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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
5.1-a1 5.1-a 4.4.6224.1 \( 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $731.1061852$ 2.316784691 \( -\frac{604416}{5} a^{3} - 93120 a^{2} + \frac{3226496}{5} a + \frac{3740672}{5} \) \( \bigl[a^{2} - 3\) , \( a^{3} - 2 a^{2} - 2 a + 5\) , \( 1\) , \( 2 a^{3} + 4 a^{2} - 12 a - 17\) , \( 11 a^{3} + 8 a^{2} - 57 a - 65\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(2a^{3}+4a^{2}-12a-17\right){x}+11a^{3}+8a^{2}-57a-65$
5.1-a2 5.1-a 4.4.6224.1 \( 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $365.5530926$ 2.316784691 \( -\frac{7911068128}{625} a^{3} + \frac{2962822216}{125} a^{2} + \frac{19735294968}{625} a - \frac{21127867224}{625} \) \( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 6 a - 1\) , \( a^{3} - 5 a - 1\) , \( -4 a - 3\) , \( -a^{3} - a^{2} - a - 4\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-6a-1\right){x}^{2}+\left(-4a-3\right){x}-a^{3}-a^{2}-a-4$
5.1-a3 5.1-a 4.4.6224.1 \( 5 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1462.212370$ 2.316784691 \( \frac{50048}{25} a^{3} - \frac{9536}{5} a^{2} - \frac{252288}{25} a + \frac{256384}{25} \) \( \bigl[a^{3} - 5 a\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( 4 a^{3} - 8 a^{2} - 10 a + 11\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(-a^{3}+a^{2}+3a-3\right){x}+4a^{3}-8a^{2}-10a+11$
5.1-a4 5.1-a 4.4.6224.1 \( 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $731.1061852$ 2.316784691 \( \frac{257761376}{5} a^{3} - 69689384 a^{2} - \frac{1057259736}{5} a + \frac{933687368}{5} \) \( \bigl[a + 1\) , \( -a^{2} + 2 a + 2\) , \( 1\) , \( 20 a^{3} + 19 a^{2} - 111 a - 134\) , \( -89 a^{3} - 73 a^{2} + 480 a + 565\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(20a^{3}+19a^{2}-111a-134\right){x}-89a^{3}-73a^{2}+480a+565$
5.1-b1 5.1-b 4.4.6224.1 \( 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.153729580$ $420.2841305$ 0.818965565 \( -\frac{604416}{5} a^{3} - 93120 a^{2} + \frac{3226496}{5} a + \frac{3740672}{5} \) \( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{2} - a - 1\) , \( a - 1\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(a^{2}-a-1\right){x}+a-1$
5.1-b2 5.1-b 4.4.6224.1 \( 5 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $0.038432395$ $1681.136522$ 0.818965565 \( -\frac{7911068128}{625} a^{3} + \frac{2962822216}{125} a^{2} + \frac{19735294968}{625} a - \frac{21127867224}{625} \) \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} - 2\) , \( -5 a^{3} - 3 a^{2} + 26 a + 29\) , \( 4 a^{3} + 3 a^{2} - 23 a - 26\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-5a^{3}-3a^{2}+26a+29\right){x}+4a^{3}+3a^{2}-23a-26$
5.1-b3 5.1-b 4.4.6224.1 \( 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.153729580$ $420.2841305$ 0.818965565 \( \frac{257761376}{5} a^{3} - 69689384 a^{2} - \frac{1057259736}{5} a + \frac{933687368}{5} \) \( \bigl[a^{3} - 4 a - 1\) , \( -a^{2} + 2 a + 3\) , \( a^{3} - 4 a\) , \( 7 a^{3} + 13 a^{2} - 43 a - 74\) , \( -20 a^{3} - 14 a^{2} + 107 a + 115\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(7a^{3}+13a^{2}-43a-74\right){x}-20a^{3}-14a^{2}+107a+115$
5.1-b4 5.1-b 4.4.6224.1 \( 5 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $0.076864790$ $1681.136522$ 0.818965565 \( \frac{50048}{25} a^{3} - \frac{9536}{5} a^{2} - \frac{252288}{25} a + \frac{256384}{25} \) \( \bigl[a^{3} - 5 a\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 5 a - 1\) , \( -5 a^{3} - 11 a^{2} - 5 a + 6\) , \( -16 a^{3} - 39 a^{2} - 4 a + 31\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-5a^{3}-11a^{2}-5a+6\right){x}-16a^{3}-39a^{2}-4a+31$
