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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
1.1-a1 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\Z/7\Z$ $-147$ $1$ \( -7604567359488000 a^{5} - 7604567359488000 a^{4} + 38022836797440000 a^{3} + 30418269437952000 a^{2} - 45627404156928000 a - 28831103815680000 \) \( \bigl[0\) , \( -a^{4} + 3 a^{2} - a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 171 a^{5} + 81 a^{4} - 934 a^{3} - 192 a^{2} + 1093 a - 400\) , \( -1717 a^{5} - 882 a^{4} + 9032 a^{3} + 2384 a^{2} - 9721 a + 2928\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(-a^{4}+3a^{2}-a-1\right){x}^{2}+\left(171a^{5}+81a^{4}-934a^{3}-192a^{2}+1093a-400\right){x}-1717a^{5}-882a^{4}+9032a^{3}+2384a^{2}-9721a+2928$
1.1-a2 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\mathsf{trivial}$ $-147$ $1$ \( -7604567359488000 a^{5} - 7604567359488000 a^{4} + 38022836797440000 a^{3} + 30418269437952000 a^{2} - 45627404156928000 a - 28831103815680000 \) \( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 681 a^{5} - 1179 a^{4} - 4774 a^{3} + 5988 a^{2} + 7513 a - 6850\) , \( 23972 a^{5} - 32308 a^{4} - 159295 a^{3} + 171151 a^{2} + 242140 a - 203233\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(681a^{5}-1179a^{4}-4774a^{3}+5988a^{2}+7513a-6850\right){x}+23972a^{5}-32308a^{4}-159295a^{3}+171151a^{2}+242140a-203233$
1.1-a3 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\Z/7\Z$ $-147$ $1$ \( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \) \( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( 131 a^{5} - 79 a^{4} - 814 a^{3} + 488 a^{2} + 1133 a - 800\) , \( -1218 a^{5} + 802 a^{4} + 7465 a^{3} - 4959 a^{2} - 10420 a + 7417\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(131a^{5}-79a^{4}-814a^{3}+488a^{2}+1133a-800\right){x}-1218a^{5}+802a^{4}+7465a^{3}-4959a^{2}-10420a+7417$
1.1-a4 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\mathsf{trivial}$ $-147$ $1$ \( 7604567359488000 a^{5} + 7604567359488000 a^{4} - 38022836797440000 a^{3} - 30418269437952000 a^{2} + 45627404156928000 a - 6017401737216000 \) \( \bigl[0\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a + 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( -188 a^{5} + 589 a^{4} + 447 a^{3} - 2545 a^{2} + 1720 a - 203\) , \( -6938 a^{5} + 17357 a^{4} + 18336 a^{3} - 72391 a^{2} + 45500 a - 5253\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a+1\right){x}^{2}+\left(-188a^{5}+589a^{4}+447a^{3}-2545a^{2}+1720a-203\right){x}-6938a^{5}+17357a^{4}+18336a^{3}-72391a^{2}+45500a-5253$
1.1-a5 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\Z/7\Z$ $-3$ $1$ \( 0 \) \( \bigl[0\) , \( a^{4} - 3 a^{2} + a + 1\) , \( a^{5} - 4 a^{3} + a^{2} + 2 a - 2\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a\) , \( a^{5} - 6 a^{3} + 2 a^{2} + 9 a - 6\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}+a+1\right){x}^{2}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a\right){x}+a^{5}-6a^{3}+2a^{2}+9a-6$
1.1-a6 1.1-a \(\Q(\zeta_{21})^+\) \( 1 \) $0$ $\Z/7\Z$ $-3$ $1$ \( 0 \) \( \bigl[0\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a + 1\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( 2 a^{5} - a^{4} - 13 a^{3} + 5 a^{2} + 20 a - 3\) , \( -2 a^{5} + a^{4} + 11 a^{3} - 5 a^{2} - 13 a + 2\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a+1\right){x}^{2}+\left(2a^{5}-a^{4}-13a^{3}+5a^{2}+20a-3\right){x}-2a^{5}+a^{4}+11a^{3}-5a^{2}-13a+2$
27.1-a1 27.1-a \(\Q(\zeta_{21})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1713910976512}{1594323} \) \( \bigl[0\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a - 2\) , \( a^{3} - 2 a\) , \( 652 a^{5} + 912 a^{4} - 2478 a^{3} - 3387 a^{2} + 131 a - 130\) , \( -26388 a^{5} - 27297 a^{4} + 115694 a^{3} + 96838 a^{2} - 65135 a + 8321\bigr] \) ${y}^2+\left(a^{3}-2a\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){x}^{2}+\left(652a^{5}+912a^{4}-2478a^{3}-3387a^{2}+131a-130\right){x}-26388a^{5}-27297a^{4}+115694a^{3}+96838a^{2}-65135a+8321$
