277824 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -1\) , \( a^{3} - 4 a\) , \( -14 a^{3} + 14 a^{2} + 62 a - 57\) , \( -50 a^{3} + 43 a^{2} + 216 a - 183\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}-{x}^{2}+\left(-14a^{3}+14a^{2}+62a-57\right){x}-50a^{3}+43a^{2}+216a-183$
9.1-c4 9.1-c 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.6566600$ 2.235326260 \( -75904 a^{3} - 59264 a^{2} + 336384 a + 277824 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -1\) , \( 1\) , \( 15 a^{3} + 13 a^{2} - 63 a - 53\) , \( 63 a^{3} + 53 a^{2} - 270 a - 226\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(15a^{3}+13a^{2}-63a-53\right){x}+63a^{3}+53a^{2}-270a-226$
9.1-d1 9.1-d 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $183.9710567$ 1.021305183 \( -1286156672 a^{3} - 1073723264 a^{2} + 5534602752 a + 4621152576 \) \( \bigl[a^{2} + a - 3\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 3 a^{3} - 4 a^{2} - 9 a - 3\) , \( 4 a^{3} - 14 a^{2} + 12\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-3a-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(3a^{3}-4a^{2}-9a-3\right){x}+4a^{3}-14a^{2}+12$
9.1-d2 9.1-d 4.4.8112.1 \( 3^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $183.9710567$ 1.021305183 \( 1286156672 a^{3} - 1073723264 a^{2} - 5534602752 a + 4621152576 \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} - a^{2} + 3 a + 2\) , \( a^{2} + a - 2\) , \( -17 a^{3} + 10 a^{2} + 70 a - 50\) , \( -55 a^{3} + 44 a^{2} + 235 a - 193\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+3a+2\right){x}^{2}+\left(-17a^{3}+10a^{2}+70a-50\right){x}-55a^{3}+44a^{2}+235a-193$
12.1-a1 12.1-a 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $0.756981788$ 2.042338545 \( -\frac{62524091407434778074210211}{18} a^{3} - 7205252996802039348824820 a^{2} + \frac{21796659878967593207125582}{9} a + \frac{10047355851349872602451729}{2} \) \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -178 a^{3} + 138 a^{2} + 778 a - 621\) , \( 10380 a^{3} - 24728 a^{2} + 2046 a + 9509\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-178a^{3}+138a^{2}+778a-621\right){x}+10380a^{3}-24728a^{2}+2046a+9509$
12.1-a2 12.1-a 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $61.31552486$ 2.042338545 \( -\frac{24247709737}{972} a^{3} - \frac{620961913}{12} a^{2} + \frac{16905633437}{972} a + \frac{216481639}{6} \) \( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 3\) , \( -a^{3} + a^{2} + 3 a\) , \( -738 a^{3} + 1530 a^{2} + 516 a - 1068\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a^{3}+a^{2}+3a\right){x}-738a^{3}+1530a^{2}+516a-1068$
12.1-a3 12.1-a 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\Z/9\Z$ $\mathrm{SU}(2)$ $1$ $551.8397237$ 2.042338545 \( \frac{224286677}{288} a^{3} - \frac{155128391}{96} a^{2} - \frac{155762893}{288} a + \frac{53943593}{48} \) \( \bigl[a + 1\) , \( a^{3} - 5 a\) , \( a^{2} - 3\) , \( -a^{3} - 2 a^{2} + 3 a + 4\) , \( -a^{3} - 2 a^{2}\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-a^{3}-2a^{2}+3a+4\right){x}-a^{3}-2a^{2}$
12.1-b1 12.1-b 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $9.492828837$ 0.948580491 \( \frac{62524091407434778074210211}{18} a^{3} - 7205252996802039348824820 a^{2} - \frac{21796659878967593207125582}{9} a + \frac{10047355851349872602451729}{2} \) \( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a + 1\) , \( -34 a^{3} + 29 a^{2} + 144 a - 127\) , \( -313 a^{3} - 1350 a^{2} - 646 a + 1667\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-34a^{3}+29a^{2}+144a-127\right){x}-313a^{3}-1350a^{2}-646a+1667$
