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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 25350ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25350.cc4 | 25350ch1 | \([1, 1, 1, -12763, -760219]\) | \(-24389/12\) | \(-113128335937500\) | \([2]\) | \(76800\) | \(1.4019\) | \(\Gamma_0(N)\)-optimal |
| 25350.cc2 | 25350ch2 | \([1, 1, 1, -224013, -40897719]\) | \(131872229/18\) | \(169692503906250\) | \([2]\) | \(153600\) | \(1.7485\) | |
| 25350.cc3 | 25350ch3 | \([1, 1, 1, -118388, 75289781]\) | \(-19465109/248832\) | \(-2345829174000000000\) | \([2]\) | \(384000\) | \(2.2066\) | |
| 25350.cc1 | 25350ch4 | \([1, 1, 1, -3498388, 2508889781]\) | \(502270291349/1889568\) | \(17813640290062500000\) | \([2]\) | \(768000\) | \(2.5532\) |
Rank
sage: E.rank()
The elliptic curves in class 25350ch have rank \(1\).
Complex multiplication
The elliptic curves in class 25350ch do not have complex multiplication.Modular form 25350.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
