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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 206310ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206310.j4 | 206310ca1 | \([1, 1, 0, -1535962, -1115900396]\) | \(-2707064176380409/2063100000000\) | \(-305412842595900000000\) | \([2]\) | \(15679488\) | \(2.6297\) | \(\Gamma_0(N)\)-optimal |
| 206310.j3 | 206310ca2 | \([1, 1, 0, -27985962, -56983590396]\) | \(16374854154935580409/4256381610000\) | \(630097235559601290000\) | \([2, 2]\) | \(31358976\) | \(2.9762\) | |
| 206310.j2 | 206310ca3 | \([1, 1, 0, -31424462, -42102450096]\) | \(23182500134142276409/8246146750089300\) | \(1220725664973930354887700\) | \([2]\) | \(62717952\) | \(3.3228\) | |
| 206310.j1 | 206310ca4 | \([1, 1, 0, -447747462, -3646867890696]\) | \(67058849150792292084409/4532630700\) | \(670992015183192300\) | \([2]\) | \(62717952\) | \(3.3228\) |
Rank
sage: E.rank()
The elliptic curves in class 206310ca have rank \(0\).
Complex multiplication
The elliptic curves in class 206310ca do not have complex multiplication.Modular form 206310.2.a.ca
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.
