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SageMath
E = EllipticCurve("mc1")
E.isogeny_class()
Elliptic curves in class 141120.mc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.mc1 | 141120jf7 | \([0, 0, 0, -182073612, 594925855216]\) | \(29689921233686449/10380965400750\) | \(233396013539665639047168000\) | \([2]\) | \(42467328\) | \(3.7610\) | |
| 141120.mc2 | 141120jf4 | \([0, 0, 0, -162599052, 798040774384]\) | \(21145699168383889/2593080\) | \(58300409588660305920\) | \([2]\) | \(14155776\) | \(3.2117\) | |
| 141120.mc3 | 141120jf6 | \([0, 0, 0, -76233612, -249380992784]\) | \(2179252305146449/66177562500\) | \(1487875036377268224000000\) | \([2, 2]\) | \(21233664\) | \(3.4144\) | |
| 141120.mc4 | 141120jf3 | \([0, 0, 0, -75669132, -253352899856]\) | \(2131200347946769/2058000\) | \(46270166340206592000\) | \([2]\) | \(10616832\) | \(3.0679\) | |
| 141120.mc5 | 141120jf2 | \([0, 0, 0, -10189452, 12399768304]\) | \(5203798902289/57153600\) | \(1284988619505165926400\) | \([2, 2]\) | \(7077888\) | \(2.8651\) | |
| 141120.mc6 | 141120jf5 | \([0, 0, 0, -2286732, 31141859056]\) | \(-58818484369/18600435000\) | \(-418194956972886589440000\) | \([2]\) | \(14155776\) | \(3.2117\) | |
| 141120.mc7 | 141120jf1 | \([0, 0, 0, -1157772, -168717584]\) | \(7633736209/3870720\) | \(87025684283947745280\) | \([2]\) | \(3538944\) | \(2.5186\) | \(\Gamma_0(N)\)-optimal |
| 141120.mc8 | 141120jf8 | \([0, 0, 0, 20574708, -839485788176]\) | \(42841933504271/13565917968750\) | \(-305003537887104000000000000\) | \([2]\) | \(42467328\) | \(3.7610\) |
Rank
sage: E.rank()
The elliptic curves in class 141120.mc have rank \(0\).
Complex multiplication
The elliptic curves in class 141120.mc do not have complex multiplication.Modular form 141120.2.a.mc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
