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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 21315.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21315.p1 | 21315c4 | \([1, 1, 0, -12618, 532323]\) | \(1888690601881/31827645\) | \(3744490606605\) | \([2]\) | \(55296\) | \(1.2106\) | |
| 21315.p2 | 21315c2 | \([1, 1, 0, -1593, -12312]\) | \(3803721481/1703025\) | \(200359188225\) | \([2, 2]\) | \(27648\) | \(0.86403\) | |
| 21315.p3 | 21315c1 | \([1, 1, 0, -1348, -19613]\) | \(2305199161/1305\) | \(153531945\) | \([2]\) | \(13824\) | \(0.51746\) | \(\Gamma_0(N)\)-optimal |
| 21315.p4 | 21315c3 | \([1, 1, 0, 5512, -84783]\) | \(157376536199/118918125\) | \(-13990598488125\) | \([2]\) | \(55296\) | \(1.2106\) |
Rank
sage: E.rank()
The elliptic curves in class 21315.p have rank \(1\).
Complex multiplication
The elliptic curves in class 21315.p do not have complex multiplication.Modular form 21315.2.a.p
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.
