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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 334400dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 334400.dm3 | 334400dm1 | \([0, 0, 0, -1571854700, 23986490514000]\) | \(104857852278310619039721/47155625\) | \(193149440000000000\) | \([2]\) | \(54263808\) | \(3.5603\) | \(\Gamma_0(N)\)-optimal |
| 334400.dm2 | 334400dm2 | \([0, 0, 0, -1571862700, 23986234146000]\) | \(104859453317683374662841/2223652969140625\) | \(9108082561600000000000000\) | \([2, 2]\) | \(108527616\) | \(3.9068\) | |
| 334400.dm4 | 334400dm3 | \([0, 0, 0, -1516990700, 25738516594000]\) | \(-94256762600623910012361/15323275604248046875\) | \(-62764136875000000000000000000\) | \([2]\) | \(217055232\) | \(4.2534\) | |
| 334400.dm1 | 334400dm4 | \([0, 0, 0, -1626862700, 22217544146000]\) | \(116256292809537371612841/15216540068579856875\) | \(62326948120903093760000000000\) | \([2]\) | \(217055232\) | \(4.2534\) |
Rank
sage: E.rank()
The elliptic curves in class 334400dm have rank \(1\).
Complex multiplication
The elliptic curves in class 334400dm do not have complex multiplication.Modular form 334400.2.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 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.
