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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 112710cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112710.cn6 | 112710cr1 | \([1, 0, 0, 4329, 44265]\) | \(371694959/249600\) | \(-6024737222400\) | \([2]\) | \(294912\) | \(1.1410\) | \(\Gamma_0(N)\)-optimal |
| 112710.cn5 | 112710cr2 | \([1, 0, 0, -18791, 363321]\) | \(30400540561/15210000\) | \(367132424490000\) | \([2, 2]\) | \(589824\) | \(1.4875\) | |
| 112710.cn3 | 112710cr3 | \([1, 0, 0, -163291, -25155379]\) | \(19948814692561/231344100\) | \(5584084176492900\) | \([2, 2]\) | \(1179648\) | \(1.8341\) | |
| 112710.cn2 | 112710cr4 | \([1, 0, 0, -244211, 46394085]\) | \(66730743078481/60937500\) | \(1470883110937500\) | \([2]\) | \(1179648\) | \(1.8341\) | |
| 112710.cn4 | 112710cr5 | \([1, 0, 0, -33241, -64092349]\) | \(-168288035761/73415764890\) | \(-1772078090720152410\) | \([2]\) | \(2359296\) | \(2.1807\) | |
| 112710.cn1 | 112710cr6 | \([1, 0, 0, -2605341, -1618837209]\) | \(81025909800741361/11088090\) | \(267639537453210\) | \([2]\) | \(2359296\) | \(2.1807\) |
Rank
sage: E.rank()
The elliptic curves in class 112710cr have rank \(0\).
Complex multiplication
The elliptic curves in class 112710cr do not have complex multiplication.Modular form 112710.2.a.cr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
