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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 3696.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3696.i1 | 3696n4 | \([0, -1, 0, -226128, 41461056]\) | \(312196988566716625/25367712678\) | \(103906151129088\) | \([2]\) | \(13824\) | \(1.7347\) | |
| 3696.i2 | 3696n3 | \([0, -1, 0, -13168, 743104]\) | \(-61653281712625/21875235228\) | \(-89600963493888\) | \([2]\) | \(6912\) | \(1.3881\) | |
| 3696.i3 | 3696n2 | \([0, -1, 0, -5808, -83520]\) | \(5290763640625/2291573592\) | \(9386285432832\) | \([2]\) | \(4608\) | \(1.1854\) | |
| 3696.i4 | 3696n1 | \([0, -1, 0, 1232, -10304]\) | \(50447927375/39517632\) | \(-161864220672\) | \([2]\) | \(2304\) | \(0.83882\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3696.i have rank \(1\).
Complex multiplication
The elliptic curves in class 3696.i do not have complex multiplication.Modular form 3696.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
