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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 125970.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 125970.w1 | 125970bf8 | \([1, 0, 1, -2276618533, -41787805454704]\) | \(1304951925893757739492873298865481/820026567584782632177192960\) | \(820026567584782632177192960\) | \([2]\) | \(127401984\) | \(4.1059\) | |
| 125970.w2 | 125970bf6 | \([1, 0, 1, -169533733, -385274802544]\) | \(538880022089281144229433470281/247731924436944317684121600\) | \(247731924436944317684121600\) | \([2, 2]\) | \(63700992\) | \(3.7593\) | |
| 125970.w3 | 125970bf5 | \([1, 0, 1, -90456358, 264814254056]\) | \(81854205816402281283637688281/17072920834174703222946000\) | \(17072920834174703222946000\) | \([6]\) | \(42467328\) | \(3.5566\) | |
| 125970.w4 | 125970bf3 | \([1, 0, 1, -85647653, 300946886288]\) | \(69481646983829384005526777161/1081610344208033843773440\) | \(1081610344208033843773440\) | \([2]\) | \(31850496\) | \(3.4128\) | |
| 125970.w5 | 125970bf2 | \([1, 0, 1, -85326358, 303346710056]\) | \(68702625114973398760876568281/4294127689804836000000\) | \(4294127689804836000000\) | \([2, 6]\) | \(21233664\) | \(3.2100\) | |
| 125970.w6 | 125970bf1 | \([1, 0, 1, -85325078, 303356267048]\) | \(68699533290471196810719174361/268409217024000\) | \(268409217024000\) | \([6]\) | \(10616832\) | \(2.8634\) | \(\Gamma_0(N)\)-optimal |
| 125970.w7 | 125970bf4 | \([1, 0, 1, -80216838, 341267523688]\) | \(-57084778602589569170800944601/17279040062689488281250000\) | \(-17279040062689488281250000\) | \([12]\) | \(42467328\) | \(3.5566\) | |
| 125970.w8 | 125970bf7 | \([1, 0, 1, 595373787, -2900596691312]\) | \(23339569558208438764535804105399/17141434910434216505111040000\) | \(-17141434910434216505111040000\) | \([4]\) | \(127401984\) | \(4.1059\) |
Rank
sage: E.rank()
The elliptic curves in class 125970.w have rank \(0\).
Complex multiplication
The elliptic curves in class 125970.w do not have complex multiplication.Modular form 125970.2.a.w
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 6 & 12 & 12 & 4 \\ 2 & 1 & 6 & 2 & 3 & 6 & 6 & 2 \\ 3 & 6 & 1 & 12 & 2 & 4 & 4 & 12 \\ 4 & 2 & 12 & 1 & 6 & 3 & 12 & 4 \\ 6 & 3 & 2 & 6 & 1 & 2 & 2 & 6 \\ 12 & 6 & 4 & 3 & 2 & 1 & 4 & 12 \\ 12 & 6 & 4 & 12 & 2 & 4 & 1 & 3 \\ 4 & 2 & 12 & 4 & 6 & 12 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
