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Label Class Base field Conductor norm Rank Torsion CM Regulator Period Leading coeff j-invariant Weierstrass coefficients Weierstrass equation
525.1-a1 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z$ $1$ $0.182944114$ 1.437180430 \( -\frac{4163452939458406987237}{35888671875} a + \frac{11621395514320863711122}{35888671875} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -1073 a - 3918\) , \( -59174 a - 139962\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1073a-3918\right){x}-59174a-139962$
525.1-a2 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/6\Z$ $1$ $1.646497027$ 1.437180430 \( -\frac{1116422858941}{661775625} a + \frac{3869831168171}{661775625} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( 152 a + 247\) , \( 1243 a + 2187\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(152a+247\right){x}+1243a+2187$
525.1-a3 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/6\Z$ $1$ $6.585988108$ 1.437180430 \( -\frac{1968209867}{1929375} a + \frac{894136682}{165375} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -43 a - 83\) , \( 82 a + 153\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-43a-83\right){x}+82a+153$
525.1-a4 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z$ $1$ $0.365888228$ 1.437180430 \( \frac{3286347314559095552233}{3755092620849609375} a + \frac{942789826966264542718}{751018524169921875} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -1763 a - 3248\) , \( -47532 a - 85968\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1763a-3248\right){x}-47532a-85968$
525.1-a5 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/6\Z$ $1$ $13.17197621$ 1.437180430 \( -\frac{100272266627}{11025} a + \frac{11213978329}{441} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -18 a - 38\) , \( -103 a - 178\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-18a-38\right){x}-103a-178$
525.1-a6 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/6\Z$ $1$ $3.292994054$ 1.437180430 \( \frac{2903214315038687}{4651171875} a + \frac{3121066865015681}{2790703125} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -638 a - 1133\) , \( 12381 a + 22203\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-638a-1133\right){x}+12381a+22203$
525.1-a7 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $0.731776456$ 1.437180430 \( \frac{8485224242586689}{5126953125} a + \frac{1262766196701523}{341796875} \) \( \bigl[1\) , \( a + 1\) , \( 0\) , \( -1618 a - 3023\) , \( -57143 a - 102873\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1618a-3023\right){x}-57143a-102873$
525.1-a8 525.1-a \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z$ $1$ $1.463552913$ 1.437180430 \( \frac{4948646226462039127}{328125} a + \frac{1772889969507804542}{65625} \) \( \bigl[a + 1\) , \( -a\) , \( a\) , \( 1147 a - 3221\) , \( 9298 a - 25980\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1147a-3221\right){x}+9298a-25980$
525.1-b1 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $5.769014758$ $0.894985729$ 3.380098590 \( \frac{1259048601200177}{35888671875} a - \frac{702888547069358}{7177734375} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 35 a - 92\) , \( 130 a - 394\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(35a-92\right){x}+130a-394$
525.1-b2 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.923004919$ $8.054871564$ 3.380098590 \( -\frac{6103957}{39375} a + \frac{3424171}{7875} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 8\) , \( 4 a - 4\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+8{x}+4a-4$
525.1-b3 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $5.769014792$ $0.447492864$ 3.380098590 \( -\frac{2728519945338440457167911}{18375} a + \frac{4569650739014584468883452}{11025} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 8740 a - 24697\) , \( 673232 a - 1878956\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8740a-24697\right){x}+673232a-1878956$
525.1-b4 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $0.480751229$ $4.027435782$ 3.380098590 \( -\frac{409618804330271}{107666015625} a + \frac{1442218014473851}{107666015625} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 35 a - 127\) , \( -276 a + 656\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(35a-127\right){x}-276a+656$
525.1-b5 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/6\Z$ $0.961502459$ $8.054871564$ 3.380098590 \( \frac{2235488219}{328125} a + \frac{47198349574}{2953125} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 5 a - 22\) , \( 6 a - 16\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a-22\right){x}+6a-16$
525.1-b6 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $2.884507379$ $0.894985729$ 3.380098590 \( -\frac{30507583626037513}{5359375} a + \frac{7299077445885083}{459375} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 515 a - 1597\) , \( 9912 a - 28786\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(515a-1597\right){x}+9912a-28786$
525.1-b7 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.923004919$ $4.027435782$ 3.380098590 \( \frac{18006302700317}{70875} a + \frac{295901264858807}{637875} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 55 a - 397\) , \( 1076 a - 1916\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(55a-397\right){x}+1076a-1916$
525.1-b8 525.1-b \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $1.442253689$ $0.447492864$ 3.380098590 \( \frac{342545846355593443}{220591875} a + \frac{613634004205251772}{220591875} \) \( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -30 a - 2577\) , \( -2500 a - 50864\bigr] \) ${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-30a-2577\right){x}-2500a-50864$
