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SageMath
E = EllipticCurve("hq1")
E.isogeny_class()
Elliptic curves in class 369600hq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.hq4 | 369600hq1 | \([0, -1, 0, -424033, 106419937]\) | \(2058561081361/12705\) | \(52039680000000\) | \([2]\) | \(3145728\) | \(1.8186\) | \(\Gamma_0(N)\)-optimal |
| 369600.hq3 | 369600hq2 | \([0, -1, 0, -432033, 102203937]\) | \(2177286259681/161417025\) | \(661164134400000000\) | \([2, 2]\) | \(6291456\) | \(2.1652\) | |
| 369600.hq5 | 369600hq3 | \([0, -1, 0, 407967, 450803937]\) | \(1833318007919/22507682505\) | \(-92191467540480000000\) | \([2]\) | \(12582912\) | \(2.5117\) | |
| 369600.hq2 | 369600hq4 | \([0, -1, 0, -1400033, -516348063]\) | \(74093292126001/14707625625\) | \(60242434560000000000\) | \([2, 2]\) | \(12582912\) | \(2.5117\) | |
| 369600.hq6 | 369600hq5 | \([0, -1, 0, 2911967, -3073364063]\) | \(666688497209279/1381398046875\) | \(-5658206400000000000000\) | \([2]\) | \(25165824\) | \(2.8583\) | |
| 369600.hq1 | 369600hq6 | \([0, -1, 0, -21200033, -37562148063]\) | \(257260669489908001/14267882475\) | \(58441246617600000000\) | \([2]\) | \(25165824\) | \(2.8583\) |
Rank
sage: E.rank()
The elliptic curves in class 369600hq have rank \(0\).
Complex multiplication
The elliptic curves in class 369600hq do not have complex multiplication.Modular form 369600.2.a.hq
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
