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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
19.1-a1 19.1-a 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $8.546425211$ 2.378562937 \( -\frac{16075947372441447683071219604597}{2476099} a^{3} - \frac{3746856964444343086453903380667}{2476099} a^{2} + \frac{75759591575353944746601364499055}{2476099} a + \frac{13037311952911337243612323131579}{2476099} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} + a - 3\) , \( a^{2} - 3\) , \( -416 a^{3} + 73 a^{2} + 1594 a - 1363\) , \( -4 a^{3} - 6042 a^{2} - 5981 a + 16343\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-416a^{3}+73a^{2}+1594a-1363\right){x}-4a^{3}-6042a^{2}-5981a+16343$
19.1-a2 19.1-a 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $213.6606302$ 2.378562937 \( \frac{1121092698094}{19} a^{3} - \frac{61710073984}{19} a^{2} - \frac{4164205346487}{19} a + \frac{3126720543036}{19} \) \( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( 0\) , \( 4 a^{3} - 16 a - 8\) , \( 22 a^{3} + 6 a^{2} - 98 a - 29\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(4a^{3}-16a-8\right){x}+22a^{3}+6a^{2}-98a-29$
19.1-a3 19.1-a 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $427.3212605$ 2.378562937 \( -\frac{4504636}{361} a^{3} + \frac{51759581}{361} a^{2} + \frac{66818656}{361} a - \frac{151977092}{361} \) \( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 3\) , \( 0\) , \( -a^{3} + 4 a + 2\) , \( 0\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}-a^{2}+4a+3\right){x}^{2}+\left(-a^{3}+4a+2\right){x}$
19.1-a4 19.1-a 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $17.09285042$ 2.378562937 \( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \) \( \bigl[1\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} + a - 2\) , \( 659 a^{3} - 621 a^{2} - 3152 a + 3119\) , \( 10968 a^{3} - 11477 a^{2} - 52551 a + 57385\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(659a^{3}-621a^{2}-3152a+3119\right){x}+10968a^{3}-11477a^{2}-52551a+57385$
19.1-b1 19.1-b 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.9403342$ 2.242853406 \( \frac{921050783480}{361} a^{3} + \frac{1052547388417}{361} a^{2} - \frac{2349627001920}{361} a - \frac{429808776872}{361} \) \( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( -a - 1\) , \( a\) , \( -11 a^{2} - 8 a + 29\) , \( -14 a^{3} - 5 a^{2} + 45 a - 22\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-2\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a^{2}-8a+29\right){x}-14a^{3}-5a^{2}+45a-22$
19.1-b2 19.1-b 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $402.9403342$ 2.242853406 \( -\frac{10354365223022612}{130321} a^{3} + \frac{21330013069525904}{130321} a^{2} + \frac{1995387782333409}{130321} a - \frac{338656947464676}{130321} \) \( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 4 a + 2\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 32 a^{3} + 19 a^{2} - 155 a - 95\) , \( -218 a^{3} - 25 a^{2} + 1023 a + 41\bigr] \) ${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+2\right){x}^{2}+\left(32a^{3}+19a^{2}-155a-95\right){x}-218a^{3}-25a^{2}+1023a+41$
19.1-c1 19.1-c 4.4.8069.1 \( 19 \) $0 \le r \le 1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $7.687728166$ 2.402020858 \( \frac{22194249947282956334538362641}{6859} a^{3} - \frac{26005091804226993659513655139}{6859} a^{2} - \frac{106506071026766490646787756502}{6859} a + \frac{129258769902742562322216519975}{6859} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 85 a^{3} + 459 a^{2} - 366 a - 2135\) , \( -10862 a^{3} + 5071 a^{2} + 51661 a - 27097\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(85a^{3}+459a^{2}-366a-2135\right){x}-10862a^{3}+5071a^{2}+51661a-27097$
