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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
841.8-a1 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( 732862 a^{3} + 606139 a^{2} - 1823977 a - 401107\) , \( 617507703 a^{3} + 510735017 a^{2} - 1536871501 a - 337973166\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(732862a^{3}+606139a^{2}-1823977a-401107\right){x}+617507703a^{3}+510735017a^{2}-1536871501a-337973166$
841.8-a2 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 1\) , \( -4944643 a^{3} - 4089726 a^{2} + 12306253 a + 2706273\) , \( 6260338923 a^{3} + 5177869806 a^{2} - 15580916689 a - 3426396906\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-4944643a^{3}-4089726a^{2}+12306253a+2706273\right){x}+6260338923a^{3}+5177869806a^{2}-15580916689a-3426396906$
841.8-a3 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -16554983445 a^{3} + 49664950335 a + 10231546590 \) \( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 46 a^{3} + 3 a^{2} - 179 a - 37\) , \( 315 a^{3} + 91 a^{2} - 1189 a - 251\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(46a^{3}+3a^{2}-179a-37\right){x}+315a^{3}+91a^{2}-1189a-251$
841.8-a4 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 85995 a^{3} - 257985 a - 52515 \) \( \bigl[a^{3} - 3 a\) , \( -a^{2} + a + 2\) , \( a^{2} + a - 2\) , \( 6 a^{3} + 3 a^{2} - 19 a - 2\) , \( 3 a^{3} - a^{2} - 16 a - 4\bigr] \) ${y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(6a^{3}+3a^{2}-19a-2\right){x}+3a^{3}-a^{2}-16a-4$
841.8-a5 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -378019 a^{3} - 312648 a^{2} + 940838 a + 206900\) , \( -174410939 a^{3} - 144253725 a^{2} + 434079076 a + 95458261\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-378019a^{3}-312648a^{2}+940838a+206900\right){x}-174410939a^{3}-144253725a^{2}+434079076a+95458261$
841.8-a6 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[a\) , \( a^{2} - a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -6055524 a^{3} - 5008513 a^{2} + 15071068 a + 3314280\) , \( -11131060964 a^{3} - 9206399587 a^{2} + 27703314743 a + 6092231498\bigr] \) ${y}^2+a{x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-6055524a^{3}-5008513a^{2}+15071068a+3314280\right){x}-11131060964a^{3}-9206399587a^{2}+27703314743a+6092231498$
841.8-a7 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-15$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( -85995 a^{3} + 257985 a - 138510 \) \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -70 a^{3} - 57 a^{2} + 174 a + 37\) , \( 432 a^{3} + 358 a^{2} - 1075 a - 238\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-70a^{3}-57a^{2}+174a+37\right){x}+432a^{3}+358a^{2}-1075a-238$
841.8-a8 841.8-a \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\Z/2\Z$ $-60$ $N(\mathrm{U}(1))$ $1$ $103.6460095$ 1.545063486 \( 16554983445 a^{3} - 49664950335 a + 26786530035 \) \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 3\) , \( a^{2} + a - 2\) , \( -1125 a^{3} - 897 a^{2} + 2819 a + 527\) , \( 27556 a^{3} + 22887 a^{2} - 68523 a - 15332\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-3\right){x}^{2}+\left(-1125a^{3}-897a^{2}+2819a+527\right){x}+27556a^{3}+22887a^{2}-68523a-15332$
841.8-b1 841.8-b \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $47.68821090$ 1.421787750 \( 0 \) \( \bigl[0\) , \( -a^{3} + a^{2} + 3 a - 2\) , \( a^{2} - 2\) , \( -a + 2\) , \( 133196725 a^{3} + 110165536 a^{2} - 331504541 a - 72901094\bigr] \) ${y}^2+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-2\right){x}^{2}+\left(-a+2\right){x}+133196725a^{3}+110165536a^{2}-331504541a-72901094$
841.8-b2 841.8-b \(\Q(\zeta_{15})^+\) \( 29^{2} \) 0 $\mathsf{trivial}$ $-3$ $N(\mathrm{U}(1))$ $1$ $47.68821090$ 1.421787750 \( 0 \) \( \bigl[0\) , \( a^{2} + a - 1\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 2 a\) , \( -32 a^{3} - 20 a^{2} + 94 a + 20\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(a^{3}+a^{2}-2a\right){x}-32a^{3}-20a^{2}+94a+20$
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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.