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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 13005.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13005.p1 | 13005n7 | \([1, -1, 0, -5618214, 5127014475]\) | \(1114544804970241/405\) | \(7126496559405\) | \([2]\) | \(163840\) | \(2.2568\) | |
| 13005.p2 | 13005n5 | \([1, -1, 0, -351189, 80151120]\) | \(272223782641/164025\) | \(2886231106559025\) | \([2, 2]\) | \(81920\) | \(1.9102\) | |
| 13005.p3 | 13005n8 | \([1, -1, 0, -286164, 110699865]\) | \(-147281603041/215233605\) | \(-3787312458026752605\) | \([2]\) | \(163840\) | \(2.2568\) | |
| 13005.p4 | 13005n3 | \([1, -1, 0, -208134, -36495927]\) | \(56667352321/15\) | \(263944317015\) | \([2]\) | \(40960\) | \(1.5636\) | |
| 13005.p5 | 13005n4 | \([1, -1, 0, -26064, 755595]\) | \(111284641/50625\) | \(890812069925625\) | \([2, 2]\) | \(40960\) | \(1.5636\) | |
| 13005.p6 | 13005n2 | \([1, -1, 0, -13059, -563112]\) | \(13997521/225\) | \(3959164755225\) | \([2, 2]\) | \(20480\) | \(1.2171\) | |
| 13005.p7 | 13005n1 | \([1, -1, 0, -54, -24705]\) | \(-1/15\) | \(-263944317015\) | \([2]\) | \(10240\) | \(0.87049\) | \(\Gamma_0(N)\)-optimal |
| 13005.p8 | 13005n6 | \([1, -1, 0, 90981, 5601258]\) | \(4733169839/3515625\) | \(-61861949300390625\) | \([2]\) | \(81920\) | \(1.9102\) |
Rank
sage: E.rank()
The elliptic curves in class 13005.p have rank \(1\).
Complex multiplication
The elliptic curves in class 13005.p do not have complex multiplication.Modular form 13005.2.a.p
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
