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SageMath
E = EllipticCurve("pi1")
E.isogeny_class()
Elliptic curves in class 369600.pi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.pi1 | 369600pi6 | \([0, 1, 0, -168488033, -841818407937]\) | \(258286045443018193442/8440380939375\) | \(17285900163840000000000\) | \([2]\) | \(50331648\) | \(3.3609\) | |
| 369600.pi2 | 369600pi3 | \([0, 1, 0, -47600033, 126386400063]\) | \(11647843478225136004/128410942275\) | \(131492804889600000000\) | \([2]\) | \(25165824\) | \(3.0144\) | |
| 369600.pi3 | 369600pi4 | \([0, 1, 0, -10988033, -11950907937]\) | \(143279368983686884/22699269140625\) | \(23244051600000000000000\) | \([2, 2]\) | \(25165824\) | \(3.0144\) | |
| 369600.pi4 | 369600pi2 | \([0, 1, 0, -3050033, 1869150063]\) | \(12257375872392016/1191317675625\) | \(304977324960000000000\) | \([2, 2]\) | \(12582912\) | \(2.6678\) | |
| 369600.pi5 | 369600pi1 | \([0, 1, 0, 230467, 140326563]\) | \(84611246065664/580054565475\) | \(-9280873047600000000\) | \([2]\) | \(6291456\) | \(2.3212\) | \(\Gamma_0(N)\)-optimal |
| 369600.pi6 | 369600pi5 | \([0, 1, 0, 19503967, -66440111937]\) | \(400647648358480318/1163177490234375\) | \(-2382187500000000000000000\) | \([2]\) | \(50331648\) | \(3.3609\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.pi have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.pi do not have complex multiplication.Modular form 369600.2.a.pi
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