8.1-a1 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $451.9341989$ 1.432123342 \( 524288 a^{3} + 1023968 a^{2} - 286688 a - 683152 \) \( \bigl[a^{3} - 5 a\) , \( a\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 23 a^{3} - 45 a^{2} - 55 a + 66\) , \( -46 a^{3} + 85 a^{2} + 115 a - 121\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+a{x}^{2}+\left(23a^{3}-45a^{2}-55a+66\right){x}-46a^{3}+85a^{2}+115a-121$
8.1-a2 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $903.8683978$ 1.432123342 \( -2048 a^{3} - 2048 a^{2} + 10240 a + 14336 \) \( \bigl[0\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a^{3} - 4 a - 1\) , \( -a + 3\) , \( -a^{3} + 2 a^{2} + 2 a - 6\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(-a+3\right){x}-a^{3}+2a^{2}+2a-6$
8.1-a3 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $56.49177486$ 1.432123342 \( -8484050373122 a^{3} - 5950704522102 a^{2} + 44647055214736 a + 50524147464286 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 68 a^{3} + 51 a^{2} - 361 a - 422\) , \( 620 a^{3} + 504 a^{2} - 3336 a - 3927\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(68a^{3}+51a^{2}-361a-422\right){x}+620a^{3}+504a^{2}-3336a-3927$
8.1-a4 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $14.12294371$ 1.432123342 \( 1537921833441718305282 a^{3} - 2107454190479225749642 a^{2} - 6339631844534479587472 a + 5611518334448113253746 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( -22 a^{3} + 41 a^{2} + 89 a - 122\) , \( -88 a^{3} + 144 a^{2} + 352 a - 423\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-22a^{3}+41a^{2}+89a-122\right){x}-88a^{3}+144a^{2}+352a-423$
8.1-a5 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $225.9670994$ 1.432123342 \( 19246710912 a^{3} - 26376158916 a^{2} - 79338143744 a + 70236176888 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( 0\) , \( 3 a^{3} + 6 a^{2} - 16 a - 32\) , \( 7 a^{3} + 8 a^{2} - 39 a - 55\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(3a^{3}+6a^{2}-16a-32\right){x}+7a^{3}+8a^{2}-39a-55$
8.1-a6 8.1-a 4.4.6224.1 \( 2^{3} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $56.49177486$ 1.432123342 \( 5668523114368 a^{3} + 13864000438468 a^{2} - 102746160640 a - 11588341595944 \) \( \bigl[a + 1\) , \( a^{2} - 3\) , \( a^{3} - a^{2} - 3 a + 3\) , \( -2 a^{3} - 5 a^{2} + 3 a + 9\) , \( -7 a^{3} - 18 a^{2} - 5 a + 7\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-2a^{3}-5a^{2}+3a+9\right){x}-7a^{3}-18a^{2}-5a+7$
8.1-b1 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/2\Z\oplus\Z/4\Z$ $\mathrm{SU}(2)$ $0.992443989$ $1047.332123$ 1.646894014 \( 524288 a^{3} + 1023968 a^{2} - 286688 a - 683152 \) \( \bigl[a^{2} - 3\) , \( -a\) , \( a^{3} - 5 a\) , \( 11 a^{3} - 20 a^{2} - 29 a + 26\) , \( 19 a^{3} - 36 a^{2} - 45 a + 46\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}-a{x}^{2}+\left(11a^{3}-20a^{2}-29a+26\right){x}+19a^{3}-36a^{2}-45a+46$
8.1-b2 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $3.969775957$ $8.182282215$ 1.646894014 \( 1537921833441718305282 a^{3} - 2107454190479225749642 a^{2} - 6339631844534479587472 a + 5611518334448113253746 \) \( \bigl[a^{2} - a - 2\) , \( 1\) , \( a^{2} - 3\) , \( 11 a^{3} + 17 a^{2} - 38 a - 56\) , \( 62 a^{3} + 93 a^{2} - 192 a - 280\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(11a^{3}+17a^{2}-38a-56\right){x}+62a^{3}+93a^{2}-192a-280$
