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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 324870.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 324870.cm1 | 324870cm3 | \([1, 0, 1, -68528, 6653756]\) | \(302503589987689/12214946250\) | \(1437076211366250\) | \([2]\) | \(2359296\) | \(1.6745\) | |
| 324870.cm2 | 324870cm2 | \([1, 0, 1, -11198, -317572]\) | \(1319778683209/395612100\) | \(46543367952900\) | \([2, 2]\) | \(1179648\) | \(1.3280\) | |
| 324870.cm3 | 324870cm1 | \([1, 0, 1, -10218, -398324]\) | \(1002702430729/159120\) | \(18720308880\) | \([2]\) | \(589824\) | \(0.98138\) | \(\Gamma_0(N)\)-optimal |
| 324870.cm4 | 324870cm4 | \([1, 0, 1, 30452, -2116852]\) | \(26546265663191/31856082570\) | \(-3747836258277930\) | \([2]\) | \(2359296\) | \(1.6745\) |
Rank
sage: E.rank()
The elliptic curves in class 324870.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 324870.cm do not have complex multiplication.Modular form 324870.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
