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SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 182070di
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 182070.k4 | 182070di1 | \([1, -1, 0, 36360, -50065344]\) | \(302111711/61689600\) | \(-1085507955928569600\) | \([2]\) | \(2359296\) | \(2.1399\) | \(\Gamma_0(N)\)-optimal |
| 182070.k3 | 182070di2 | \([1, -1, 0, -1836360, -929869200]\) | \(38920307374369/1274490000\) | \(22426292839496490000\) | \([2, 2]\) | \(4718592\) | \(2.4865\) | |
| 182070.k2 | 182070di3 | \([1, -1, 0, -4489380, 2369957076]\) | \(568671957006049/191329687500\) | \(3366692246125392187500\) | \([2]\) | \(9437184\) | \(2.8331\) | |
| 182070.k1 | 182070di4 | \([1, -1, 0, -29146860, -60559614900]\) | \(155624507032726369/175394100\) | \(3086285062197374100\) | \([2]\) | \(9437184\) | \(2.8331\) |
Rank
sage: E.rank()
The elliptic curves in class 182070di have rank \(0\).
Complex multiplication
The elliptic curves in class 182070di do not have complex multiplication.Modular form 182070.2.a.di
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.
