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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 2175.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2175.h1 | 2175b3 | \([1, 1, 0, -174000, -28009125]\) | \(37286818682653441/1305\) | \(20390625\) | \([2]\) | \(7680\) | \(1.3488\) | |
| 2175.h2 | 2175b2 | \([1, 1, 0, -10875, -441000]\) | \(9104453457841/1703025\) | \(26609765625\) | \([2, 2]\) | \(3840\) | \(1.0023\) | |
| 2175.h3 | 2175b4 | \([1, 1, 0, -9750, -534375]\) | \(-6561258219361/3978455625\) | \(-62163369140625\) | \([2]\) | \(7680\) | \(1.3488\) | |
| 2175.h4 | 2175b1 | \([1, 1, 0, -750, -5625]\) | \(2992209121/951345\) | \(14864765625\) | \([2]\) | \(1920\) | \(0.65569\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2175.h have rank \(0\).
Complex multiplication
The elliptic curves in class 2175.h do not have complex multiplication.Modular form 2175.2.a.h
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.
