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SageMath
E = EllipticCurve("hl1")
E.isogeny_class()
Elliptic curves in class 193200.hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.hl1 | 193200ft5 | \([0, 1, 0, -35554008, 81586415988]\) | \(155324313723954725282/13018359375\) | \(416587500000000000\) | \([2]\) | \(11010048\) | \(2.8234\) | |
| 193200.hl2 | 193200ft4 | \([0, 1, 0, -3060008, -2059146012]\) | \(198048499826486404/242568272835\) | \(3881092365360000000\) | \([2]\) | \(5505024\) | \(2.4768\) | |
| 193200.hl3 | 193200ft3 | \([0, 1, 0, -2227008, 1268345988]\) | \(76343005935514084/694180580625\) | \(11106889290000000000\) | \([2, 2]\) | \(5505024\) | \(2.4768\) | |
| 193200.hl4 | 193200ft6 | \([0, 1, 0, -652008, 3029195988]\) | \(-957928673903042/123339801817575\) | \(-3946873658162400000000\) | \([2]\) | \(11010048\) | \(2.8234\) | |
| 193200.hl5 | 193200ft2 | \([0, 1, 0, -242508, -13641012]\) | \(394315384276816/208332909225\) | \(833331636900000000\) | \([2, 2]\) | \(2752512\) | \(2.1302\) | |
| 193200.hl6 | 193200ft1 | \([0, 1, 0, 57617, -1636012]\) | \(84611246065664/53699121315\) | \(-13424780328750000\) | \([2]\) | \(1376256\) | \(1.7837\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193200.hl have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.hl do not have complex multiplication.Modular form 193200.2.a.hl
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
