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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 3315.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3315.a1 | 3315b3 | \([1, 1, 1, -1046, -13456]\) | \(126574061279329/16286595\) | \(16286595\) | \([2]\) | \(1792\) | \(0.40495\) | |
3315.a2 | 3315b2 | \([1, 1, 1, -71, -196]\) | \(39616946929/10989225\) | \(10989225\) | \([2, 2]\) | \(896\) | \(0.058381\) | |
3315.a3 | 3315b1 | \([1, 1, 1, -26, 38]\) | \(1948441249/89505\) | \(89505\) | \([2]\) | \(448\) | \(-0.28819\) | \(\Gamma_0(N)\)-optimal |
3315.a4 | 3315b4 | \([1, 1, 1, 184, -1012]\) | \(688699320191/910381875\) | \(-910381875\) | \([2]\) | \(1792\) | \(0.40495\) |
Rank
sage: E.rank()
The elliptic curves in class 3315.a have rank \(2\).
Complex multiplication
The elliptic curves in class 3315.a do not have complex multiplication.Modular form 3315.2.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 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.