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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 190575eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190575.dd3 | 190575eb1 | \([1, -1, 0, -300042, -50373009]\) | \(148035889/31185\) | \(629287947082265625\) | \([2]\) | \(2211840\) | \(2.1300\) | \(\Gamma_0(N)\)-optimal |
| 190575.dd2 | 190575eb2 | \([1, -1, 0, -1525167, 681026616]\) | \(19443408769/1334025\) | \(26919539958519140625\) | \([2, 2]\) | \(4423680\) | \(2.4765\) | |
| 190575.dd1 | 190575eb3 | \([1, -1, 0, -23985792, 45220445991]\) | \(75627935783569/396165\) | \(7994287624045078125\) | \([2]\) | \(8847360\) | \(2.8231\) | |
| 190575.dd4 | 190575eb4 | \([1, -1, 0, 1333458, 2936481741]\) | \(12994449551/192163125\) | \(-3877695636881923828125\) | \([2]\) | \(8847360\) | \(2.8231\) |
Rank
sage: E.rank()
The elliptic curves in class 190575eb have rank \(1\).
Complex multiplication
The elliptic curves in class 190575eb do not have complex multiplication.Modular form 190575.2.a.eb
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.