27.1-a2 27.1-a \(\Q(\zeta_{21})^+\) \( 3^{3} \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{28672}{3} \) \( \bigl[0\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{4} - 3 a^{2} + 1\) , \( -2 a^{5} + 7 a^{4} + 4 a^{3} - 28 a^{2} + 17 a + 4\) , \( -6 a^{5} + 13 a^{4} + 15 a^{3} - 52 a^{2} + 33 a - 3\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-2a^{5}+7a^{4}+4a^{3}-28a^{2}+17a+4\right){x}-6a^{5}+13a^{4}+15a^{3}-52a^{2}+33a-3$
27.1-b1 27.1-b \(\Q(\zeta_{21})^+\) \( 3^{3} \) $1$ $\mathsf{trivial}$ $0.991510447$ \( -\frac{1713910976512}{1594323} \) \( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -391 a^{5} + 2607 a^{3} - 391 a^{2} - 3911 a + 522\) , \( -5979 a^{5} + 40423 a^{3} - 5979 a^{2} - 61479 a + 8968\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-391a^{5}+2607a^{3}-391a^{2}-3911a+522\right){x}-5979a^{5}+40423a^{3}-5979a^{2}-61479a+8968$
27.1-b2 27.1-b \(\Q(\zeta_{21})^+\) \( 3^{3} \) $1$ $\Z/13\Z$ $0.076270034$ \( -\frac{28672}{3} \) \( \bigl[0\) , \( -a^{5} + 5 a^{3} - a^{2} - 5 a + 2\) , \( a^{5} - 5 a^{3} + a^{2} + 5 a - 2\) , \( -a^{5} + 7 a^{3} - a^{2} - 11 a + 2\) , \( a^{5} - 7 a^{3} + a^{2} + 11 a - 2\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-a^{5}+7a^{3}-a^{2}-11a+2\right){x}+a^{5}-7a^{3}+a^{2}+11a-2$
41.1-a1 41.1-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( -\frac{18790953}{41} a^{5} + \frac{6125244}{41} a^{4} + \frac{114705520}{41} a^{3} - \frac{31448339}{41} a^{2} - \frac{163512831}{41} a + \frac{24344025}{41} \) \( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a^{5} + 6 a^{3} - a^{2} - 7 a + 2\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} + a^{3} - a - 3\) , \( 2 a^{5} + 2 a^{4} - 7 a^{3} - 5 a^{2} + 5 a + 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-a^{2}-7a+2\right){x}^{2}+\left(a^{4}+a^{3}-a-3\right){x}+2a^{5}+2a^{4}-7a^{3}-5a^{2}+5a+1$
41.1-b1 41.1-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{12961040607}{41} a^{5} + \frac{9890721675}{41} a^{4} - \frac{57231718290}{41} a^{3} - \frac{19855931193}{41} a^{2} + \frac{55985643342}{41} a - \frac{13499009721}{41} \) \( \bigl[a^{3} - 2 a + 1\) , \( a^{5} - 5 a^{3} + 6 a - 1\) , \( a^{4} - 4 a^{2} + 3\) , \( 4 a^{5} + 3 a^{4} - 20 a^{3} - 8 a^{2} + 19 a - 4\) , \( 28 a^{5} + 18 a^{4} - 137 a^{3} - 58 a^{2} + 124 a - 19\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-4a^{2}+3\right){y}={x}^{3}+\left(a^{5}-5a^{3}+6a-1\right){x}^{2}+\left(4a^{5}+3a^{4}-20a^{3}-8a^{2}+19a-4\right){x}+28a^{5}+18a^{4}-137a^{3}-58a^{2}+124a-19$
41.1-b2 41.1-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{1692444184785}{68921} a^{5} + \frac{1104206748561}{68921} a^{4} - \frac{8329796043201}{68921} a^{3} - \frac{3610036693062}{68921} a^{2} + \frac{7573650019722}{68921} a - \frac{1024189061643}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( a^{5} - a^{4} - 5 a^{3} + 5 a^{2} + 5 a - 4\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{4} + a^{3} + 2 a^{2} - 4 a + 3\) , \( -a^{4} + 3 a^{2} - a - 1\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+5a^{2}+5a-4\right){x}^{2}+\left(-a^{4}+a^{3}+2a^{2}-4a+3\right){x}-a^{4}+3a^{2}-a-1$
41.1-c1 41.1-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( \frac{12961040607}{41} a^{5} + \frac{9890721675}{41} a^{4} - \frac{57231718290}{41} a^{3} - \frac{19855931193}{41} a^{2} + \frac{55985643342}{41} a - \frac{13499009721}{41} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 6 a\) , \( -a^{5} + 5 a^{3} - 2 a^{2} - 4 a + 5\) , \( a^{5} - 5 a^{3} + a^{2} + 6 a - 1\) , \( -3 a^{5} - 2 a^{4} + 13 a^{3} + 5 a^{2} - 12 a + 4\) , \( -a^{5} - a^{4} + 5 a^{3} + 2 a^{2} - 5 a + 1\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+6a\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+6a-1\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-2a^{2}-4a+5\right){x}^{2}+\left(-3a^{5}-2a^{4}+13a^{3}+5a^{2}-12a+4\right){x}-a^{5}-a^{4}+5a^{3}+2a^{2}-5a+1$
41.1-c2 41.1-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( \frac{1692444184785}{68921} a^{5} + \frac{1104206748561}{68921} a^{4} - \frac{8329796043201}{68921} a^{3} - \frac{3610036693062}{68921} a^{2} + \frac{7573650019722}{68921} a - \frac{1024189061643}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 4 a - 3\) , \( a^{3} - 3 a\) , \( 8 a^{5} + 3 a^{4} - 45 a^{3} - 2 a^{2} + 62 a - 8\) , \( -4 a^{5} + 9 a^{4} + 37 a^{3} - 30 a^{2} - 71 a + 11\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+4a-3\right){x}^{2}+\left(8a^{5}+3a^{4}-45a^{3}-2a^{2}+62a-8\right){x}-4a^{5}+9a^{4}+37a^{3}-30a^{2}-71a+11$