12.1-b2 12.1-b 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $85.43545953$ 0.948580491 \( \frac{24247709737}{972} a^{3} - \frac{620961913}{12} a^{2} - \frac{16905633437}{972} a + \frac{216481639}{6} \) \( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a + 1\) , \( a^{3} - a^{2} - 6 a + 8\) , \( 20 a^{3} + 43 a^{2} - 13 a - 31\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(a^{3}-a^{2}-6a+8\right){x}+20a^{3}+43a^{2}-13a-31$
12.1-b3 12.1-b 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $85.43545953$ 0.948580491 \( -\frac{224286677}{288} a^{3} - \frac{155128391}{96} a^{2} + \frac{155762893}{288} a + \frac{53943593}{48} \) \( \bigl[a^{2} + a - 2\) , \( a + 1\) , \( a^{2} + a - 3\) , \( 7 a^{3} + 8 a^{2} - 27 a - 28\) , \( 25 a^{3} + 23 a^{2} - 104 a - 92\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a^{3}+8a^{2}-27a-28\right){x}+25a^{3}+23a^{2}-104a-92$
12.1-c1 12.1-c 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $0.756981788$ 2.042338545 \( \frac{62524091407434778074210211}{18} a^{3} - 7205252996802039348824820 a^{2} - \frac{21796659878967593207125582}{9} a + \frac{10047355851349872602451729}{2} \) \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -a^{3} - a^{2} + 4 a + 4\) , \( a^{2} + a - 3\) , \( 167 a^{3} + 135 a^{2} - 732 a - 609\) , \( -9900 a^{3} - 24318 a^{2} - 4104 a + 7761\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+4\right){x}^{2}+\left(167a^{3}+135a^{2}-732a-609\right){x}-9900a^{3}-24318a^{2}-4104a+7761$
12.1-c2 12.1-c 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $61.31552486$ 2.042338545 \( \frac{24247709737}{972} a^{3} - \frac{620961913}{12} a^{2} - \frac{16905633437}{972} a + \frac{216481639}{6} \) \( \bigl[a\) , \( -a + 1\) , \( a^{2} + a - 3\) , \( a^{2}\) , \( 737 a^{3} + 1530 a^{2} - 513 a - 1068\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+a^{2}{x}+737a^{3}+1530a^{2}-513a-1068$
12.1-c3 12.1-c 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\Z/9\Z$ $\mathrm{SU}(2)$ $1$ $551.8397237$ 2.042338545 \( -\frac{224286677}{288} a^{3} - \frac{155128391}{96} a^{2} + \frac{155762893}{288} a + \frac{53943593}{48} \) \( \bigl[a + 1\) , \( -a^{3} + 4 a\) , \( a^{2} - 3\) , \( -2 a^{2} + 4\) , \( a^{3} - 2 a^{2}\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-2a^{2}+4\right){x}+a^{3}-2a^{2}$
12.1-d1 12.1-d 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $9.492828837$ 0.948580491 \( -\frac{62524091407434778074210211}{18} a^{3} - 7205252996802039348824820 a^{2} + \frac{21796659878967593207125582}{9} a + \frac{10047355851349872602451729}{2} \) \( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a + 1\) , \( 33 a^{3} + 29 a^{2} - 142 a - 127\) , \( 313 a^{3} - 1350 a^{2} + 645 a + 1667\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-2\right){x}^{2}+\left(33a^{3}+29a^{2}-142a-127\right){x}+313a^{3}-1350a^{2}+645a+1667$
12.1-d2 12.1-d 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $85.43545953$ 0.948580491 \( -\frac{24247709737}{972} a^{3} - \frac{620961913}{12} a^{2} + \frac{16905633437}{972} a + \frac{216481639}{6} \) \( \bigl[a^{2} - 2\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a + 1\) , \( -2 a^{3} - a^{2} + 8 a + 8\) , \( -20 a^{3} + 43 a^{2} + 12 a - 31\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-2\right){x}^{2}+\left(-2a^{3}-a^{2}+8a+8\right){x}-20a^{3}+43a^{2}+12a-31$
