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SageMath
sage: E = EllipticCurve("hl1")
sage: E.isogeny_class()
Elliptic curves in class 369600.hl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.hl1 | 369600hl5 | \([0, -1, 0, -4878720033, 131163302591937]\) | \(3135316978843283198764801/571725\) | \(2341785600000000\) | \([2]\) | \(94371840\) | \(3.7436\) | |
| 369600.hl2 | 369600hl3 | \([0, -1, 0, -304920033, 2049502391937]\) | \(765458482133960722801/326869475625\) | \(1338857372160000000000\) | \([2, 2]\) | \(47185920\) | \(3.3970\) | |
| 369600.hl3 | 369600hl6 | \([0, -1, 0, -303408033, 2070832175937]\) | \(-754127868744065783521/15825714261328125\) | \(-64822125614400000000000000\) | \([2]\) | \(94371840\) | \(3.7436\) | |
| 369600.hl4 | 369600hl4 | \([0, -1, 0, -40712033, -52712856063]\) | \(1821931919215868881/761147600816295\) | \(3117660572943544320000000\) | \([2]\) | \(47185920\) | \(3.3970\) | |
| 369600.hl5 | 369600hl2 | \([0, -1, 0, -19152033, 31694543937]\) | \(189674274234120481/3859869269025\) | \(15810024525926400000000\) | \([2, 2]\) | \(23592960\) | \(3.0504\) | |
| 369600.hl6 | 369600hl1 | \([0, -1, 0, 55967, 1480359937]\) | \(4733169839/231139696095\) | \(-946748195205120000000\) | \([2]\) | \(11796480\) | \(2.7039\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.hl have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.hl do not have complex multiplication.Modular form 369600.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