41.1-d1 41.1-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{18790953}{41} a^{5} + \frac{6125244}{41} a^{4} + \frac{114705520}{41} a^{3} - \frac{31448339}{41} a^{2} - \frac{163512831}{41} a + \frac{24344025}{41} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 5\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{5} + 3 a^{4} + a^{3} - 14 a^{2} + 10 a + 7\) , \( -a^{5} + 4 a^{4} + 2 a^{3} - 17 a^{2} + 8 a + 4\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-a^{5}+3a^{4}+a^{3}-14a^{2}+10a+7\right){x}-a^{5}+4a^{4}+2a^{3}-17a^{2}+8a+4$
41.2-a1 41.2-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( -\frac{7678148}{41} a^{5} - \frac{7305801}{41} a^{4} + \frac{31443377}{41} a^{3} + \frac{15419812}{41} a^{2} - \frac{29392241}{41} a + \frac{4246630}{41} \) \( \bigl[a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 4 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} - 4 a^{4} + 6 a^{3} + 14 a^{2} - 13 a - 2\) , \( -a^{5} - 3 a^{4} + 6 a^{3} + 9 a^{2} - 12 a + 3\bigr] \) ${y}^2+\left(a^{4}+a^{3}-4a^{2}-2a+2\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+4a-1\right){x}^{2}+\left(-a^{5}-4a^{4}+6a^{3}+14a^{2}-13a-2\right){x}-a^{5}-3a^{4}+6a^{3}+9a^{2}-12a+3$
41.2-b1 41.2-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{164991566655}{68921} a^{5} - \frac{491296262031}{68921} a^{4} - \frac{18167532093}{68921} a^{3} + \frac{1025969866218}{68921} a^{2} - \frac{709759186668}{68921} a + \frac{83606184531}{68921} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( -a^{5} + 4 a^{3} - a^{2} - 3 a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -a^{5} + 4 a^{3} - 3 a - 1\) , \( -1\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}{y}+\left(a^{4}+a^{3}-4a^{2}-3a+2\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-a^{5}+4a^{3}-3a-1\right){x}-1$
41.2-b2 41.2-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{17389718577}{41} a^{5} + \frac{13900894542}{41} a^{4} + \frac{106655548392}{41} a^{3} - \frac{82884018420}{41} a^{2} - \frac{152815374306}{41} a + \frac{108203083038}{41} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 3 a^{2} + 5 a + 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 8 a + 6\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 5 a - 3\) , \( -3 a^{5} - a^{4} + 19 a^{3} - 4 a^{2} - 27 a + 15\) , \( a^{5} - 8 a^{4} + 8 a^{3} + 14 a^{2} - 20 a + 6\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-3a^{2}+5a+1\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+5a-3\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-8a+6\right){x}^{2}+\left(-3a^{5}-a^{4}+19a^{3}-4a^{2}-27a+15\right){x}+a^{5}-8a^{4}+8a^{3}+14a^{2}-20a+6$
41.2-c1 41.2-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( \frac{164991566655}{68921} a^{5} - \frac{491296262031}{68921} a^{4} - \frac{18167532093}{68921} a^{3} + \frac{1025969866218}{68921} a^{2} - \frac{709759186668}{68921} a + \frac{83606184531}{68921} \) \( \bigl[a^{4} - 3 a^{2} + 1\) , \( -a^{5} - a^{4} + 4 a^{3} + 2 a^{2} - a + 1\) , \( a + 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 9 a + 1\) , \( 3 a^{5} - a^{4} - 20 a^{3} + 5 a^{2} + 31 a - 5\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+4a^{3}+2a^{2}-a+1\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+3a^{2}-9a+1\right){x}+3a^{5}-a^{4}-20a^{3}+5a^{2}+31a-5$
41.2-c2 41.2-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( -\frac{17389718577}{41} a^{5} + \frac{13900894542}{41} a^{4} + \frac{106655548392}{41} a^{3} - \frac{82884018420}{41} a^{2} - \frac{152815374306}{41} a + \frac{108203083038}{41} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} - a^{3} - 5 a^{2} + 3 a + 4\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( -2 a^{4} - a^{3} + 8 a^{2} + a - 3\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+4\right){x}^{2}+\left(-2a^{4}-a^{3}+8a^{2}+a-3\right){x}+a^{4}-a^{3}-4a^{2}+3a$