8.1-b3 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1.984887978$ $130.9165154$ 1.646894014 \( 19246710912 a^{3} - 26376158916 a^{2} - 79338143744 a + 70236176888 \) \( \bigl[a^{2} - a - 2\) , \( 1\) , \( a^{2} - 3\) , \( 6 a^{3} + 2 a^{2} - 43 a - 46\) , \( 35 a^{3} + 32 a^{2} - 172 a - 206\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+{x}^{2}+\left(6a^{3}+2a^{2}-43a-46\right){x}+35a^{3}+32a^{2}-172a-206$
8.1-b4 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $3.969775957$ $8.182282215$ 1.646894014 \( -8484050373122 a^{3} - 5950704522102 a^{2} + 44647055214736 a + 50524147464286 \) \( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 5 a - 2\) , \( a + 1\) , \( -74 a^{3} + 178 a^{2} + 262 a - 625\) , \( -810 a^{3} + 1616 a^{2} + 3050 a - 5278\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-74a^{3}+178a^{2}+262a-625\right){x}-810a^{3}+1616a^{2}+3050a-5278$
8.1-b5 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $0.496221994$ $523.6660617$ 1.646894014 \( 5668523114368 a^{3} + 13864000438468 a^{2} - 102746160640 a - 11588341595944 \) \( \bigl[a^{3} - 4 a - 1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 4 a + 2\) , \( -4 a^{3} + 5 a^{2} + 17 a - 12\) , \( -31 a^{3} + 44 a^{2} + 127 a - 120\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-4a^{3}+5a^{2}+17a-12\right){x}-31a^{3}+44a^{2}+127a-120$
8.1-b6 8.1-b 4.4.6224.1 \( 2^{3} \) $1$ $\Z/8\Z$ $\mathrm{SU}(2)$ $0.496221994$ $2094.664247$ 1.646894014 \( -2048 a^{3} - 2048 a^{2} + 10240 a + 14336 \) \( \bigl[0\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - a - 2\) , \( -a^{2} - a + 2\) , \( a^{3} - 2 a\bigr] \) ${y}^2+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(-a^{2}-a+2\right){x}+a^{3}-2a$
10.1-a1 10.1-a 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $1.261507002$ 0.783521564 \( \frac{54525693103349555799801104349341851}{20} a^{3} - \frac{20417431637346804286338828785973231}{4} a^{2} - \frac{136018800709295526260888946535110001}{20} a + \frac{145613378862132049509045981818993263}{20} \) \( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 4\) , \( a^{3} - 5 a\) , \( 14013 a^{3} - 31184 a^{2} - 37811 a + 42712\) , \( -1637998 a^{3} + 3402030 a^{2} + 4278012 a - 4741198\bigr] \) ${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-4\right){x}^{2}+\left(14013a^{3}-31184a^{2}-37811a+42712\right){x}-1637998a^{3}+3402030a^{2}+4278012a-4741198$
10.1-a2 10.1-a 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $61.81384313$ 0.783521564 \( \frac{70983281829}{5000000} a^{3} - \frac{23784646563}{1000000} a^{2} - \frac{1537011898717}{40000000} a + \frac{1141561933231}{40000000} \) \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - a^{2} - 4 a + 3\) , \( 7 a^{3} - 10 a^{2} - 28 a + 24\) , \( -562 a^{3} + 771 a^{2} + 2313 a - 2048\bigr] \) ${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a^{3}-a^{2}-4a+3\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+4\right){x}^{2}+\left(7a^{3}-10a^{2}-28a+24\right){x}-562a^{3}+771a^{2}+2313a-2048$
10.1-b1 10.1-b 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $19.33687172$ 2.205941208 \( \frac{5586707071641887510419}{250} a^{3} + \frac{445157264262336585691}{25} a^{2} - \frac{14986581029407284866432}{125} a - \frac{35056482742175374115673}{250} \) \( \bigl[a\) , \( a^{3} - 6 a - 1\) , \( a^{3} - 5 a\) , \( 13 a^{3} - 9 a^{2} - 54 a - 5\) , \( 16 a^{3} - 3 a^{2} - 82 a - 18\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-6a-1\right){x}^{2}+\left(13a^{3}-9a^{2}-54a-5\right){x}+16a^{3}-3a^{2}-82a-18$
