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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 31680.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
31680.i1 | 31680co4 | \([0, 0, 0, -8999868, -10392076208]\) | \(6749703004355978704/5671875\) | \(67744512000000\) | \([2]\) | \(442368\) | \(2.3894\) | |
31680.i2 | 31680co3 | \([0, 0, 0, -562368, -162451208]\) | \(-26348629355659264/24169921875\) | \(-18042750000000000\) | \([2]\) | \(221184\) | \(2.0429\) | |
31680.i3 | 31680co2 | \([0, 0, 0, -113628, -13575152]\) | \(13584145739344/1195803675\) | \(14282602562764800\) | \([2]\) | \(147456\) | \(1.8401\) | |
31680.i4 | 31680co1 | \([0, 0, 0, 7872, -987752]\) | \(72268906496/606436875\) | \(-452702701440000\) | \([2]\) | \(73728\) | \(1.4936\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31680.i have rank \(1\).
Complex multiplication
The elliptic curves in class 31680.i do not have complex multiplication.Modular form 31680.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.