41.2-d1 41.2-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{7678148}{41} a^{5} - \frac{7305801}{41} a^{4} + \frac{31443377}{41} a^{3} + \frac{15419812}{41} a^{2} - \frac{29392241}{41} a + \frac{4246630}{41} \) \( \bigl[a^{3} + a^{2} - 2 a - 1\) , \( -a^{5} - a^{4} + 5 a^{3} + 4 a^{2} - 5 a - 1\) , \( a^{2} + a - 1\) , \( -a^{5} + 7 a^{3} - 7 a + 1\) , \( 2 a^{5} - 2 a^{4} - 2 a^{3} + 3 a^{2} - 2 a\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+5a^{3}+4a^{2}-5a-1\right){x}^{2}+\left(-a^{5}+7a^{3}-7a+1\right){x}+2a^{5}-2a^{4}-2a^{3}+3a^{2}-2a$
41.3-a1 41.3-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( \frac{358438}{41} a^{5} - \frac{6125244}{41} a^{4} + \frac{5527520}{41} a^{3} + \frac{13015824}{41} a^{2} - \frac{12861139}{41} a + \frac{1612826}{41} \) \( \bigl[a^{2} + a - 2\) , \( a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a - 2\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 5 a - 3\) , \( a^{5} - a^{4} - 6 a^{3} + 4 a^{2} + 6 a - 3\) , \( -a^{5} - a^{4} + 4 a^{3} + 3 a^{2} - 2 a - 1\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+5a-3\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+5a-2\right){x}^{2}+\left(a^{5}-a^{4}-6a^{3}+4a^{2}+6a-3\right){x}-a^{5}-a^{4}+4a^{3}+3a^{2}-2a-1$
41.3-b1 41.3-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{1298086563213}{68921} a^{5} - \frac{1104206748561}{68921} a^{4} - \frac{7953510945933}{68921} a^{3} + \frac{6600567441060}{68921} a^{2} + \frac{11370963467700}{68921} a - \frac{8686212609186}{68921} \) \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{5} - 4 a^{3} + a^{2} + a - 2\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{5} - 2 a^{4} + 6 a^{3} + 7 a^{2} - 10 a\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 10 a\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a\right){y}={x}^{3}+\left(a^{5}-4a^{3}+a^{2}+a-2\right){x}^{2}+\left(-a^{5}-2a^{4}+6a^{3}+7a^{2}-10a\right){x}-a^{5}-a^{4}+6a^{3}+3a^{2}-10a$
41.3-b2 41.3-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{5806060965}{41} a^{5} - \frac{9890721675}{41} a^{4} - \frac{17446647213}{41} a^{3} + \frac{38623032765}{41} a^{2} - \frac{19621562553}{41} a + \frac{2132617005}{41} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a - 1\) , \( a^{5} - 6 a^{3} + a^{2} + 9 a - 3\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 2\) , \( 15 a^{5} + 15 a^{4} - 66 a^{3} - 37 a^{2} + 68 a - 10\) , \( 41 a^{5} + 38 a^{4} - 177 a^{3} - 90 a^{2} + 170 a - 26\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-2\right){y}={x}^{3}+\left(a^{5}-6a^{3}+a^{2}+9a-3\right){x}^{2}+\left(15a^{5}+15a^{4}-66a^{3}-37a^{2}+68a-10\right){x}+41a^{5}+38a^{4}-177a^{3}-90a^{2}+170a-26$
41.3-c1 41.3-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( \frac{1298086563213}{68921} a^{5} - \frac{1104206748561}{68921} a^{4} - \frac{7953510945933}{68921} a^{3} + \frac{6600567441060}{68921} a^{2} + \frac{11370963467700}{68921} a - \frac{8686212609186}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 7 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( 3 a^{5} + 5 a^{4} - 15 a^{3} - 11 a^{2} + 17 a - 1\) , \( 9 a^{5} + 3 a^{4} - 33 a^{3} - 8 a^{2} + 25 a - 4\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){y}={x}^{3}+\left(-a^{5}-a^{4}+6a^{3}+3a^{2}-7a-1\right){x}^{2}+\left(3a^{5}+5a^{4}-15a^{3}-11a^{2}+17a-1\right){x}+9a^{5}+3a^{4}-33a^{3}-8a^{2}+25a-4$
41.3-c2 41.3-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( \frac{5806060965}{41} a^{5} - \frac{9890721675}{41} a^{4} - \frac{17446647213}{41} a^{3} + \frac{38623032765}{41} a^{2} - \frac{19621562553}{41} a + \frac{2132617005}{41} \) \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( -a^{4} + a^{3} + 5 a^{2} - 2 a - 5\) , \( a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{5} + 2 a^{4} - 7 a^{3} - 8 a^{2} + 11 a + 5\) , \( -a^{5} - a^{4} + 6 a^{3} + 3 a^{2} - 9 a - 1\bigr] \) ${y}^2+\left(a^{4}-4a^{2}+a+3\right){x}{y}+\left(a^{5}-4a^{3}+2a^{2}+3a-3\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-2a-5\right){x}^{2}+\left(a^{5}+2a^{4}-7a^{3}-8a^{2}+11a+5\right){x}-a^{5}-a^{4}+6a^{3}+3a^{2}-9a-1$