19.1-c2 19.1-c 4.4.8069.1 \( 19 \) $0 \le r \le 1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $7.687728166$ 2.402020858 \( -\frac{32920179219067742335}{47045881} a^{3} + \frac{38573217360313423603}{47045881} a^{2} + \frac{157978303983605228544}{47045881} a - \frac{191727709378707559964}{47045881} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + 5 a\) , \( 1\) , \( 40 a^{3} + 39 a^{2} - 181 a - 160\) , \( 261 a^{3} + 179 a^{2} - 1219 a - 762\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(40a^{3}+39a^{2}-181a-160\right){x}+261a^{3}+179a^{2}-1219a-762$
19.1-c3 19.1-c 4.4.8069.1 \( 19 \) $0 \le r \le 1$ $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $622.7059815$ 2.402020858 \( -\frac{140184437194705}{19} a^{3} - \frac{32674219622507}{19} a^{2} + \frac{660633828065334}{19} a + \frac{113692420421896}{19} \) \( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} + 11 a^{2} + 19 a - 45\) , \( -13 a^{3} + 30 a^{2} + 74 a - 109\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(-a^{3}+11a^{2}+19a-45\right){x}-13a^{3}+30a^{2}+74a-109$
19.1-c4 19.1-c 4.4.8069.1 \( 19 \) $0 \le r \le 1$ $\Z/6\Z$ $\mathrm{SU}(2)$ $1$ $622.7059815$ 2.402020858 \( -\frac{468732461}{361} a^{3} - \frac{107902627}{361} a^{2} + \frac{2209071420}{361} a + \frac{374364855}{361} \) \( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 4 a^{3} + 6 a^{2} - 11 a - 5\) , \( 7 a^{3} + 9 a^{2} - 18 a - 5\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(4a^{3}+6a^{2}-11a-5\right){x}+7a^{3}+9a^{2}-18a-5$
19.1-d1 19.1-d 4.4.8069.1 \( 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.422148619$ $189.7385750$ 2.675053088 \( \frac{22194249947282956334538362641}{6859} a^{3} - \frac{26005091804226993659513655139}{6859} a^{2} - \frac{106506071026766490646787756502}{6859} a + \frac{129258769902742562322216519975}{6859} \) \( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -406 a^{3} + 495 a^{2} + 1969 a - 2505\) , \( 8935 a^{3} - 10618 a^{2} - 42945 a + 52855\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-406a^{3}+495a^{2}+1969a-2505\right){x}+8935a^{3}-10618a^{2}-42945a+52855$
19.1-d2 19.1-d 4.4.8069.1 \( 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.211074309$ $189.7385750$ 2.675053088 \( -\frac{32920179219067742335}{47045881} a^{3} + \frac{38573217360313423603}{47045881} a^{2} + \frac{157978303983605228544}{47045881} a - \frac{191727709378707559964}{47045881} \) \( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -26 a^{3} + 30 a^{2} + 124 a - 155\) , \( 153 a^{3} - 184 a^{2} - 737 a + 908\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-26a^{3}+30a^{2}+124a-155\right){x}+153a^{3}-184a^{2}-737a+908$
19.1-d3 19.1-d 4.4.8069.1 \( 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.140716206$ $1707.647175$ 2.675053088 \( -\frac{140184437194705}{19} a^{3} - \frac{32674219622507}{19} a^{2} + \frac{660633828065334}{19} a + \frac{113692420421896}{19} \) \( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( 14 a^{3} + 10 a^{2} - 66 a - 45\) , \( -101 a^{3} - 46 a^{2} + 475 a + 186\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(14a^{3}+10a^{2}-66a-45\right){x}-101a^{3}-46a^{2}+475a+186$
19.1-d4 19.1-d 4.4.8069.1 \( 19 \) $1$ $\Z/2\Z$ $\mathrm{SU}(2)$ $0.070358103$ $1707.647175$ 2.675053088 \( -\frac{468732461}{361} a^{3} - \frac{107902627}{361} a^{2} + \frac{2209071420}{361} a + \frac{374364855}{361} \) \( \bigl[a\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a\) , \( -a^{3} + 4 a\) , \( -2 a^{3} - a^{2} + 9 a + 4\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(-a^{3}+4a\right){x}-2a^{3}-a^{2}+9a+4$
