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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 306735p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 306735.p6 | 306735p1 | \([1, 0, 0, -2249816, -1299026145]\) | \(147281603041/5265\) | \(45020944337639985\) | \([2]\) | \(5160960\) | \(2.2845\) | \(\Gamma_0(N)\)-optimal |
| 306735.p5 | 306735p2 | \([1, 0, 0, -2352061, -1174512184]\) | \(168288035761/27720225\) | \(237035271937674521025\) | \([2, 2]\) | \(10321920\) | \(2.6311\) | |
| 306735.p4 | 306735p3 | \([1, 0, 0, -10633906, 12227169395]\) | \(15551989015681/1445900625\) | \(12363876838724380880625\) | \([2, 2]\) | \(20643840\) | \(2.9777\) | |
| 306735.p7 | 306735p4 | \([1, 0, 0, 4293864, -6606891279]\) | \(1023887723039/2798036865\) | \(-23925975679739731268385\) | \([2]\) | \(20643840\) | \(2.9777\) | |
| 306735.p2 | 306735p5 | \([1, 0, 0, -166148551, 824293542656]\) | \(59319456301170001/594140625\) | \(5080488510323956640625\) | \([2, 2]\) | \(41287680\) | \(3.3243\) | |
| 306735.p8 | 306735p6 | \([1, 0, 0, 12371219, 57901544570]\) | \(24487529386319/183539412225\) | \(-1569443050625809077029025\) | \([2]\) | \(41287680\) | \(3.3243\) | |
| 306735.p1 | 306735p7 | \([1, 0, 0, -2658370426, 52755717640781]\) | \(242970740812818720001/24375\) | \(208430297859444375\) | \([2]\) | \(82575360\) | \(3.6708\) | |
| 306735.p3 | 306735p8 | \([1, 0, 0, -162160996, 865739391815]\) | \(-55150149867714721/5950927734375\) | \(-50886303188340911865234375\) | \([2]\) | \(82575360\) | \(3.6708\) |
Rank
sage: E.rank()
The elliptic curves in class 306735p have rank \(1\).
Complex multiplication
The elliptic curves in class 306735p do not have complex multiplication.Modular form 306735.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