41.3-d1 41.3-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{358438}{41} a^{5} - \frac{6125244}{41} a^{4} + \frac{5527520}{41} a^{3} + \frac{13015824}{41} a^{2} - \frac{12861139}{41} a + \frac{1612826}{41} \) \( \bigl[a^{4} - 3 a^{2} + a + 1\) , \( -a^{5} + 6 a^{3} - 8 a\) , \( a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( 6 a^{5} - 4 a^{4} - 36 a^{3} + 23 a^{2} + 49 a - 30\) , \( 37 a^{5} - 35 a^{4} - 231 a^{3} + 201 a^{2} + 333 a - 260\bigr] \) ${y}^2+\left(a^{4}-3a^{2}+a+1\right){x}{y}+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-8a\right){x}^{2}+\left(6a^{5}-4a^{4}-36a^{3}+23a^{2}+49a-30\right){x}+37a^{5}-35a^{4}-231a^{3}+201a^{2}+333a-260$
41.4-a1 41.4-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( \frac{14625511}{41} a^{5} + \frac{11843590}{41} a^{4} - \frac{68962113}{41} a^{3} - \frac{40054650}{41} a^{2} + \frac{54505985}{41} a - \frac{7378940}{41} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 2\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{4} - 4 a^{2} + 2\) , \( 3 a^{5} + a^{4} - 15 a^{3} - 3 a^{2} + 17 a + 1\) , \( 2 a^{5} - 10 a^{3} + 2 a^{2} + 10 a - 2\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-2\right){x}{y}+\left(a^{4}-4a^{2}+2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(3a^{5}+a^{4}-15a^{3}-3a^{2}+17a+1\right){x}+2a^{5}-10a^{3}+2a^{2}+10a-2$
41.4-b1 41.4-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{2317236930}{41} a^{5} + \frac{6745914900}{41} a^{4} + \frac{942380973}{41} a^{3} - \frac{15400001988}{41} a^{2} + \frac{10454504706}{41} a - \frac{1224790551}{41} \) \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( a^{4} - 5 a^{2} + a + 3\) , \( a^{2} + a - 2\) , \( 3 a^{5} - 14 a^{3} + 15 a + 1\) , \( 5 a^{5} - 28 a^{3} + 2 a^{2} + 38 a - 5\bigr] \) ${y}^2+\left(a^{4}+a^{3}-3a^{2}-2a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{4}-5a^{2}+a+3\right){x}^{2}+\left(3a^{5}-14a^{3}+15a+1\right){x}+5a^{5}-28a^{3}+2a^{2}+38a-5$
41.4-b2 41.4-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{971781867837}{68921} a^{5} - \frac{885653883603}{68921} a^{4} + \frac{4138247022237}{68921} a^{3} + \frac{2079537404544}{68921} a^{2} - \frac{3801129136902}{68921} a + \frac{507608403966}{68921} \) \( \bigl[a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( -a^{4} - a^{3} + 4 a^{2} + 2 a - 1\) , \( a + 1\) , \( 2 a^{5} - 10 a^{3} + 3 a^{2} + 10 a - 5\) , \( 2 a^{5} - 10 a^{3} + 3 a^{2} + 11 a - 7\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+4a^{2}+2a-1\right){x}^{2}+\left(2a^{5}-10a^{3}+3a^{2}+10a-5\right){x}+2a^{5}-10a^{3}+3a^{2}+11a-7$
41.4-c1 41.4-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( -\frac{2317236930}{41} a^{5} + \frac{6745914900}{41} a^{4} + \frac{942380973}{41} a^{3} - \frac{15400001988}{41} a^{2} + \frac{10454504706}{41} a - \frac{1224790551}{41} \) \( \bigl[a^{5} - 5 a^{3} + 2 a^{2} + 6 a - 3\) , \( -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - a + 2\) , \( a^{5} - 4 a^{3} + 2 a^{2} + 2 a - 4\) , \( 3 a^{5} + 2 a^{4} - 11 a^{3} - 4 a^{2} + 8 a + 1\) , \( 6 a^{5} + 6 a^{4} - 24 a^{3} - 14 a^{2} + 19 a - 2\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+2a^{2}+6a-3\right){x}{y}+\left(a^{5}-4a^{3}+2a^{2}+2a-4\right){y}={x}^{3}+\left(-a^{5}+a^{4}+4a^{3}-4a^{2}-a+2\right){x}^{2}+\left(3a^{5}+2a^{4}-11a^{3}-4a^{2}+8a+1\right){x}+6a^{5}+6a^{4}-24a^{3}-14a^{2}+19a-2$
41.4-c2 41.4-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( -\frac{971781867837}{68921} a^{5} - \frac{885653883603}{68921} a^{4} + \frac{4138247022237}{68921} a^{3} + \frac{2079537404544}{68921} a^{2} - \frac{3801129136902}{68921} a + \frac{507608403966}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( a^{3} - a^{2} - 4 a + 2\) , \( a^{2} + a - 1\) , \( -3 a^{5} - a^{4} + 18 a^{3} + 5 a^{2} - 20 a + 3\) , \( 17 a^{5} + 11 a^{4} - 82 a^{3} - 31 a^{2} + 79 a - 11\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+2\right){x}^{2}+\left(-3a^{5}-a^{4}+18a^{3}+5a^{2}-20a+3\right){x}+17a^{5}+11a^{4}-82a^{3}-31a^{2}+79a-11$
