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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 73920dc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73920.gq4 | 73920dc1 | \([0, 1, 0, 1455, 20943]\) | \(20777545136/23059575\) | \(-377808076800\) | \([2]\) | \(65536\) | \(0.90798\) | \(\Gamma_0(N)\)-optimal |
73920.gq3 | 73920dc2 | \([0, 1, 0, -8225, 189375]\) | \(939083699236/300155625\) | \(19670999040000\) | \([2, 2]\) | \(131072\) | \(1.2545\) | |
73920.gq2 | 73920dc3 | \([0, 1, 0, -52225, -4465825]\) | \(120186986927618/4332064275\) | \(567812328652800\) | \([2]\) | \(262144\) | \(1.6011\) | |
73920.gq1 | 73920dc4 | \([0, 1, 0, -119105, 15779103]\) | \(1425631925916578/270703125\) | \(35481600000000\) | \([2]\) | \(262144\) | \(1.6011\) |
Rank
sage: E.rank()
The elliptic curves in class 73920dc have rank \(1\).
Complex multiplication
The elliptic curves in class 73920dc do not have complex multiplication.Modular form 73920.2.a.dc
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.