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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 37440dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37440.bt3 | 37440dr1 | \([0, 0, 0, -57658143, 168515203808]\) | \(454357982636417669333824/3003024375\) | \(140109105240000\) | \([2]\) | \(1720320\) | \(2.7734\) | \(\Gamma_0(N)\)-optimal |
| 37440.bt2 | 37440dr2 | \([0, 0, 0, -57659268, 168508299008]\) | \(7099759044484031233216/577161945398025\) | \(1723396334367376281600\) | \([2, 2]\) | \(3440640\) | \(3.1200\) | |
| 37440.bt4 | 37440dr3 | \([0, 0, 0, -53722668, 192504237968]\) | \(-717825640026599866952/254764560814329735\) | \(-6085783218868924475473920\) | \([2]\) | \(6881280\) | \(3.4666\) | |
| 37440.bt1 | 37440dr4 | \([0, 0, 0, -61613868, 144070452848]\) | \(1082883335268084577352/251301565117746585\) | \(6003059620932395350917120\) | \([2]\) | \(6881280\) | \(3.4666\) |
Rank
sage: E.rank()
The elliptic curves in class 37440dr have rank \(1\).
Complex multiplication
The elliptic curves in class 37440dr do not have complex multiplication.Modular form 37440.2.a.dr
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