19.1-e1 19.1-e 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $99.74404697$ 0.555197026 \( -\frac{10354365223022612}{130321} a^{3} + \frac{21330013069525904}{130321} a^{2} + \frac{1995387782333409}{130321} a - \frac{338656947464676}{130321} \) \( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -16 a^{3} + 12 a^{2} + 72 a - 84\) , \( 11 a^{3} - 32 a^{2} - 66 a + 111\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-16a^{3}+12a^{2}+72a-84\right){x}+11a^{3}-32a^{2}-66a+111$
19.1-e2 19.1-e 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $99.74404697$ 0.555197026 \( \frac{921050783480}{361} a^{3} + \frac{1052547388417}{361} a^{2} - \frac{2349627001920}{361} a - \frac{429808776872}{361} \) \( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a - 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{3} - 8 a^{2} - 3 a + 21\) , \( -5 a^{3} - 12 a^{2} + 7 a + 20\bigr] \) ${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+5a-1\right){x}^{2}+\left(-a^{3}-8a^{2}-3a+21\right){x}-5a^{3}-12a^{2}+7a+20$
19.1-f1 19.1-f 4.4.8069.1 \( 19 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $2054.093936$ 0.457341317 \( -\frac{4504636}{361} a^{3} + \frac{51759581}{361} a^{2} + \frac{66818656}{361} a - \frac{151977092}{361} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - a - 4\) , \( a^{2} - 3\) , \( 14 a^{3} + 5 a^{2} - 64 a - 10\) , \( -18 a^{3} - 2 a^{2} + 88 a + 12\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(14a^{3}+5a^{2}-64a-10\right){x}-18a^{3}-2a^{2}+88a+12$
19.1-f2 19.1-f 4.4.8069.1 \( 19 \) 0 $\Z/10\Z$ $\mathrm{SU}(2)$ $1$ $1027.046968$ 0.457341317 \( \frac{1121092698094}{19} a^{3} - \frac{61710073984}{19} a^{2} - \frac{4164205346487}{19} a + \frac{3126720543036}{19} \) \( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 2 a^{3} - 18 a^{2} + 24 a - 5\) , \( -14 a^{3} + 48 a^{2} - 41 a + 2\bigr] \) ${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-4a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(2a^{3}-18a^{2}+24a-5\right){x}-14a^{3}+48a^{2}-41a+2$
19.1-f3 19.1-f 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $1.643275149$ 0.457341317 \( -\frac{16075947372441447683071219604597}{2476099} a^{3} - \frac{3746856964444343086453903380667}{2476099} a^{2} + \frac{75759591575353944746601364499055}{2476099} a + \frac{13037311952911337243612323131579}{2476099} \) \( \bigl[a\) , \( -a^{3} + a^{2} + 3 a - 4\) , \( 0\) , \( -565 a^{3} + 38 a^{2} + 2304 a - 1843\) , \( -5666 a^{3} + 20 a^{2} + 23247 a - 18075\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(-a^{3}+a^{2}+3a-4\right){x}^{2}+\left(-565a^{3}+38a^{2}+2304a-1843\right){x}-5666a^{3}+20a^{2}+23247a-18075$
19.1-f4 19.1-f 4.4.8069.1 \( 19 \) 0 $\Z/2\Z$ $\mathrm{SU}(2)$ $1$ $3.286550298$ 0.457341317 \( -\frac{7453708100651023724731227}{6131066257801} a^{3} - \frac{1737250893512074261087546}{6131066257801} a^{2} + \frac{35126388668209027651898288}{6131066257801} a + \frac{6044827780477046155354082}{6131066257801} \) \( \bigl[a^{3} - 4 a + 1\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 763 a^{3} - 898 a^{2} - 3634 a + 4396\) , \( -17404 a^{3} + 20349 a^{2} + 83667 a - 101492\bigr] \) ${y}^2+\left(a^{3}-4a+1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(763a^{3}-898a^{2}-3634a+4396\right){x}-17404a^{3}+20349a^{2}+83667a-101492$
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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.