Show commands:
SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 162240df
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162240.ec4 | 162240df1 | \([0, -1, 0, 2204380, 895320282]\) | \(3834800837445824/3342041015625\) | \(-1032409193765625000000\) | \([2]\) | \(8257536\) | \(2.7197\) | \(\Gamma_0(N)\)-optimal |
| 162240.ec3 | 162240df2 | \([0, -1, 0, -10998745, 7948429657]\) | \(7442744143086784/2927948765625\) | \(57887332161362496000000\) | \([2, 2]\) | \(16515072\) | \(3.0663\) | |
| 162240.ec1 | 162240df3 | \([0, -1, 0, -153803745, 733997610657]\) | \(2543984126301795848/909361981125\) | \(143829126176832884736000\) | \([2]\) | \(33030144\) | \(3.4128\) | |
| 162240.ec2 | 162240df4 | \([0, -1, 0, -79443745, -266913001343]\) | \(350584567631475848/8259273550125\) | \(1306326987741767307264000\) | \([2]\) | \(33030144\) | \(3.4128\) |
Rank
sage: E.rank()
The elliptic curves in class 162240df have rank \(1\).
Complex multiplication
The elliptic curves in class 162240df do not have complex multiplication.Modular form 162240.2.a.df
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.