12.1-d3 12.1-d 4.4.8112.1 \( 2^{2} \cdot 3 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $85.43545953$ 0.948580491 \( \frac{224286677}{288} a^{3} - \frac{155128391}{96} a^{2} - \frac{155762893}{288} a + \frac{53943593}{48} \) \( \bigl[a^{2} + a - 2\) , \( -a^{3} + 4 a + 1\) , \( a^{3} - 3 a + 1\) , \( -7 a^{3} + 5 a^{2} + 30 a - 21\) , \( -19 a^{3} + 16 a^{2} + 82 a - 69\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(-7a^{3}+5a^{2}+30a-21\right){x}-19a^{3}+16a^{2}+82a-69$
13.1-a1 13.1-a 4.4.8112.1 \( 13 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $128.4199871$ 2.851665927 \( \frac{108479}{13} a^{3} - \frac{2895807}{169} a^{2} - \frac{1061238}{169} a + \frac{2065005}{169} \) \( \bigl[a^{2} - 2\) , \( a^{2} - a - 4\) , \( a^{3} - 3 a + 1\) , \( 4 a^{3} - 5 a^{2} - 19 a + 21\) , \( -15 a^{3} + 12 a^{2} + 65 a - 56\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(4a^{3}-5a^{2}-19a+21\right){x}-15a^{3}+12a^{2}+65a-56$
13.1-b1 13.1-b 4.4.8112.1 \( 13 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.021640836$ $333.8983841$ 1.283643360 \( \frac{108479}{13} a^{3} - \frac{2895807}{169} a^{2} - \frac{1061238}{169} a + \frac{2065005}{169} \) \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( a^{3} - 3 a + 2\) , \( -a^{3} + 5 a + 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(a^{3}-3a+2\right){x}-a^{3}+5a+1$
13.2-a1 13.2-a 4.4.8112.1 \( 13 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $128.4199871$ 2.851665927 \( -\frac{108479}{13} a^{3} - \frac{2895807}{169} a^{2} + \frac{1061238}{169} a + \frac{2065005}{169} \) \( \bigl[a^{2} - 2\) , \( a^{2} + a - 4\) , \( a^{3} - 3 a + 1\) , \( -4 a^{3} - 5 a^{2} + 16 a + 21\) , \( 14 a^{3} + 12 a^{2} - 62 a - 56\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a+1\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-4a^{3}-5a^{2}+16a+21\right){x}+14a^{3}+12a^{2}-62a-56$
13.2-b1 13.2-b 4.4.8112.1 \( 13 \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.021640836$ $333.8983841$ 1.283643360 \( -\frac{108479}{13} a^{3} - \frac{2895807}{169} a^{2} + \frac{1061238}{169} a + \frac{2065005}{169} \) \( \bigl[a^{3} + a^{2} - 4 a - 3\) , \( -a^{3} + a^{2} + 5 a - 3\) , \( a^{3} - 4 a + 1\) , \( -a^{3} - 3 a^{2} + 5 a + 14\) , \( -3 a^{3} - 2 a^{2} + 13 a + 9\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-3\right){x}{y}+\left(a^{3}-4a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+5a-3\right){x}^{2}+\left(-a^{3}-3a^{2}+5a+14\right){x}-3a^{3}-2a^{2}+13a+9$
17.1-a1 17.1-a 4.4.8112.1 \( 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1111.470398$ 3.085133334 \( \frac{1552890}{17} a^{3} - \frac{3079198}{17} a^{2} - \frac{1001764}{17} a + \frac{2243389}{17} \) \( \bigl[a\) , \( a^{2} + a - 4\) , \( a^{3} - 4 a\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( -a^{3} + a^{2} + 2 a - 4\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(a^{3}-2a^{2}-3a+6\right){x}-a^{3}+a^{2}+2a-4$
17.1-a2 17.1-a 4.4.8112.1 \( 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $555.7351991$ 3.085133334 \( -\frac{34758114803134}{289} a^{3} + \frac{72088194197379}{289} a^{2} + \frac{24247868402117}{289} a - \frac{50250235305016}{289} \) \( \bigl[a\) , \( a^{2} + a - 4\) , \( a^{3} - 4 a\) , \( 6 a^{3} - 12 a^{2} - 8 a + 11\) , \( -19 a^{3} + 39 a^{2} + 14 a - 30\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(6a^{3}-12a^{2}-8a+11\right){x}-19a^{3}+39a^{2}+14a-30$
17.1-b1 17.1-b 4.4.8112.1 \( 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.031436091$ $1555.350655$ 1.085732606 \( -\frac{34758114803134}{289} a^{3} + \frac{72088194197379}{289} a^{2} + \frac{24247868402117}{289} a - \frac{50250235305016}{289} \) \( \bigl[a^{2} - 2\) , \( a^{3} - 4 a\) , \( a^{2} - 2\) , \( 6 a^{3} + 4 a^{2} - 24 a - 23\) , \( -21 a^{3} - 15 a^{2} + 89 a + 66\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(6a^{3}+4a^{2}-24a-23\right){x}-21a^{3}-15a^{2}+89a+66$
