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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 441.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
441.c1 | 441d4 | \([1, -1, 1, -16400, -804212]\) | \(16581375\) | \(29417779503\) | \([2]\) | \(448\) | \(1.0697\) | \(-28\) | |
441.c2 | 441d3 | \([1, -1, 1, -965, -13940]\) | \(-3375\) | \(-29417779503\) | \([2]\) | \(224\) | \(0.72312\) | \(-7\) | |
441.c3 | 441d2 | \([1, -1, 1, -335, 2440]\) | \(16581375\) | \(250047\) | \([2]\) | \(64\) | \(0.096741\) | \(-28\) | |
441.c4 | 441d1 | \([1, -1, 1, -20, 46]\) | \(-3375\) | \(-250047\) | \([2]\) | \(32\) | \(-0.24983\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
Rank
sage: E.rank()
The elliptic curves in class 441.c have rank \(1\).
Complex multiplication
Each elliptic curve in class 441.c has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 441.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.