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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3312.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3312.e1 | 3312g3 | \([0, 0, 0, -6771, 181474]\) | \(45989074372/7555707\) | \(5640305052672\) | \([4]\) | \(6144\) | \(1.1684\) | |
3312.e2 | 3312g2 | \([0, 0, 0, -1911, -29450]\) | \(4135597648/385641\) | \(71969865984\) | \([2, 2]\) | \(3072\) | \(0.82180\) | |
3312.e3 | 3312g1 | \([0, 0, 0, -1866, -31025]\) | \(61604313088/621\) | \(7243344\) | \([2]\) | \(1536\) | \(0.47523\) | \(\Gamma_0(N)\)-optimal |
3312.e4 | 3312g4 | \([0, 0, 0, 2229, -139574]\) | \(1640689628/12223143\) | \(-9124527356928\) | \([2]\) | \(6144\) | \(1.1684\) |
Rank
sage: E.rank()
The elliptic curves in class 3312.e have rank \(1\).
Complex multiplication
The elliptic curves in class 3312.e do not have complex multiplication.Modular form 3312.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.