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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 163254ds
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163254.s3 | 163254ds1 | \([1, 1, 0, -11537143114, -476979551006252]\) | \(35185071518153330625706314097/14069338492530327552\) | \(67910009659791817800941568\) | \([2]\) | \(235339776\) | \(4.3038\) | \(\Gamma_0(N)\)-optimal |
| 163254.s2 | 163254ds2 | \([1, 1, 0, -11592521034, -472169391648300]\) | \(35694169041755352991632836977/703246476580672029917184\) | \(3394436422377876980052532985856\) | \([2, 2]\) | \(470679552\) | \(4.6504\) | |
| 163254.s1 | 163254ds3 | \([1, 1, 0, -24424407114, 761988579640020]\) | \(333837938020256819250317130097/141225711267619951003309824\) | \(681669534177949388082334888271616\) | \([2]\) | \(941359104\) | \(4.9969\) | |
| 163254.s4 | 163254ds4 | \([1, 1, 0, 353318326, -1398476056469292]\) | \(1010559964403977354667663/175039170153957612660987648\) | \(-844880641851653990410569128255232\) | \([2]\) | \(941359104\) | \(4.9969\) |
Rank
sage: E.rank()
The elliptic curves in class 163254ds have rank \(0\).
Complex multiplication
The elliptic curves in class 163254ds do not have complex multiplication.Modular form 163254.2.a.ds
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.
