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SageMath
E = EllipticCurve("em1")
E.isogeny_class()
Elliptic curves in class 72450.em
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.em1 | 72450ec4 | \([1, -1, 1, -97375730, -369824444103]\) | \(8964546681033941529169/31696875000\) | \(361047216796875000\) | \([2]\) | \(7077888\) | \(3.0113\) | |
| 72450.em2 | 72450ec3 | \([1, -1, 1, -8113730, -1600388103]\) | \(5186062692284555089/2903809817953800\) | \(33076208707630003125000\) | \([2]\) | \(7077888\) | \(3.0113\) | |
| 72450.em3 | 72450ec2 | \([1, -1, 1, -6088730, -5771888103]\) | \(2191574502231419089/4115217960000\) | \(46874904575625000000\) | \([2, 2]\) | \(3538944\) | \(2.6647\) | |
| 72450.em4 | 72450ec1 | \([1, -1, 1, -256730, -149840103]\) | \(-164287467238609/757170892800\) | \(-8624649700800000000\) | \([2]\) | \(1769472\) | \(2.3182\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.em have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.em do not have complex multiplication.Modular form 72450.2.a.em
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.
