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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 19600.dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19600.dl1 | 19600cp6 | \([0, -1, 0, -53518208, -150677721088]\) | \(2251439055699625/25088\) | \(188900999168000000\) | \([2]\) | \(995328\) | \(2.8839\) | |
| 19600.dl2 | 19600cp5 | \([0, -1, 0, -3342208, -2357465088]\) | \(-548347731625/1835008\) | \(-13816758796288000000\) | \([2]\) | \(497664\) | \(2.5373\) | |
| 19600.dl3 | 19600cp4 | \([0, -1, 0, -696208, -183041088]\) | \(4956477625/941192\) | \(7086739046912000000\) | \([2]\) | \(331776\) | \(2.3346\) | |
| 19600.dl4 | 19600cp2 | \([0, -1, 0, -206208, 36086912]\) | \(128787625/98\) | \(737894528000000\) | \([2]\) | \(110592\) | \(1.7853\) | |
| 19600.dl5 | 19600cp1 | \([0, -1, 0, -10208, 806912]\) | \(-15625/28\) | \(-210827008000000\) | \([2]\) | \(55296\) | \(1.4387\) | \(\Gamma_0(N)\)-optimal |
| 19600.dl6 | 19600cp3 | \([0, -1, 0, 87792, -16833088]\) | \(9938375/21952\) | \(-165288374272000000\) | \([2]\) | \(165888\) | \(1.9880\) |
Rank
sage: E.rank()
The elliptic curves in class 19600.dl have rank \(1\).
Complex multiplication
The elliptic curves in class 19600.dl do not have complex multiplication.Modular form 19600.2.a.dl
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.
