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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
128.5-a1 128.5-a \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/4\Z$ $1$ $8.823675287$ 2.140055600 \( -\frac{217}{16} a - \frac{339}{16} \) \( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 2 a + 3\) , \( a + 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a+3\right){x}+a+3$
128.5-a2 128.5-a \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $35.29470114$ 2.140055600 \( -\frac{1592342311}{2} a + \frac{4078872403}{2} \) \( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -a - 9\) , \( -2 a + 3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-9\right){x}-2a+3$
128.5-a3 128.5-a \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $17.64735057$ 2.140055600 \( \frac{159495}{4} a + \frac{481229}{4} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 47 a - 126\) , \( -221 a + 563\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(47a-126\right){x}-221a+563$
128.5-a4 128.5-a \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.411837643$ 2.140055600 \( \frac{23841914775}{2} a + \frac{37230413581}{2} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 97 a - 256\) , \( 459 a - 1181\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(97a-256\right){x}+459a-1181$
128.5-b1 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( -\frac{55573026649}{16777216} a - \frac{85738862931}{16777216} \) \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -23 a + 42\) , \( -329 a + 864\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-23a+42\right){x}-329a+864$
128.5-b2 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( -\frac{110887}{256} a + \frac{66933}{64} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -89 a + 228\) , \( -57 a + 146\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-89a+228\right){x}-57a+146$
128.5-b3 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 274 a + 429\) , \( 18518 a + 28917\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(274a+429\right){x}+18518a+28917$
128.5-b4 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( \frac{110887}{256} a + \frac{156845}{256} \) \( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( 2 a - 3\) , \( 14 a - 35\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-3\right){x}+14a-35$
128.5-b5 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $8.369837957$ 1.014991940 \( -\frac{915957}{16} a + \frac{2374013}{16} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -33 a - 52\) , \( 13 a + 20\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-33a-52\right){x}+13a+20$
128.5-b6 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 1996 a - 5262\) , \( -72821 a + 185664\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(1996a-5262\right){x}-72821a+185664$
128.5-b7 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $8.369837957$ 1.014991940 \( -\frac{203862548967}{4096} a + \frac{130566616997}{1024} \) \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 116 a - 342\) , \( -1173 a + 3040\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(116a-342\right){x}-1173a+3040$
128.5-b8 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $8.369837957$ 1.014991940 \( \frac{915957}{16} a + \frac{182257}{2} \) \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( a - 7\) , \( -7 a - 6\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(a-7\right){x}-7a-6$
128.5-b9 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -373 a - 592\) , \( 5721 a + 8920\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-373a-592\right){x}+5721a+8920$
128.5-b10 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $8.369837957$ 1.014991940 \( \frac{203862548967}{4096} a + \frac{318403919021}{4096} \) \( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( -1898 a - 2967\) , \( 64912 a + 101365\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1898a-2967\right){x}+64912a+101365$
128.5-b11 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( \frac{54503407609}{4} a + \frac{42555672073}{2} \) \( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -9 a - 117\) , \( -339 a - 74\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-9a-117\right){x}-339a-74$
128.5-b12 128.5-b \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $4.184918978$ 1.014991940 \( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) \( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 10858 a - 27827\) , \( 698508 a - 1789259\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(10858a-27827\right){x}+698508a-1789259$
128.5-c1 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/4\Z$ $1$ $1.373831790$ 1.999218911 \( -\frac{55573026649}{16777216} a - \frac{85738862931}{16777216} \) \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -285 a - 443\) , \( -4041 a - 6307\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-285a-443\right){x}-4041a-6307$
128.5-c2 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $8.242990742$ 1.999218911 \( -\frac{110887}{256} a + \frac{66933}{64} \) \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -a + 4\) , \( a + 2\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+4\right){x}+a+2$
128.5-c3 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $2.747663580$ 1.999218911 \( \frac{55573026649}{16777216} a - \frac{35327972395}{4194304} \) \( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 24 a - 41\) , \( -98 a + 129\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(24a-41\right){x}-98a+129$
128.5-c4 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/4\Z$ $1$ $4.121495371$ 1.999218911 \( \frac{110887}{256} a + \frac{156845}{256} \) \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 20 a + 32\) , \( 0\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(20a+32\right){x}$
128.5-c5 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $8.242990742$ 1.999218911 \( -\frac{915957}{16} a + \frac{2374013}{16} \) \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 9 a - 24\) , \( 15 a - 41\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(9a-24\right){x}+15a-41$
128.5-c6 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $5.495327161$ 1.999218911 \( -\frac{653762688677050897}{64} a + \frac{418661913522614865}{16} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -2431 a - 3923\) , \( 73584 a + 115495\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2431a-3923\right){x}+73584a+115495$
128.5-c7 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $5.495327161$ 1.999218911 \( -\frac{203862548967}{4096} a + \frac{130566616997}{1024} \) \( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -791 a - 1243\) , \( -16960 a - 26473\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-791a-1243\right){x}-16960a-26473$
128.5-c8 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $16.48598148$ 1.999218911 \( \frac{915957}{16} a + \frac{182257}{2} \) \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 146 a - 374\) , \( 0\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(146a-374\right){x}$
128.5-c9 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $2.060747685$ 1.999218911 \( -\frac{54503407609}{4} a + \frac{139614751755}{4} \) \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 149 a - 404\) , \( 1359 a - 3561\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(149a-404\right){x}+1359a-3561$
128.5-c10 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $2.747663580$ 1.999218911 \( \frac{203862548967}{4096} a + \frac{318403919021}{4096} \) \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( 24 a - 179\) , \( -493 a + 579\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(24a-179\right){x}-493a+579$
128.5-c11 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $16.48598148$ 1.999218911 \( \frac{54503407609}{4} a + \frac{42555672073}{2} \) \( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 1656 a - 4244\) , \( -51666 a + 132346\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(1656a-4244\right){x}-51666a+132346$
128.5-c12 128.5-c \(\Q(\sqrt{17}) \) \( 2^{7} \) $0$ $\Z/2\Z$ $1$ $0.686915895$ 1.999218911 \( \frac{653762688677050897}{64} a + \frac{1020884965413408563}{64} \) \( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -296 a - 1139\) , \( -7277 a - 16189\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-296a-1139\right){x}-7277a-16189$
128.5-d1 128.5-d \(\Q(\sqrt{17}) \) \( 2^{7} \) $1$ $\Z/2\Z$ $0.081969010$ $17.28603663$ 1.374613658 \( -\frac{217}{16} a - \frac{339}{16} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( a - 1\) , \( -197 a + 505\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(a-1\right){x}-197a+505$
128.5-d2 128.5-d \(\Q(\sqrt{17}) \) \( 2^{7} \) $1$ $\Z/2\Z$ $0.327876043$ $34.57207326$ 1.374613658 \( -\frac{1592342311}{2} a + \frac{4078872403}{2} \) \( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -85 a - 133\) , \( -137 a - 214\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-85a-133\right){x}-137a-214$
128.5-d3 128.5-d \(\Q(\sqrt{17}) \) \( 2^{7} \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.163938021$ $34.57207326$ 1.374613658 \( \frac{159495}{4} a + \frac{481229}{4} \) \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a - 7\) , \( a + 2\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a-7\right){x}+a+2$
128.5-d4 128.5-d \(\Q(\sqrt{17}) \) \( 2^{7} \) $1$ $\Z/2\Z$ $0.327876043$ $17.28603663$ 1.374613658 \( \frac{23841914775}{2} a + \frac{37230413581}{2} \) \( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -33 a - 57\) , \( 141 a + 214\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-33a-57\right){x}+141a+214$
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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.