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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 33810ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33810.cm4 | 33810ch1 | \([1, 1, 1, -55910, -15258013]\) | \(-164287467238609/757170892800\) | \(-89080398367027200\) | \([2]\) | \(442368\) | \(1.9371\) | \(\Gamma_0(N)\)-optimal |
| 33810.cm3 | 33810ch2 | \([1, 1, 1, -1325990, -587302045]\) | \(2191574502231419089/4115217960000\) | \(484151277776040000\) | \([2, 2]\) | \(884736\) | \(2.2837\) | |
| 33810.cm2 | 33810ch3 | \([1, 1, 1, -1766990, -163589245]\) | \(5186062692284555089/2903809817953800\) | \(341630321272446616200\) | \([2]\) | \(1769472\) | \(2.6302\) | |
| 33810.cm1 | 33810ch4 | \([1, 1, 1, -21206270, -37596431293]\) | \(8964546681033941529169/31696875000\) | \(3729105646875000\) | \([2]\) | \(1769472\) | \(2.6302\) |
Rank
sage: E.rank()
The elliptic curves in class 33810ch have rank \(0\).
Complex multiplication
The elliptic curves in class 33810ch do not have complex multiplication.Modular form 33810.2.a.ch
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.
