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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 137904.cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 137904.cv1 | 137904p5 | \([0, 1, 0, -54550552, 155034754772]\) | \(908031902324522977/161726530797\) | \(3197432112700361822208\) | \([4]\) | \(11010048\) | \(3.1308\) | |
| 137904.cv2 | 137904p3 | \([0, 1, 0, -3755912, 1899074100]\) | \(296380748763217/92608836489\) | \(1830933157661219229696\) | \([2, 2]\) | \(5505024\) | \(2.7842\) | |
| 137904.cv3 | 137904p2 | \([0, 1, 0, -1471032, -664561260]\) | \(17806161424897/668584449\) | \(13218322168599515136\) | \([2, 2]\) | \(2752512\) | \(2.4376\) | |
| 137904.cv4 | 137904p1 | \([0, 1, 0, -1457512, -677762188]\) | \(17319700013617/25857\) | \(511208654082048\) | \([2]\) | \(1376256\) | \(2.0911\) | \(\Gamma_0(N)\)-optimal |
| 137904.cv5 | 137904p4 | \([0, 1, 0, 597528, -2383120908]\) | \(1193377118543/124806800313\) | \(-2467506532401116024832\) | \([2]\) | \(5505024\) | \(2.7842\) | |
| 137904.cv6 | 137904p6 | \([0, 1, 0, 10480648, 12872614548]\) | \(6439735268725823/7345472585373\) | \(-145224471283022498844672\) | \([2]\) | \(11010048\) | \(3.1308\) |
Rank
sage: E.rank()
The elliptic curves in class 137904.cv have rank \(0\).
Complex multiplication
The elliptic curves in class 137904.cv do not have complex multiplication.Modular form 137904.2.a.cv
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
