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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 121275ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.fs4 | 121275ek1 | \([1, -1, 0, -34811667, 5410616616]\) | \(3481467828171481/2005331497785\) | \(2687335998189680343515625\) | \([2]\) | \(17694720\) | \(3.3771\) | \(\Gamma_0(N)\)-optimal |
| 121275.fs2 | 121275ek2 | \([1, -1, 0, -396486792, 3031546387491]\) | \(5143681768032498601/14238434358225\) | \(19080863812782542112890625\) | \([2, 2]\) | \(35389440\) | \(3.7237\) | |
| 121275.fs3 | 121275ek3 | \([1, -1, 0, -240207417, 5445593893116]\) | \(-1143792273008057401/8897444448004035\) | \(-11923426517473316785909921875\) | \([2]\) | \(70778880\) | \(4.0703\) | |
| 121275.fs1 | 121275ek4 | \([1, -1, 0, -6339568167, 194285848116366]\) | \(21026497979043461623321/161783881875\) | \(216805874824077451171875\) | \([2]\) | \(70778880\) | \(4.0703\) |
Rank
sage: E.rank()
The elliptic curves in class 121275ek have rank \(0\).
Complex multiplication
The elliptic curves in class 121275ek do not have complex multiplication.Modular form 121275.2.a.ek
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.