17.1-b2 17.1-b 4.4.8112.1 \( 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.062872182$ $1555.350655$ 1.085732606 \( \frac{1552890}{17} a^{3} - \frac{3079198}{17} a^{2} - \frac{1001764}{17} a + \frac{2243389}{17} \) \( \bigl[a^{2} - 2\) , \( a^{3} - 4 a\) , \( a^{2} - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -a^{2} + 3\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-4a\right){x}^{2}+\left(a^{3}-a^{2}-4a+2\right){x}-a^{2}+3$
17.2-a1 17.2-a 4.4.8112.1 \( 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1111.470398$ 3.085133334 \( -\frac{1552890}{17} a^{3} - \frac{3079198}{17} a^{2} + \frac{1001764}{17} a + \frac{2243389}{17} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{3} - 4 a\) , \( -a^{3} - 2 a^{2} + 3 a + 6\) , \( a^{3} + a^{2} - 2 a - 4\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{3}-2a^{2}+3a+6\right){x}+a^{3}+a^{2}-2a-4$
17.2-a2 17.2-a 4.4.8112.1 \( 17 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $555.7351991$ 3.085133334 \( \frac{34758114803134}{289} a^{3} + \frac{72088194197379}{289} a^{2} - \frac{24247868402117}{289} a - \frac{50250235305016}{289} \) \( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{3} - 4 a\) , \( -6 a^{3} - 12 a^{2} + 8 a + 11\) , \( 19 a^{3} + 39 a^{2} - 14 a - 30\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-6a^{3}-12a^{2}+8a+11\right){x}+19a^{3}+39a^{2}-14a-30$
17.2-b1 17.2-b 4.4.8112.1 \( 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.031436091$ $1555.350655$ 1.085732606 \( \frac{34758114803134}{289} a^{3} + \frac{72088194197379}{289} a^{2} - \frac{24247868402117}{289} a - \frac{50250235305016}{289} \) \( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a\) , \( a^{2} - 2\) , \( -6 a^{3} + 4 a^{2} + 24 a - 23\) , \( 21 a^{3} - 15 a^{2} - 89 a + 66\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-6a^{3}+4a^{2}+24a-23\right){x}+21a^{3}-15a^{2}-89a+66$
17.2-b2 17.2-b 4.4.8112.1 \( 17 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.062872182$ $1555.350655$ 1.085732606 \( -\frac{1552890}{17} a^{3} - \frac{3079198}{17} a^{2} + \frac{1001764}{17} a + \frac{2243389}{17} \) \( \bigl[a^{2} - 2\) , \( -a^{3} + 4 a\) , \( a^{2} - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( -a^{2} + 3\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+4a\right){x}^{2}+\left(-a^{3}-a^{2}+4a+2\right){x}-a^{2}+3$
27.1-a1 27.1-a 4.4.8112.1 \( 3^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $79.91285306$ 2.661790801 \( -\frac{1171418240}{9} a^{3} + \frac{673500736}{3} a^{2} + \frac{1324309504}{9} a - 109490816 \) \( \bigl[a^{2} + a - 3\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a\) , \( -4 a^{3} - 8 a^{2} + 2 a + 3\) , \( 4 a^{3} + 13 a^{2} + 3 a - 15\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-4a-2\right){x}^{2}+\left(-4a^{3}-8a^{2}+2a+3\right){x}+4a^{3}+13a^{2}+3a-15$
27.1-a2 27.1-a 4.4.8112.1 \( 3^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $79.91285306$ 2.661790801 \( \frac{4610920576}{9} a^{3} - \frac{11550519616}{27} a^{2} - \frac{6613222784}{3} a + \frac{49698746816}{27} \) \( \bigl[a^{2} + a - 3\) , \( -a^{3} + 4 a + 1\) , \( a + 1\) , \( -a^{2} + 4 a + 4\) , \( 5 a^{3} + 6 a^{2} - 3 a - 4\bigr] \) ${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+4a+1\right){x}^{2}+\left(-a^{2}+4a+4\right){x}+5a^{3}+6a^{2}-3a-4$
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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.