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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 3192.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3192.m1 | 3192i4 | \([0, 1, 0, -2864, 58032]\) | \(2538016415428/872613\) | \(893555712\) | \([2]\) | \(2048\) | \(0.68876\) | |
3192.m2 | 3192i3 | \([0, 1, 0, -1464, -21600]\) | \(339112345828/8210223\) | \(8407268352\) | \([2]\) | \(2048\) | \(0.68876\) | |
3192.m3 | 3192i2 | \([0, 1, 0, -204, 576]\) | \(3685542352/1432809\) | \(366799104\) | \([2, 2]\) | \(1024\) | \(0.34218\) | |
3192.m4 | 3192i1 | \([0, 1, 0, 41, 86]\) | \(464857088/410571\) | \(-6569136\) | \([4]\) | \(512\) | \(-0.0043898\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3192.m have rank \(1\).
Complex multiplication
The elliptic curves in class 3192.m do not have complex multiplication.Modular form 3192.2.a.m
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.