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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5054c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5054.c5 | 5054c1 | \([1, 1, 1, -188, -2087]\) | \(-15625/28\) | \(-1317284668\) | \([2]\) | \(2376\) | \(0.44013\) | \(\Gamma_0(N)\)-optimal |
| 5054.c4 | 5054c2 | \([1, 1, 1, -3798, -91615]\) | \(128787625/98\) | \(4610496338\) | \([2]\) | \(4752\) | \(0.78671\) | |
| 5054.c6 | 5054c3 | \([1, 1, 1, 1617, 42677]\) | \(9938375/21952\) | \(-1032751179712\) | \([2]\) | \(7128\) | \(0.98944\) | |
| 5054.c3 | 5054c4 | \([1, 1, 1, -12823, 452773]\) | \(4956477625/941192\) | \(44279206830152\) | \([2]\) | \(14256\) | \(1.3360\) | |
| 5054.c2 | 5054c5 | \([1, 1, 1, -61558, 5869939]\) | \(-548347731625/1835008\) | \(-86329568002048\) | \([2]\) | \(21384\) | \(1.5387\) | |
| 5054.c1 | 5054c6 | \([1, 1, 1, -985718, 376273267]\) | \(2251439055699625/25088\) | \(1180287062528\) | \([2]\) | \(42768\) | \(1.8853\) |
Rank
sage: E.rank()
The elliptic curves in class 5054c have rank \(0\).
Complex multiplication
The elliptic curves in class 5054c do not have complex multiplication.Modular form 5054.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
