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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 287490ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 287490.ca7 | 287490ca1 | \([1, 0, 1, -56158, -1807024]\) | \(7633736209/3870720\) | \(9931208525844480\) | \([2]\) | \(2488320\) | \(1.7620\) | \(\Gamma_0(N)\)-optimal |
| 287490.ca5 | 287490ca2 | \([1, 0, 1, -494238, 132420688]\) | \(5203798902289/57153600\) | \(146640500889422400\) | \([2, 2]\) | \(4976640\) | \(2.1086\) | |
| 287490.ca4 | 287490ca3 | \([1, 0, 1, -3670318, -2706775792]\) | \(2131200347946769/2058000\) | \(5280264949722000\) | \([2]\) | \(7464960\) | \(2.3113\) | |
| 287490.ca2 | 287490ca4 | \([1, 0, 1, -7886838, 8524500208]\) | \(21145699168383889/2593080\) | \(6653133836649720\) | \([2]\) | \(9953280\) | \(2.4552\) | |
| 287490.ca6 | 287490ca5 | \([1, 0, 1, -110918, 332667056]\) | \(-58818484369/18600435000\) | \(-47723627298387915000\) | \([2]\) | \(9953280\) | \(2.4552\) | |
| 287490.ca3 | 287490ca6 | \([1, 0, 1, -3697698, -2664347744]\) | \(2179252305146449/66177562500\) | \(169793519789498062500\) | \([2, 2]\) | \(14929920\) | \(2.6579\) | |
| 287490.ca1 | 287490ca7 | \([1, 0, 1, -8831448, 6354624256]\) | \(29689921233686449/10380965400750\) | \(26634717079619543406750\) | \([2]\) | \(29859840\) | \(3.0045\) | |
| 287490.ca8 | 287490ca8 | \([1, 0, 1, 997972, -8967815152]\) | \(42841933504271/13565917968750\) | \(-34806433994749511718750\) | \([2]\) | \(29859840\) | \(3.0045\) |
Rank
sage: E.rank()
The elliptic curves in class 287490ca have rank \(0\).
Complex multiplication
The elliptic curves in class 287490ca do not have complex multiplication.Modular form 287490.2.a.ca
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