41.4-d1 41.4-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{14625511}{41} a^{5} + \frac{11843590}{41} a^{4} - \frac{68962113}{41} a^{3} - \frac{40054650}{41} a^{2} + \frac{54505985}{41} a - \frac{7378940}{41} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( a^{5} - a^{4} - 6 a^{3} + 5 a^{2} + 7 a - 4\) , \( a^{2} + a - 2\) , \( -6 a^{5} + 36 a^{3} - 3 a^{2} - 53 a + 7\) , \( -14 a^{5} + 3 a^{4} + 86 a^{3} - 18 a^{2} - 126 a + 18\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+7a-4\right){x}^{2}+\left(-6a^{5}+36a^{3}-3a^{2}-53a+7\right){x}-14a^{5}+3a^{4}+86a^{3}-18a^{2}-126a+18$
41.5-a1 41.5-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( -\frac{1959802}{41} a^{5} + \frac{7305801}{41} a^{4} - \frac{1686142}{41} a^{3} - \frac{25057762}{41} a^{2} + \frac{36500036}{41} a - \frac{14311424}{41} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 3 a - 1\) , \( -a^{4} + a^{3} + 4 a^{2} - 2 a - 1\) , \( a^{5} - 4 a^{3} + a^{2} + 3 a - 2\) , \( 17 a^{5} + 13 a^{4} - 68 a^{3} - 31 a^{2} + 59 a - 8\) , \( 42 a^{5} + 54 a^{4} - 196 a^{3} - 123 a^{2} + 199 a - 27\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+3a-1\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+3a-2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-2a-1\right){x}^{2}+\left(17a^{5}+13a^{4}-68a^{3}-31a^{2}+59a-8\right){x}+42a^{5}+54a^{4}-196a^{3}-123a^{2}+199a-27$
41.5-b1 41.5-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1824869065509}{68921} a^{5} + \frac{491296262031}{68921} a^{4} + \frac{11308085774361}{68921} a^{3} - \frac{2685847365072}{68921} a^{2} - \frac{16561220551596}{68921} a + \frac{2497339386342}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\) , \( 4 a^{5} + 6 a^{4} - 20 a^{3} - 14 a^{2} + 21 a - 3\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-1\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a\right){y}={x}^{3}+\left(a^{4}+a^{3}-3a^{2}-3a+1\right){x}^{2}+\left(4a^{5}+6a^{4}-20a^{3}-14a^{2}+21a-3\right){x}+4a^{5}+6a^{4}-20a^{3}-14a^{2}+21a-3$
41.5-b2 41.5-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{20534525352}{41} a^{5} - \frac{13900894542}{41} a^{4} + \frac{101732772825}{41} a^{3} + \frac{44959774491}{41} a^{2} - \frac{93107150055}{41} a + \frac{12580316820}{41} \) \( \bigl[a^{5} + a^{4} - 4 a^{3} - 3 a^{2} + 2 a + 1\) , \( a^{5} + a^{4} - 6 a^{3} - 3 a^{2} + 7 a\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( -4 a^{5} + 2 a^{4} + 23 a^{3} - 8 a^{2} - 32 a + 6\) , \( -32 a^{5} + 10 a^{4} + 198 a^{3} - 51 a^{2} - 290 a + 44\bigr] \) ${y}^2+\left(a^{5}+a^{4}-4a^{3}-3a^{2}+2a+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-3a\right){y}={x}^{3}+\left(a^{5}+a^{4}-6a^{3}-3a^{2}+7a\right){x}^{2}+\left(-4a^{5}+2a^{4}+23a^{3}-8a^{2}-32a+6\right){x}-32a^{5}+10a^{4}+198a^{3}-51a^{2}-290a+44$
41.5-c1 41.5-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( -\frac{1824869065509}{68921} a^{5} + \frac{491296262031}{68921} a^{4} + \frac{11308085774361}{68921} a^{3} - \frac{2685847365072}{68921} a^{2} - \frac{16561220551596}{68921} a + \frac{2497339386342}{68921} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 1\) , \( a^{4} - 3 a^{2} + 1\) , \( 17 a^{5} + 11 a^{4} - 72 a^{3} - 20 a^{2} + 60 a - 11\) , \( 34 a^{5} + 34 a^{4} - 144 a^{3} - 84 a^{2} + 133 a - 13\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-1\right){x}^{2}+\left(17a^{5}+11a^{4}-72a^{3}-20a^{2}+60a-11\right){x}+34a^{5}+34a^{4}-144a^{3}-84a^{2}+133a-13$
41.5-c2 41.5-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( -\frac{20534525352}{41} a^{5} - \frac{13900894542}{41} a^{4} + \frac{101732772825}{41} a^{3} + \frac{44959774491}{41} a^{2} - \frac{93107150055}{41} a + \frac{12580316820}{41} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 5 a - 1\) , \( -a^{5} + 6 a^{3} - 2 a^{2} - 9 a + 4\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{5} + a^{4} - 5 a^{3} - 5 a^{2} + 4 a + 2\) , \( -4 a^{5} - 2 a^{4} + 20 a^{3} + 6 a^{2} - 19 a + 3\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+5a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-2a^{2}-9a+4\right){x}^{2}+\left(a^{5}+a^{4}-5a^{3}-5a^{2}+4a+2\right){x}-4a^{5}-2a^{4}+20a^{3}+6a^{2}-19a+3$
