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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 400554.dh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400554.dh1 | 400554dh4 | \([1, -1, 1, -178289501, -916253482543]\) | \(35618855581745079337/188166132\) | \(3311025413072955732\) | \([2]\) | \(42467328\) | \(3.1702\) | |
400554.dh2 | 400554dh2 | \([1, -1, 1, -11149241, -14297783479]\) | \(8710408612492777/19986042384\) | \(351680153791848157584\) | \([2, 2]\) | \(21233664\) | \(2.8236\) | |
400554.dh3 | 400554dh3 | \([1, -1, 1, -7143701, -24713789695]\) | \(-2291249615386537/13671036998388\) | \(-240559501561754421064788\) | \([2]\) | \(42467328\) | \(3.1702\) | |
400554.dh4 | 400554dh1 | \([1, -1, 1, -953321, -43887319]\) | \(5445273626857/3103398144\) | \(54608286902913241344\) | \([4]\) | \(10616832\) | \(2.4770\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400554.dh have rank \(0\).
Complex multiplication
The elliptic curves in class 400554.dh do not have complex multiplication.Modular form 400554.2.a.dh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.