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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 312d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 312.a3 | 312d1 | \([0, -1, 0, -39, 108]\) | \(420616192/117\) | \(1872\) | \([4]\) | \(32\) | \(-0.39031\) | \(\Gamma_0(N)\)-optimal |
| 312.a2 | 312d2 | \([0, -1, 0, -44, 84]\) | \(37642192/13689\) | \(3504384\) | \([2, 2]\) | \(64\) | \(-0.043740\) | |
| 312.a1 | 312d3 | \([0, -1, 0, -304, -1892]\) | \(3044193988/85293\) | \(87340032\) | \([2]\) | \(128\) | \(0.30283\) | |
| 312.a4 | 312d4 | \([0, -1, 0, 136, 444]\) | \(269676572/257049\) | \(-263218176\) | \([4]\) | \(128\) | \(0.30283\) |
Rank
sage: E.rank()
The elliptic curves in class 312d have rank \(0\).
Complex multiplication
The elliptic curves in class 312d do not have complex multiplication.Modular form 312.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