10.1-b2 10.1-b 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $174.0318455$ 2.205941208 \( \frac{2292449}{10} a^{3} + \frac{2048911}{2} a^{2} - \frac{17433329}{10} a - \frac{53500293}{10} \) \( \bigl[a\) , \( a^{3} - 6 a - 1\) , \( a^{3} - 5 a\) , \( -2 a^{3} + a^{2} + 6 a + 5\) , \( -a^{3} + 3 a^{2} + 2 a - 10\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-6a-1\right){x}^{2}+\left(-2a^{3}+a^{2}+6a+5\right){x}-a^{3}+3a^{2}+2a-10$
10.1-c1 10.1-c 4.4.6224.1 \( 2 \cdot 5 \) 0 $\Z/3\Z$ $\mathrm{SU}(2)$ $1$ $498.7982668$ 0.702502112 \( \frac{2292449}{10} a^{3} + \frac{2048911}{2} a^{2} - \frac{17433329}{10} a - \frac{53500293}{10} \) \( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a - 6\) , \( a^{2} - a - 2\) , \( -a^{2} - 2 a - 1\) , \( a^{3} - 2 a - 1\bigr] \) ${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+4a-6\right){x}^{2}+\left(-a^{2}-2a-1\right){x}+a^{3}-2a-1$
10.1-c2 10.1-c 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $6.158003294$ 0.702502112 \( \frac{5586707071641887510419}{250} a^{3} + \frac{445157264262336585691}{25} a^{2} - \frac{14986581029407284866432}{125} a - \frac{35056482742175374115673}{250} \) \( \bigl[a^{3} - 4 a\) , \( -a^{3} + 2 a^{2} + 2 a - 5\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 20 a^{3} + 31 a^{2} - 23 a - 13\) , \( 211 a^{3} + 256 a^{2} - 283 a - 44\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-5\right){x}^{2}+\left(20a^{3}+31a^{2}-23a-13\right){x}+211a^{3}+256a^{2}-283a-44$
10.1-d1 10.1-d 4.4.6224.1 \( 2 \cdot 5 \) 0 $\Z/7\Z$ $\mathrm{SU}(2)$ $1$ $32.38793466$ 2.052666841 \( \frac{70983281829}{5000000} a^{3} - \frac{23784646563}{1000000} a^{2} - \frac{1537011898717}{40000000} a + \frac{1141561933231}{40000000} \) \( \bigl[a\) , \( -a^{3} + 5 a\) , \( a^{2} - 3\) , \( 3 a^{3} - 7 a^{2} - 9 a + 13\) , \( 2 a^{3} - 2\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(3a^{3}-7a^{2}-9a+13\right){x}+2a^{3}-2$
10.1-d2 10.1-d 4.4.6224.1 \( 2 \cdot 5 \) 0 $\mathsf{trivial}$ $\mathrm{SU}(2)$ $1$ $0.013489352$ 2.052666841 \( \frac{54525693103349555799801104349341851}{20} a^{3} - \frac{20417431637346804286338828785973231}{4} a^{2} - \frac{136018800709295526260888946535110001}{20} a + \frac{145613378862132049509045981818993263}{20} \) \( \bigl[a\) , \( -a^{3} + 5 a\) , \( a^{2} - 3\) , \( 5293 a^{3} - 10877 a^{2} - 11229 a + 13113\) , \( 465056 a^{3} - 894282 a^{2} - 1103684 a + 1211518\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(5293a^{3}-10877a^{2}-11229a+13113\right){x}+465056a^{3}-894282a^{2}-1103684a+1211518$
20.1-a1 20.1-a 4.4.6224.1 \( 2^{2} \cdot 5 \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $585.2367558$ 1.854542587 \( -\frac{21636032}{625} a^{3} - \frac{2985696}{125} a^{2} + \frac{116788992}{625} a + \frac{133704944}{625} \) \( \bigl[a^{3} - 5 a\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} - 3\) , \( 3 a^{3} - 2 a^{2} - 13 a - 4\) , \( 5 a^{3} - 23 a - 19\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(3a^{3}-2a^{2}-13a-4\right){x}+5a^{3}-23a-19$
20.1-a2 20.1-a 4.4.6224.1 \( 2^{2} \cdot 5 \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $1170.473511$ 1.854542587 \( \frac{434176}{25} a^{3} - \frac{120832}{5} a^{2} - \frac{1785856}{25} a + \frac{1630208}{25} \) \( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a^{3} - 4 a - 1\) , \( -2 a^{3} + 4 a^{2} + 7 a - 10\) , \( 2 a^{3} - 2 a^{2} - 10 a + 3\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-2a^{3}+4a^{2}+7a-10\right){x}+2a^{3}-2a^{2}-10a+3$
