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SageMath
E = EllipticCurve("mr1")
E.isogeny_class()
Elliptic curves in class 187200mr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.jf4 | 187200mr1 | \([0, 0, 0, 75300, -5614000]\) | \(253012016/219375\) | \(-40940640000000000\) | \([2]\) | \(1179648\) | \(1.8751\) | \(\Gamma_0(N)\)-optimal |
| 187200.jf3 | 187200mr2 | \([0, 0, 0, -374700, -49714000]\) | \(7793764996/3080025\) | \(2299226342400000000\) | \([2, 2]\) | \(2359296\) | \(2.2217\) | |
| 187200.jf2 | 187200mr3 | \([0, 0, 0, -2714700, 1686566000]\) | \(1481943889298/34543665\) | \(51573415495680000000\) | \([2]\) | \(4718592\) | \(2.5683\) | |
| 187200.jf1 | 187200mr4 | \([0, 0, 0, -5234700, -4608394000]\) | \(10625310339698/3855735\) | \(5756581509120000000\) | \([2]\) | \(4718592\) | \(2.5683\) |
Rank
sage: E.rank()
The elliptic curves in class 187200mr have rank \(1\).
Complex multiplication
The elliptic curves in class 187200mr do not have complex multiplication.Modular form 187200.2.a.mr
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.
