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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 111600eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111600.gn2 | 111600eh1 | \([0, 0, 0, -49131075, 132741405250]\) | \(-281115640967896441/468084326400\) | \(-21838942332518400000000\) | \([2]\) | \(9584640\) | \(3.1823\) | \(\Gamma_0(N)\)-optimal |
111600.gn1 | 111600eh2 | \([0, 0, 0, -786411075, 8488335645250]\) | \(1152829477932246539641/3188367360\) | \(148756467548160000000\) | \([2]\) | \(19169280\) | \(3.5289\) |
Rank
sage: E.rank()
The elliptic curves in class 111600eh have rank \(1\).
Complex multiplication
The elliptic curves in class 111600eh do not have complex multiplication.Modular form 111600.2.a.eh
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.