41.5-d1 41.5-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{1959802}{41} a^{5} + \frac{7305801}{41} a^{4} - \frac{1686142}{41} a^{3} - \frac{25057762}{41} a^{2} + \frac{36500036}{41} a - \frac{14311424}{41} \) \( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 2 a - 4\) , \( a^{5} - a^{4} - 4 a^{3} + 6 a^{2} + a - 7\) , \( a^{5} - 5 a^{3} + a^{2} + 6 a - 2\) , \( 6 a^{5} - 6 a^{4} - 32 a^{3} + 31 a^{2} + 35 a - 26\) , \( 4 a^{5} - 5 a^{4} - 20 a^{3} + 25 a^{2} + 18 a - 20\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a^{2}+2a-4\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+6a^{2}+a-7\right){x}^{2}+\left(6a^{5}-6a^{4}-32a^{3}+31a^{2}+35a-26\right){x}+4a^{5}-5a^{4}-20a^{3}+25a^{2}+18a-20$
41.6-a1 41.6-a \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.000898331$ \( \frac{13444954}{41} a^{5} - \frac{11843590}{41} a^{4} - \frac{81028162}{41} a^{3} + \frac{68125115}{41} a^{2} + \frac{114760190}{41} a - \frac{87740634}{41} \) \( \bigl[a^{4} - 3 a^{2}\) , \( a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 3 a - 5\) , \( a^{5} + a^{4} - 4 a^{3} - 2 a^{2} + 2 a - 2\) , \( -2 a^{5} + 13 a^{3} - 3 a^{2} - 21 a + 8\) , \( -a^{4} - a^{3} + 3 a^{2} + 2 a - 1\bigr] \) ${y}^2+\left(a^{4}-3a^{2}\right){x}{y}+\left(a^{5}+a^{4}-4a^{3}-2a^{2}+2a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+3a-5\right){x}^{2}+\left(-2a^{5}+13a^{3}-3a^{2}-21a+8\right){x}-a^{4}-a^{3}+3a^{2}+2a-1$
41.6-b1 41.6-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{21474379287}{41} a^{5} - \frac{6745914900}{41} a^{4} - \frac{134652336687}{41} a^{3} + \frac{34557144345}{41} a^{2} + \frac{199103938866}{41} a - \frac{30078826092}{41} \) \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{5} - 6 a^{3} + 8 a - 1\) , \( a + 1\) , \( a^{5} - 11 a^{3} + 26 a - 3\) , \( -4 a^{5} + 14 a^{4} + 3 a^{3} - 53 a^{2} + 49 a - 8\bigr] \) ${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+8a-1\right){x}^{2}+\left(a^{5}-11a^{3}+26a-3\right){x}-4a^{5}+14a^{4}+3a^{3}-53a^{2}+49a-8$
41.6-b2 41.6-b \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( -\frac{358871381307}{68921} a^{5} + \frac{885653883603}{68921} a^{4} + \frac{855141724629}{68921} a^{3} - \frac{3410190653688}{68921} a^{2} + \frac{2127495387744}{68921} a - \frac{244988831367}{68921} \) \( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{4} - a^{3} - 5 a^{2} + 4 a + 5\) , \( a^{3} + a^{2} - 2 a - 2\) , \( 4 a^{5} + 7 a^{4} - 15 a^{3} - 27 a^{2} + 9 a + 20\) , \( 6 a^{5} + 11 a^{4} - 21 a^{3} - 39 a^{2} + 10 a + 22\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a^{2}+3a-4\right){x}{y}+\left(a^{3}+a^{2}-2a-2\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+4a+5\right){x}^{2}+\left(4a^{5}+7a^{4}-15a^{3}-27a^{2}+9a+20\right){x}+6a^{5}+11a^{4}-21a^{3}-39a^{2}+10a+22$
41.6-c1 41.6-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.004176285$ \( \frac{21474379287}{41} a^{5} - \frac{6745914900}{41} a^{4} - \frac{134652336687}{41} a^{3} + \frac{34557144345}{41} a^{2} + \frac{199103938866}{41} a - \frac{30078826092}{41} \) \( \bigl[a^{5} - 4 a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{5} - a^{4} - 6 a^{3} + 6 a^{2} + 7 a - 7\) , \( a^{3} - 3 a\) , \( -a^{5} + 3 a^{4} + 11 a^{3} - 9 a^{2} - 22 a + 5\) , \( 6 a^{5} + 2 a^{4} - 31 a^{3} - 2 a^{2} + 37 a - 6\bigr] \) ${y}^2+\left(a^{5}-4a^{3}+2a^{2}+3a-4\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(a^{5}-a^{4}-6a^{3}+6a^{2}+7a-7\right){x}^{2}+\left(-a^{5}+3a^{4}+11a^{3}-9a^{2}-22a+5\right){x}+6a^{5}+2a^{4}-31a^{3}-2a^{2}+37a-6$
41.6-c2 41.6-c \(\Q(\zeta_{21})^+\) \( 41 \) $1$ $\mathsf{trivial}$ $0.001392095$ \( -\frac{358871381307}{68921} a^{5} + \frac{885653883603}{68921} a^{4} + \frac{855141724629}{68921} a^{3} - \frac{3410190653688}{68921} a^{2} + \frac{2127495387744}{68921} a - \frac{244988831367}{68921} \) \( \bigl[a^{3} + a^{2} - 2 a - 2\) , \( a^{5} - 6 a^{3} + a^{2} + 7 a - 1\) , \( a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( -a^{5} + 2 a^{3} - a^{2} + a + 3\) , \( a^{5} + a^{4} - 6 a^{3} - 2 a^{2} + 8 a - 1\bigr] \) ${y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(a^{5}-6a^{3}+a^{2}+7a-1\right){x}^{2}+\left(-a^{5}+2a^{3}-a^{2}+a+3\right){x}+a^{5}+a^{4}-6a^{3}-2a^{2}+8a-1$
