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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 14161a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
|---|---|---|---|---|---|---|---|---|---|
| 14161.d4 | 14161a1 | \([1, -1, 0, -632, -7365]\) | \(-3375\) | \(-8279186167\) | \([2]\) | \(5184\) | \(0.61747\) | \(\Gamma_0(N)\)-optimal | \(-7\) |
| 14161.d3 | 14161a2 | \([1, -1, 0, -10747, -426126]\) | \(16581375\) | \(8279186167\) | \([2]\) | \(10368\) | \(0.96404\) | \(-28\) | |
| 14161.d2 | 14161a3 | \([1, -1, 0, -30977, 2588144]\) | \(-3375\) | \(-974037973361383\) | \([2]\) | \(36288\) | \(1.5904\) | \(-7\) | |
| 14161.d1 | 14161a4 | \([1, -1, 0, -526612, 147214437]\) | \(16581375\) | \(974037973361383\) | \([2]\) | \(72576\) | \(1.9370\) | \(-28\) |
Rank
sage: E.rank()
The elliptic curves in class 14161a have rank \(0\).
Complex multiplication
Each elliptic curve in class 14161a has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).Modular form 14161.2.a.a
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 Cremona 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 Cremona labels.