525.1-c1 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.653725436$ $6.118976388$ 3.491600165 \( -\frac{2398737609666907}{180075} a + \frac{6695567161558627}{180075} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( 125 a + 194\) , \( 3909 a + 7065\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(125a+194\right){x}+3909a+7065$
525.1-c2 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $1.307450873$ $12.23795277$ 3.491600165 \( -\frac{153721}{525} a + \frac{125998}{105} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( -5 a + 14\) , \( 0\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(-5a+14\right){x}$
525.1-c3 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.653725436$ $12.23795277$ 3.491600165 \( \frac{11927641}{13125} a + \frac{8818441}{2625} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( 20 a - 56\) , \( 5 a - 14\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(20a-56\right){x}+5a-14$
525.1-c4 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.326862718$ $12.23795277$ 3.491600165 \( -\frac{14235870121}{275625} a + \frac{4755925414}{30625} \) \( \bigl[a\) , \( a + 1\) , \( 0\) , \( -100 a - 181\) , \( 594 a + 1065\bigr] \) ${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-100a-181\right){x}+594a+1065$
525.1-c5 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $1.307450873$ $3.059488194$ 3.491600165 \( \frac{46195315900021}{8203125} a + \frac{17098029643466}{1640625} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( 225 a - 631\) , \( 2804 a - 7829\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(225a-631\right){x}+2804a-7829$
525.1-c6 525.1-c \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.163431359$ $6.118976388$ 3.491600165 \( \frac{5191424415879943}{221484375} a + \frac{9299320211564417}{221484375} \) \( \bigl[1\) , \( 0\) , \( 0\) , \( 110 a - 321\) , \( -5045 a + 14101\bigr] \) ${y}^2+{x}{y}={x}^{3}+\left(110a-321\right){x}-5045a+14101$
525.1-d1 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $5.769014758$ $0.894985729$ 3.380098590 \( -\frac{1259048601200177}{35888671875} a - \frac{2255394134146613}{35888671875} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -32 a - 61\) , \( -224 a - 435\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-32a-61\right){x}-224a-435$
525.1-d2 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.923004919$ $8.054871564$ 3.380098590 \( \frac{6103957}{39375} a + \frac{11016898}{39375} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( 3 a + 4\) , \( 2 a + 4\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3a+4\right){x}+2a+4$
525.1-d3 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/6\Z$ $0.961502459$ $8.054871564$ 3.380098590 \( -\frac{2235488219}{328125} a + \frac{13463548709}{590625} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -2 a - 21\) , \( -30 a - 31\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a-21\right){x}-30a-31$
525.1-d4 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $0.480751229$ $4.027435782$ 3.380098590 \( \frac{409618804330271}{107666015625} a + \frac{206519842028716}{21533203125} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -32 a - 96\) , \( 147 a + 209\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-32a-96\right){x}+147a+209$
525.1-d5 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $1.442253689$ $0.447492864$ 3.380098590 \( -\frac{342545846355593443}{220591875} a + \frac{191235970112169043}{44118375} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( 33 a - 2611\) , \( -79 a - 53210\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(33a-2611\right){x}-79a-53210$
525.1-d6 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.923004919$ $4.027435782$ 3.380098590 \( -\frac{18006302700317}{70875} a + \frac{91591597832332}{127575} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -52 a - 346\) , \( -1475 a - 1111\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-52a-346\right){x}-1475a-1111$
525.1-d7 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $2.884507379$ $0.894985729$ 3.380098590 \( \frac{30507583626037513}{5359375} a + \frac{163944959727865366}{16078125} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -512 a - 1086\) , \( -11511 a - 21445\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-512a-1086\right){x}-11511a-21445$
525.1-d8 525.1-d \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $5.769014729$ $0.447492864$ 3.380098590 \( \frac{2728519945338440457167911}{18375} a + \frac{14662693859057600972913527}{55125} \) \( \bigl[a\) , \( a + 1\) , \( a\) , \( -8737 a - 15961\) , \( -697931 a - 1249420\bigr] \) ${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8737a-15961\right){x}-697931a-1249420$
525.1-e1 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.518830022$ $7.231173902$ 1.637397992 \( -\frac{257906497658703295754}{8203125} a + \frac{143978254535512478441}{1640625} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -509 a - 988\) , \( 12258 a + 22148\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-509a-988\right){x}+12258a+22148$
525.1-e2 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.259415011$ $3.615586951$ 1.637397992 \( \frac{590589719}{972405} \) \( \bigl[a\) , \( 1\) , \( a + 1\) , \( -88 a + 244\) , \( -970 a + 2706\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-88a+244\right){x}-970a+2706$
525.1-e3 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.129707505$ $14.46234780$ 1.637397992 \( \frac{47045881}{11025} \) \( \bigl[a\) , \( 1\) , \( a + 1\) , \( 37 a - 106\) , \( -125 a + 346\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(37a-106\right){x}-125a+346$
