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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 51870m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.l4 | 51870m1 | \([1, 1, 0, -313287, 62963829]\) | \(3400580030216483633401/248179914178560000\) | \(248179914178560000\) | \([2]\) | \(884736\) | \(2.0840\) | \(\Gamma_0(N)\)-optimal |
| 51870.l2 | 51870m2 | \([1, 1, 0, -4921287, 4200026229]\) | \(13181351126943641326385401/78687520727961600\) | \(78687520727961600\) | \([2, 2]\) | \(1769472\) | \(2.4306\) | |
| 51870.l3 | 51870m3 | \([1, 1, 0, -4830087, 4363292469]\) | \(-12462027714326806804452601/1020321931394362309440\) | \(-1020321931394362309440\) | \([2]\) | \(3538944\) | \(2.7772\) | |
| 51870.l1 | 51870m4 | \([1, 1, 0, -78740487, 268900913589]\) | \(53990582156643221755293310201/1924038392640\) | \(1924038392640\) | \([2]\) | \(3538944\) | \(2.7772\) |
Rank
sage: E.rank()
The elliptic curves in class 51870m have rank \(0\).
Complex multiplication
The elliptic curves in class 51870m do not have complex multiplication.Modular form 51870.2.a.m
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.