20.1-a3 20.1-a 4.4.6224.1 \( 2^{2} \cdot 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $292.6183779$ 1.854542587 \( -\frac{283994931876}{25} a^{3} - \frac{45126369068}{5} a^{2} + \frac{1525795637556}{25} a + \frac{1783754768992}{25} \) \( \bigl[a + 1\) , \( a^{2} - a - 4\) , \( a + 1\) , \( -3 a^{3} - 14 a^{2} - 4 a + 15\) , \( 14 a^{3} + 37 a^{2} + a - 33\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-3a^{3}-14a^{2}-4a+15\right){x}+14a^{3}+37a^{2}+a-33$
20.1-a4 20.1-a 4.4.6224.1 \( 2^{2} \cdot 5 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $73.15459448$ 1.854542587 \( -\frac{2965006839868}{390625} a^{3} + \frac{1123051193996}{78125} a^{2} + \frac{7189991108908}{390625} a - \frac{7794319232144}{390625} \) \( \bigl[a + 1\) , \( a^{2} - a - 4\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 2 a^{3} - 9 a^{2} - 3 a + 12\) , \( 9 a^{3} - 24 a^{2} - 24 a + 28\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(2a^{3}-9a^{2}-3a+12\right){x}+9a^{3}-24a^{2}-24a+28$
20.1-b1 20.1-b 4.4.6224.1 \( 2^{2} \cdot 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.389616101$ $64.87708415$ 1.922404072 \( -\frac{283994931876}{25} a^{3} - \frac{45126369068}{5} a^{2} + \frac{1525795637556}{25} a + \frac{1783754768992}{25} \) \( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( a^{3} - a^{2} - 3 a + 3\) , \( 29 a^{3} - 42 a^{2} - 121 a + 109\) , \( -86 a^{3} + 115 a^{2} + 353 a - 311\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(29a^{3}-42a^{2}-121a+109\right){x}-86a^{3}+115a^{2}+353a-311$
20.1-b2 20.1-b 4.4.6224.1 \( 2^{2} \cdot 5 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $0.194808050$ $519.0166732$ 1.922404072 \( -\frac{21636032}{625} a^{3} - \frac{2985696}{125} a^{2} + \frac{116788992}{625} a + \frac{133704944}{625} \) \( \bigl[a^{3} - 5 a\) , \( a^{2} - 2 a - 4\) , \( 0\) , \( -2 a^{3} - 6 a^{2} - 4 a + 9\) , \( -6 a^{3} - 15 a^{2} - 2 a + 14\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-2a^{3}-6a^{2}-4a+9\right){x}-6a^{3}-15a^{2}-2a+14$
20.1-b3 20.1-b 4.4.6224.1 \( 2^{2} \cdot 5 \) $1$ $\Z/4\Z$ $\mathrm{SU}(2)$ $0.097404025$ $1038.033346$ 1.922404072 \( \frac{434176}{25} a^{3} - \frac{120832}{5} a^{2} - \frac{1785856}{25} a + \frac{1630208}{25} \) \( \bigl[0\) , \( a^{3} - a^{2} - 4 a + 3\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 2 a^{3} - a^{2} - 9 a\) , \( 0\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+3\right){x}^{2}+\left(2a^{3}-a^{2}-9a\right){x}$
20.1-b4 20.1-b 4.4.6224.1 \( 2^{2} \cdot 5 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.097404025$ $259.5083366$ 1.922404072 \( -\frac{2965006839868}{390625} a^{3} + \frac{1123051193996}{78125} a^{2} + \frac{7189991108908}{390625} a - \frac{7794319232144}{390625} \) \( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{2} - 2 a - 4\) , \( a^{2} - 3\) , \( 67 a^{3} - 126 a^{2} - 167 a + 183\) , \( -437 a^{3} + 819 a^{2} + 1089 a - 1172\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(67a^{3}-126a^{2}-167a+183\right){x}-437a^{3}+819a^{2}+1089a-1172$
