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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 3234.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3234.q1 | 3234p4 | \([1, 1, 1, -692518, 221512829]\) | \(312196988566716625/25367712678\) | \(2984486028854022\) | \([2]\) | \(27648\) | \(2.0145\) | |
| 3234.q2 | 3234p3 | \([1, 1, 1, -40328, 3942245]\) | \(-61653281712625/21875235228\) | \(-2573599549338972\) | \([2]\) | \(13824\) | \(1.6679\) | |
| 3234.q3 | 3234p2 | \([1, 1, 1, -17788, -465403]\) | \(5290763640625/2291573592\) | \(269601341525208\) | \([2]\) | \(9216\) | \(1.4652\) | |
| 3234.q4 | 3234p1 | \([1, 1, 1, 3772, -51451]\) | \(50447927375/39517632\) | \(-4649209887168\) | \([2]\) | \(4608\) | \(1.1186\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3234.q have rank \(1\).
Complex multiplication
The elliptic curves in class 3234.q do not have complex multiplication.Modular form 3234.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
