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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3312.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3312.d1 | 3312p4 | \([0, 0, 0, -35571, 2581810]\) | \(1666957239793/301806\) | \(901187887104\) | \([2]\) | \(6144\) | \(1.2975\) | |
| 3312.d2 | 3312p3 | \([0, 0, 0, -15411, -712334]\) | \(135559106353/5037138\) | \(15040813473792\) | \([2]\) | \(6144\) | \(1.2975\) | |
| 3312.d3 | 3312p2 | \([0, 0, 0, -2451, 31570]\) | \(545338513/171396\) | \(511785713664\) | \([2, 2]\) | \(3072\) | \(0.95089\) | |
| 3312.d4 | 3312p1 | \([0, 0, 0, 429, 3346]\) | \(2924207/3312\) | \(-9889579008\) | \([2]\) | \(1536\) | \(0.60432\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3312.d have rank \(1\).
Complex multiplication
The elliptic curves in class 3312.d do not have complex multiplication.Modular form 3312.2.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 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.