525.1-e4 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.259415011$ $14.46234780$ 1.637397992 \( \frac{1771561}{105} \) \( \bigl[a\) , \( 1\) , \( a + 1\) , \( 12 a - 36\) , \( 42 a - 120\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(12a-36\right){x}+42a-120$
525.1-e5 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.259415011$ $14.46234780$ 1.637397992 \( \frac{157551496201}{13125} \) \( \bigl[a\) , \( 1\) , \( a + 1\) , \( 562 a - 1576\) , \( -10688 a + 29830\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(562a-1576\right){x}-10688a+29830$
525.1-e6 525.1-e \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.518830022$ $7.231173902$ 1.637397992 \( \frac{257906497658703295754}{8203125} a + \frac{461984775018859096451}{8203125} \) \( \bigl[a\) , \( 1\) , \( a + 1\) , \( 507 a - 1496\) , \( -12259 a + 34406\bigr] \) ${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(507a-1496\right){x}-12259a+34406$
525.1-f1 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z$ $1$ $1.052007738$ 1.836535274 \( -\frac{2398737609666907}{180075} a + \frac{6695567161558627}{180075} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 153 a - 410\) , \( 1627 a - 4685\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(153a-410\right){x}+1627a-4685$
525.1-f2 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/4\Z$ $1$ $16.83212381$ 1.836535274 \( -\frac{153721}{525} a + \frac{125998}{105} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -2 a\) , \( -a\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}-2a{x}-a$
525.1-f3 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/4\Z$ $1$ $16.83212381$ 1.836535274 \( \frac{11927641}{13125} a + \frac{8818441}{2625} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -2 a - 5\) , \( -3\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a-5\right){x}-3$
525.1-f4 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z\oplus\Z/2\Z$ $1$ $4.208030954$ 1.836535274 \( -\frac{14235870121}{275625} a + \frac{4755925414}{30625} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 3 a - 35\) , \( 22 a - 110\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(3a-35\right){x}+22a-110$
525.1-f5 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/4\Z$ $1$ $16.83212381$ 1.836535274 \( \frac{46195315900021}{8203125} a + \frac{17098029643466}{1640625} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -7 a - 55\) , \( 22 a + 112\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7a-55\right){x}+22a+112$
525.1-f6 525.1-f \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $0$ $\Z/2\Z$ $1$ $1.052007738$ 1.836535274 \( \frac{5191424415879943}{221484375} a + \frac{9299320211564417}{221484375} \) \( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -67 a - 140\) , \( -475 a - 1083\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-67a-140\right){x}-475a-1083$
525.1-g1 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.636183209$ $0.996753031$ 2.847081088 \( -\frac{3286347314559095552233}{3755092620849609375} a + \frac{2666765483130139421941}{1251697540283203125} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -373 a - 1024\) , \( 19356 a + 36660\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-373a-1024\right){x}+19356a+36660$
525.1-g2 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.545394403$ $0.996753031$ 2.847081088 \( -\frac{2903214315038687}{4651171875} a + \frac{24314977270194466}{13953515625} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 77 a + 11\) , \( -489 a - 1392\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(77a+11\right){x}-489a-1392$
525.1-g3 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/6\Z$ $1.636183209$ $7.974024253$ 2.847081088 \( -\frac{4948646226462039127}{328125} a + \frac{13813096074001061837}{328125} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 22 a - 289\) , \( -486 a + 1515\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(22a-289\right){x}-486a+1515$
525.1-g4 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/2\Z$ $0.272697201$ $3.987012126$ 2.847081088 \( \frac{1968209867}{1929375} a + \frac{25390154269}{5788125} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -28 a - 59\) , \( -146 a - 265\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-28a-59\right){x}-146a-265$
525.1-g5 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/4\Z$ $0.545394403$ $1.993506063$ 2.847081088 \( \frac{1116422858941}{661775625} a + \frac{550681661846}{132355125} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -133 a - 209\) , \( 961 a + 1670\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-133a-209\right){x}+961a+1670$
525.1-g6 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z\oplus\Z/6\Z$ $0.818091604$ $3.987012126$ 2.847081088 \( -\frac{8485224242586689}{5126953125} a + \frac{27426717193109534}{5126953125} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -553 a - 1319\) , \( 11803 a + 23528\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-553a-1319\right){x}+11803a+23528$
525.1-g7 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/2\Z$ $0.545394403$ $7.974024253$ 2.847081088 \( \frac{100272266627}{11025} a + \frac{180077191598}{11025} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -28 a - 54\) , \( -157 a - 277\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-28a-54\right){x}-157a-277$
525.1-g8 525.1-g \(\Q(\sqrt{21}) \) \( 3 \cdot 5^{2} \cdot 7 \) $1$ $\Z/12\Z$ $1.636183209$ $1.993506063$ 2.847081088 \( \frac{4163452939458406987237}{35888671875} a + \frac{1491588514972491344777}{7177734375} \) \( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -9933 a - 18094\) , \( 814606 a + 1461968\bigr] \) ${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-9933a-18094\right){x}+814606a+1461968$
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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.