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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 223440.fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 223440.fd1 | 223440ct3 | \([0, 1, 0, -2383376, 1415446164]\) | \(3107086841064961/570\) | \(274677473280\) | \([2]\) | \(2654208\) | \(2.0306\) | |
| 223440.fd2 | 223440ct4 | \([0, 1, 0, -172496, 14613780]\) | \(1177918188481/488703750\) | \(235501598653440000\) | \([2]\) | \(2654208\) | \(2.0306\) | |
| 223440.fd3 | 223440ct2 | \([0, 1, 0, -148976, 22074324]\) | \(758800078561/324900\) | \(156566159769600\) | \([2, 2]\) | \(1327104\) | \(1.6840\) | |
| 223440.fd4 | 223440ct1 | \([0, 1, 0, -7856, 454740]\) | \(-111284641/123120\) | \(-59330334228480\) | \([2]\) | \(663552\) | \(1.3375\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 223440.fd have rank \(0\).
Complex multiplication
The elliptic curves in class 223440.fd do not have complex multiplication.Modular form 223440.2.a.fd
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
