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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 19600.dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19600.dm1 | 19600co3 | \([0, -1, 0, -50633, -4367488]\) | \(488095744/125\) | \(3676531250000\) | \([2]\) | \(51840\) | \(1.3970\) | |
19600.dm2 | 19600co4 | \([0, -1, 0, -44508, -5469988]\) | \(-20720464/15625\) | \(-7353062500000000\) | \([2]\) | \(103680\) | \(1.7436\) | |
19600.dm3 | 19600co1 | \([0, -1, 0, -1633, 18012]\) | \(16384/5\) | \(147061250000\) | \([2]\) | \(17280\) | \(0.84772\) | \(\Gamma_0(N)\)-optimal |
19600.dm4 | 19600co2 | \([0, -1, 0, 4492, 116012]\) | \(21296/25\) | \(-11764900000000\) | \([2]\) | \(34560\) | \(1.1943\) |
Rank
sage: E.rank()
The elliptic curves in class 19600.dm have rank \(1\).
Complex multiplication
The elliptic curves in class 19600.dm do not have complex multiplication.Modular form 19600.2.a.dm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.