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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 296670.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 296670.ch1 | 296670ch7 | \([1, 0, 0, -1511970675, -21908417586165]\) | \(382255564514020240464545161057201/13864271122403467274747839590\) | \(13864271122403467274747839590\) | \([2]\) | \(356990976\) | \(4.1698\) | |
| 296670.ch2 | 296670ch6 | \([1, 0, 0, -1498556725, -22328512989475]\) | \(372171637324012917340923463688401/517402519982194923440100\) | \(517402519982194923440100\) | \([2, 2]\) | \(178495488\) | \(3.8232\) | |
| 296670.ch3 | 296670ch3 | \([1, 0, 0, -1498556225, -22328528634375]\) | \(372171264794057661066407028416401/719306972010000\) | \(719306972010000\) | \([2]\) | \(89247744\) | \(3.4766\) | |
| 296670.ch4 | 296670ch8 | \([1, 0, 0, -1485150775, -22747607117185]\) | \(-362272485498941445395063901231601/13889106446308893988425924390\) | \(-13889106446308893988425924390\) | \([2]\) | \(356990976\) | \(4.1698\) | |
| 296670.ch5 | 296670ch4 | \([1, 0, 0, -216501225, 1216577426625]\) | \(1122289206802103611076620496401/10067762037666166128999000\) | \(10067762037666166128999000\) | \([6]\) | \(118996992\) | \(3.6205\) | |
| 296670.ch6 | 296670ch2 | \([1, 0, 0, -23506225, -12762124375]\) | \(1436390022991844803901216401/760905198226037601000000\) | \(760905198226037601000000\) | \([2, 6]\) | \(59498496\) | \(3.2739\) | |
| 296670.ch7 | 296670ch1 | \([1, 0, 0, -18506225, -30611124375]\) | \(700934163081257731181216401/872298801000000000000\) | \(872298801000000000000\) | \([6]\) | \(29749248\) | \(2.9273\) | \(\Gamma_0(N)\)-optimal |
| 296670.ch8 | 296670ch5 | \([1, 0, 0, 89488775, -99745675375]\) | \(79255498364244733399698063599/50164147441411970976999000\) | \(-50164147441411970976999000\) | \([6]\) | \(118996992\) | \(3.6205\) |
Rank
sage: E.rank()
The elliptic curves in class 296670.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 296670.ch do not have complex multiplication.Modular form 296670.2.a.ch
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
