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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 21294.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21294.q1 | 21294o6 | \([1, -1, 0, -4153122, 3258731412]\) | \(2251439055699625/25088\) | \(88278243475968\) | \([2]\) | \(311040\) | \(2.2449\) | |
| 21294.q2 | 21294o5 | \([1, -1, 0, -259362, 51051924]\) | \(-548347731625/1835008\) | \(-6456922951385088\) | \([2]\) | \(155520\) | \(1.8983\) | |
| 21294.q3 | 21294o4 | \([1, -1, 0, -54027, 3975453]\) | \(4956477625/941192\) | \(3311813477903112\) | \([2]\) | \(103680\) | \(1.6956\) | |
| 21294.q4 | 21294o2 | \([1, -1, 0, -16002, -774630]\) | \(128787625/98\) | \(344836888578\) | \([2]\) | \(34560\) | \(1.1463\) | |
| 21294.q5 | 21294o1 | \([1, -1, 0, -792, -17172]\) | \(-15625/28\) | \(-98524825308\) | \([2]\) | \(17280\) | \(0.79970\) | \(\Gamma_0(N)\)-optimal |
| 21294.q6 | 21294o3 | \([1, -1, 0, 6813, 361557]\) | \(9938375/21952\) | \(-77243463041472\) | \([2]\) | \(51840\) | \(1.3490\) |
Rank
sage: E.rank()
The elliptic curves in class 21294.q have rank \(0\).
Complex multiplication
The elliptic curves in class 21294.q do not have complex multiplication.Modular form 21294.2.a.q
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
