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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 111090cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 111090.ch4 | 111090cf1 | \([1, 1, 1, 15330, 1696347]\) | \(2691419471/9891840\) | \(-1464347328245760\) | \([4]\) | \(811008\) | \(1.5929\) | \(\Gamma_0(N)\)-optimal |
| 111090.ch3 | 111090cf2 | \([1, 1, 1, -153950, 20249435]\) | \(2725812332209/373262400\) | \(55256231214273600\) | \([2, 2]\) | \(1622016\) | \(1.9395\) | |
| 111090.ch2 | 111090cf3 | \([1, 1, 1, -640630, -177147973]\) | \(196416765680689/22365315000\) | \(3310869288790035000\) | \([2]\) | \(3244032\) | \(2.2860\) | |
| 111090.ch1 | 111090cf4 | \([1, 1, 1, -2375750, 1408430075]\) | \(10017490085065009/235066440\) | \(34798269419465160\) | \([2]\) | \(3244032\) | \(2.2860\) |
Rank
sage: E.rank()
The elliptic curves in class 111090cf have rank \(0\).
Complex multiplication
The elliptic curves in class 111090cf do not have complex multiplication.Modular form 111090.2.a.cf
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.
