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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 29040dh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 29040.de6 | 29040dh1 | \([0, 1, 0, 493640, 12851540]\) | \(1833318007919/1070530560\) | \(-7768105735799439360\) | \([2]\) | \(552960\) | \(2.3141\) | \(\Gamma_0(N)\)-optimal |
| 29040.de5 | 29040dh2 | \([0, 1, 0, -1984440, 101071188]\) | \(119102750067601/68309049600\) | \(495671903102671257600\) | \([2, 2]\) | \(1105920\) | \(2.6607\) | |
| 29040.de3 | 29040dh3 | \([0, 1, 0, -20724920, -36173001900]\) | \(135670761487282321/643043610000\) | \(4666126257255260160000\) | \([2, 2]\) | \(2211840\) | \(3.0073\) | |
| 29040.de2 | 29040dh4 | \([0, 1, 0, -22893240, 42060851028]\) | \(182864522286982801/463015182960\) | \(3359783487651023093760\) | \([2]\) | \(2211840\) | \(3.0073\) | |
| 29040.de4 | 29040dh5 | \([0, 1, 0, -10076920, -73283411500]\) | \(-15595206456730321/310672490129100\) | \(-2254337094801811558809600\) | \([4]\) | \(4423680\) | \(3.3539\) | |
| 29040.de1 | 29040dh6 | \([0, 1, 0, -331220600, -2320303422252]\) | \(553808571467029327441/12529687500\) | \(90919345017600000000\) | \([2]\) | \(4423680\) | \(3.3539\) |
Rank
sage: E.rank()
The elliptic curves in class 29040dh have rank \(1\).
Complex multiplication
The elliptic curves in class 29040dh do not have complex multiplication.Modular form 29040.2.a.dh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