41.6-d1 41.6-d \(\Q(\zeta_{21})^+\) \( 41 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{13444954}{41} a^{5} - \frac{11843590}{41} a^{4} - \frac{81028162}{41} a^{3} + \frac{68125115}{41} a^{2} + \frac{114760190}{41} a - \frac{87740634}{41} \) \( \bigl[a^{3} - 2 a + 1\) , \( -a^{5} + 4 a^{3} - 2 a^{2} - 2 a + 3\) , \( a^{4} - 3 a^{2} + 1\) , \( -2 a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 3\) , \( a^{5} - 3 a^{4} + 5 a^{2} - 4 a + 1\bigr] \) ${y}^2+\left(a^{3}-2a+1\right){x}{y}+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+3\right){x}^{2}+\left(-2a^{4}+2a^{3}+4a^{2}-6a+3\right){x}+a^{5}-3a^{4}+5a^{2}-4a+1$
43.1-a1 43.1-a \(\Q(\zeta_{21})^+\) \( 43 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{122671686822496847013039511249758605577398}{929293739471222707} a^{5} + \frac{80040521603412429280297685957868481938607}{929293739471222707} a^{4} - \frac{603764956031188829103770286089077775728189}{929293739471222707} a^{3} - \frac{261677914170315830376360951082281856697755}{929293739471222707} a^{2} + \frac{548956616586298989964223373216163970685955}{929293739471222707} a - \frac{74235009600386253636048203807121847689780}{929293739471222707} \) \( \bigl[a^{2} + a - 2\) , \( a^{4} - 3 a^{2} - a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( -1815 a^{5} + 640 a^{4} + 11287 a^{3} - 3274 a^{2} - 16592 a + 2508\) , \( 61652 a^{5} - 18037 a^{4} - 382582 a^{3} + 96346 a^{2} + 561130 a - 84717\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}-a+1\right){x}^{2}+\left(-1815a^{5}+640a^{4}+11287a^{3}-3274a^{2}-16592a+2508\right){x}+61652a^{5}-18037a^{4}-382582a^{3}+96346a^{2}+561130a-84717$
43.1-a2 43.1-a \(\Q(\zeta_{21})^+\) \( 43 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{159376181}{43} a^{5} - \frac{145623752}{43} a^{4} - \frac{948152084}{43} a^{3} + \frac{815727811}{43} a^{2} + \frac{1334598975}{43} a - \frac{1032410901}{43} \) \( \bigl[a^{2} + a - 2\) , \( a^{4} - 3 a^{2} - a + 1\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( -3 a^{3} + a^{2} + 8 a - 2\) , \( -a^{5} - a^{4} + 5 a^{3} + 2 a^{2} - 5 a\bigr] \) ${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-2\right){y}={x}^{3}+\left(a^{4}-3a^{2}-a+1\right){x}^{2}+\left(-3a^{3}+a^{2}+8a-2\right){x}-a^{5}-a^{4}+5a^{3}+2a^{2}-5a$
43.1-b1 43.1-b \(\Q(\zeta_{21})^+\) \( 43 \) $0$ $\mathsf{trivial}$ $1$ \( \frac{122671686822496847013039511249758605577398}{929293739471222707} a^{5} + \frac{80040521603412429280297685957868481938607}{929293739471222707} a^{4} - \frac{603764956031188829103770286089077775728189}{929293739471222707} a^{3} - \frac{261677914170315830376360951082281856697755}{929293739471222707} a^{2} + \frac{548956616586298989964223373216163970685955}{929293739471222707} a - \frac{74235009600386253636048203807121847689780}{929293739471222707} \) \( \bigl[a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 9 a + 2\) , \( 0\) , \( 298 a^{5} - 358 a^{4} - 2027 a^{3} + 1978 a^{2} + 3192 a - 2634\) , \( 22934 a^{5} - 21647 a^{4} - 143434 a^{3} + 125585 a^{2} + 208990 a - 162582\bigr] \) ${y}^2+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-1\right){x}{y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-4a^{2}-9a+2\right){x}^{2}+\left(298a^{5}-358a^{4}-2027a^{3}+1978a^{2}+3192a-2634\right){x}+22934a^{5}-21647a^{4}-143434a^{3}+125585a^{2}+208990a-162582$
43.1-b2 43.1-b \(\Q(\zeta_{21})^+\) \( 43 \) $0$ $\Z/11\Z$ $1$ \( \frac{159376181}{43} a^{5} - \frac{145623752}{43} a^{4} - \frac{948152084}{43} a^{3} + \frac{815727811}{43} a^{2} + \frac{1334598975}{43} a - \frac{1032410901}{43} \) \( \bigl[a^{5} - 5 a^{3} + a^{2} + 5 a - 1\) , \( -a^{5} + a^{4} + 6 a^{3} - 6 a^{2} - 9 a + 6\) , \( a^{5} + a^{4} - 5 a^{3} - 2 a^{2} + 6 a - 2\) , \( -2 a^{5} - 2 a^{4} + 9 a^{3} + 4 a^{2} - 8 a + 6\) , \( a^{5} + a^{4} - 5 a^{3} - 5 a^{2} + 3 a + 1\bigr] \) ${y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{5}+a^{4}-5a^{3}-2a^{2}+6a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-9a+6\right){x}^{2}+\left(-2a^{5}-2a^{4}+9a^{3}+4a^{2}-8a+6\right){x}+a^{5}+a^{4}-5a^{3}-5a^{2}+3a+1$
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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.