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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 3120r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3120.k6 | 3120r1 | \([0, -1, 0, -1760, -27840]\) | \(147281603041/5265\) | \(21565440\) | \([2]\) | \(1536\) | \(0.49627\) | \(\Gamma_0(N)\)-optimal |
| 3120.k5 | 3120r2 | \([0, -1, 0, -1840, -25088]\) | \(168288035761/27720225\) | \(113542041600\) | \([2, 2]\) | \(3072\) | \(0.84284\) | |
| 3120.k4 | 3120r3 | \([0, -1, 0, -8320, 270400]\) | \(15551989015681/1445900625\) | \(5922408960000\) | \([2, 4]\) | \(6144\) | \(1.1894\) | |
| 3120.k7 | 3120r4 | \([0, -1, 0, 3360, -145728]\) | \(1023887723039/2798036865\) | \(-11460758999040\) | \([2]\) | \(6144\) | \(1.1894\) | |
| 3120.k2 | 3120r5 | \([0, -1, 0, -130000, 18084352]\) | \(59319456301170001/594140625\) | \(2433600000000\) | \([2, 4]\) | \(12288\) | \(1.5360\) | |
| 3120.k8 | 3120r6 | \([0, -1, 0, 9680, 1264000]\) | \(24487529386319/183539412225\) | \(-751777432473600\) | \([4]\) | \(12288\) | \(1.5360\) | |
| 3120.k1 | 3120r7 | \([0, -1, 0, -2080000, 1155324352]\) | \(242970740812818720001/24375\) | \(99840000\) | \([4]\) | \(24576\) | \(1.8826\) | |
| 3120.k3 | 3120r8 | \([0, -1, 0, -126880, 18990400]\) | \(-55150149867714721/5950927734375\) | \(-24375000000000000\) | \([4]\) | \(24576\) | \(1.8826\) |
Rank
sage: E.rank()
The elliptic curves in class 3120r have rank \(0\).
Complex multiplication
The elliptic curves in class 3120r do not have complex multiplication.Modular form 3120.2.a.r
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.