25.1-a1 25.1-a 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $360.9678543$ 1.143862295 \( -\frac{58486835328}{15625} a^{3} + \frac{21942599616}{3125} a^{2} + \frac{145990903168}{15625} a - \frac{156369149824}{15625} \) \( \bigl[a^{2} - 3\) , \( -a^{3} + 2 a^{2} + 3 a - 5\) , \( a^{2} - a - 3\) , \( -34 a^{3} + 48 a^{2} + 139 a - 130\) , \( -107 a^{3} + 147 a^{2} + 441 a - 393\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-5\right){x}^{2}+\left(-34a^{3}+48a^{2}+139a-130\right){x}-107a^{3}+147a^{2}+441a-393$
25.1-a2 25.1-a 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $90.24196359$ 1.143862295 \( \frac{31153506048}{125} a^{3} + \frac{15241204544}{25} a^{2} - \frac{527343488}{125} a - \frac{63657884416}{125} \) \( \bigl[a^{2} - 3\) , \( a^{3} - a^{2} - 4 a + 3\) , \( 1\) , \( -12 a^{3} - 47 a^{2} - 17 a + 51\) , \( -160 a^{3} - 385 a^{2} + 9 a + 318\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+{y}={x}^{3}+\left(a^{3}-a^{2}-4a+3\right){x}^{2}+\left(-12a^{3}-47a^{2}-17a+51\right){x}-160a^{3}-385a^{2}+9a+318$
25.1-a3 25.1-a 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $90.24196359$ 1.143862295 \( \frac{1521253261641712}{244140625} a^{3} - \frac{416904193452264}{48828125} a^{2} - \frac{6271166879450072}{244140625} a + \frac{5551244368587496}{244140625} \) \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -5 a^{3} + 5 a^{2} + 22 a - 15\) , \( -6 a^{3} + 8 a^{2} + 24 a - 20\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(-5a^{3}+5a^{2}+22a-15\right){x}-6a^{3}+8a^{2}+24a-20$
25.1-a4 25.1-a 4.4.6224.1 \( 5^{2} \) 0 $\Z/4\Z$ $\mathrm{SU}(2)$ $1$ $360.9678543$ 1.143862295 \( -\frac{10154599715840048}{125} a^{3} + \frac{3802443098725256}{25} a^{2} + \frac{25331479468026488}{125} a - \frac{27118326983574584}{125} \) \( \bigl[a^{3} - 4 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 29 a^{3} - 57 a^{2} - 69 a + 79\) , \( -168 a^{3} + 314 a^{2} + 418 a - 447\bigr] \) ${y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(29a^{3}-57a^{2}-69a+79\right){x}-168a^{3}+314a^{2}+418a-447$
25.1-b1 25.1-b 4.4.6224.1 \( 5^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.008786406$ $1161.425138$ 1.552204675 \( -202419 a^{3} + 284078 a^{2} + 830562 a - 769230 \) \( \bigl[a^{2} - a - 3\) , \( a^{3} - a^{2} - 5 a + 1\) , \( a^{3} - a^{2} - 3 a + 3\) , \( a^{3} + a^{2} - 6 a - 5\) , \( -2 a^{3} - a^{2} + 9 a + 8\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{3}-a^{2}-3a+3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a+1\right){x}^{2}+\left(a^{3}+a^{2}-6a-5\right){x}-2a^{3}-a^{2}+9a+8$
25.1-c1 25.1-c 4.4.6224.1 \( 5^{2} \) $1$ $\mathsf{trivial}$ $\mathrm{SU}(2)$ $0.044532329$ $1353.524284$ 3.056093711 \( -202419 a^{3} + 284078 a^{2} + 830562 a - 769230 \) \( \bigl[a^{3} - a^{2} - 4 a + 3\) , \( -a^{3} + 2 a^{2} + 3 a - 6\) , \( 1\) , \( -2 a^{3} + 2 a^{2} + 8 a - 4\) , \( a^{3} - 3 a + 1\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+3\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-6\right){x}^{2}+\left(-2a^{3}+2a^{2}+8a-4\right){x}+a^{3}-3a+1$
25.1-d1 25.1-d 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $120.9307858$ 0.766429279 \( \frac{31153506048}{125} a^{3} + \frac{15241204544}{25} a^{2} - \frac{527343488}{125} a - \frac{63657884416}{125} \) \( \bigl[a^{2} - 3\) , \( a^{2} - 2 a - 2\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 7 a^{3} - 13 a^{2} - 21 a + 20\) , \( 34 a^{3} - 64 a^{2} - 85 a + 91\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(7a^{3}-13a^{2}-21a+20\right){x}+34a^{3}-64a^{2}-85a+91$
