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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 297024p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 297024.p4 | 297024p1 | \([0, -1, 0, -153949, 3571789]\) | \(394055318218528768/222812668422981\) | \(228160172465132544\) | \([2]\) | \(2555904\) | \(2.0207\) | \(\Gamma_0(N)\)-optimal |
| 297024.p2 | 297024p2 | \([0, -1, 0, -1824369, 947359089]\) | \(40986616004177118928/85210419440601\) | \(1396087512114806784\) | \([2, 2]\) | \(5111808\) | \(2.3673\) | |
| 297024.p1 | 297024p3 | \([0, -1, 0, -29175329, 60665445153]\) | \(41907435261174342201892/181692769167\) | \(11907417320128512\) | \([2]\) | \(10223616\) | \(2.7139\) | |
| 297024.p3 | 297024p4 | \([0, -1, 0, -1200129, 1604933505]\) | \(-2916942941620850692/15282717505127163\) | \(-1001568174416013754368\) | \([2]\) | \(10223616\) | \(2.7139\) |
Rank
sage: E.rank()
The elliptic curves in class 297024p have rank \(1\).
Complex multiplication
The elliptic curves in class 297024p do not have complex multiplication.Modular form 297024.2.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 Cremona 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 Cremona labels.
