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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 8085.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.h1 | 8085w3 | \([1, 0, 0, -43170, -3455985]\) | \(75627935783569/396165\) | \(46608416085\) | \([2]\) | \(18432\) | \(1.2431\) | |
| 8085.h2 | 8085w2 | \([1, 0, 0, -2745, -52200]\) | \(19443408769/1334025\) | \(156946707225\) | \([2, 2]\) | \(9216\) | \(0.89653\) | |
| 8085.h3 | 8085w1 | \([1, 0, 0, -540, 3807]\) | \(148035889/31185\) | \(3668884065\) | \([4]\) | \(4608\) | \(0.54995\) | \(\Gamma_0(N)\)-optimal |
| 8085.h4 | 8085w4 | \([1, 0, 0, 2400, -224043]\) | \(12994449551/192163125\) | \(-22607799493125\) | \([2]\) | \(18432\) | \(1.2431\) |
Rank
sage: E.rank()
The elliptic curves in class 8085.h have rank \(0\).
Complex multiplication
The elliptic curves in class 8085.h do not have complex multiplication.Modular form 8085.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