25.1-d2 25.1-d 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $120.9307858$ 0.766429279 \( -\frac{10154599715840048}{125} a^{3} + \frac{3802443098725256}{25} a^{2} + \frac{25331479468026488}{125} a - \frac{27118326983574584}{125} \) \( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 1\) , \( a\) , \( -8 a^{3} + 3 a^{2} + 27 a - 15\) , \( 155 a^{3} + 330 a^{2} - 43 a - 255\bigr] \) ${y}^2+\left(a^{2}-a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-5a-1\right){x}^{2}+\left(-8a^{3}+3a^{2}+27a-15\right){x}+155a^{3}+330a^{2}-43a-255$
25.1-d3 25.1-d 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z\oplus\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $241.8615717$ 0.766429279 \( -\frac{58486835328}{15625} a^{3} + \frac{21942599616}{3125} a^{2} + \frac{145990903168}{15625} a - \frac{156369149824}{15625} \) \( \bigl[a^{3} - 5 a\) , \( -a^{2} + 3\) , \( a^{2} - 2\) , \( 2 a^{3} - 6 a^{2} - 9 a + 13\) , \( 2 a^{3} - 5 a\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(2a^{3}-6a^{2}-9a+13\right){x}+2a^{3}-5a$
25.1-d4 25.1-d 4.4.6224.1 \( 5^{2} \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $60.46539292$ 0.766429279 \( \frac{1521253261641712}{244140625} a^{3} - \frac{416904193452264}{48828125} a^{2} - \frac{6271166879450072}{244140625} a + \frac{5551244368587496}{244140625} \) \( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( -a\) , \( 1\) , \( 3 a^{3} - 13 a^{2} - 10 a + 21\) , \( 11 a^{3} - 21 a^{2} - 29 a + 29\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(3a^{3}-13a^{2}-10a+21\right){x}+11a^{3}-21a^{2}-29a+29$
28.1-a1 28.1-a 4.4.6224.1 \( 2^{2} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $171.2763683$ 2.171014149 \( -\frac{2200220800}{2401} a^{3} + \frac{4506771008}{2401} a^{2} + \frac{5715788416}{2401} a - \frac{6301711312}{2401} \) \( \bigl[a^{3} - 5 a\) , \( a^{3} - 2 a^{2} - 4 a + 6\) , \( 0\) , \( -260 a^{3} - 223 a^{2} + 1404 a + 1704\) , \( 8630 a^{3} + 6860 a^{2} - 46291 a - 54078\bigr] \) ${y}^2+\left(a^{3}-5a\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+6\right){x}^{2}+\left(-260a^{3}-223a^{2}+1404a+1704\right){x}+8630a^{3}+6860a^{2}-46291a-54078$
28.1-a2 28.1-a 4.4.6224.1 \( 2^{2} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $342.5527367$ 2.171014149 \( \frac{272072704}{49} a^{3} - \frac{372899840}{49} a^{2} - \frac{1121484800}{49} a + \frac{993001472}{49} \) \( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{3} - a^{2} - 4 a + 2\) , \( 0\) , \( -a^{2} - a + 1\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}-a^{2}-a+1$
28.1-b1 28.1-b 4.4.6224.1 \( 2^{2} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $255.2501755$ 1.617712204 \( \frac{1750207616680}{49} a^{3} + \frac{4273805316294}{49} a^{2} - \frac{39068998776}{49} a - \frac{3567815747260}{49} \) \( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 5 a\) , \( -15 a^{3} + 9 a^{2} + 49 a - 37\) , \( -46 a^{2} - 50 a + 66\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(-15a^{3}+9a^{2}+49a-37\right){x}-46a^{2}-50a+66$
28.1-b2 28.1-b 4.4.6224.1 \( 2^{2} \cdot 7 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $510.5003511$ 1.617712204 \( -\frac{2355168}{7} a^{3} + \frac{3301976}{7} a^{2} + \frac{5282712}{7} a - \frac{5102780}{7} \) \( \bigl[a^{3} - a^{2} - 4 a + 2\) , \( a^{3} - 2 a^{2} - 3 a + 6\) , \( a^{3} - 5 a\) , \( -a^{2} - a + 3\) , \( -a^{2} - a\bigr] \) ${y}^2+\left(a^{3}-a^{2}-4a+2\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-3a+6\right){x}^{2}+\left(-a^{2}-a+3\right){x}-a^{2}-a$
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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.