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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 12600cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12600.ck4 | 12600cb1 | \([0, 0, 0, -39450, -3015875]\) | \(37256083456/525\) | \(95681250000\) | \([2]\) | \(24576\) | \(1.2472\) | \(\Gamma_0(N)\)-optimal |
12600.ck3 | 12600cb2 | \([0, 0, 0, -40575, -2834750]\) | \(2533446736/275625\) | \(803722500000000\) | \([2, 2]\) | \(49152\) | \(1.5938\) | |
12600.ck2 | 12600cb3 | \([0, 0, 0, -153075, 20002750]\) | \(34008619684/4862025\) | \(56710659600000000\) | \([2, 2]\) | \(98304\) | \(1.9404\) | |
12600.ck5 | 12600cb4 | \([0, 0, 0, 53925, -14080250]\) | \(1486779836/8203125\) | \(-95681250000000000\) | \([2]\) | \(98304\) | \(1.9404\) | |
12600.ck1 | 12600cb5 | \([0, 0, 0, -2358075, 1393717750]\) | \(62161150998242/1607445\) | \(37498476960000000\) | \([2]\) | \(196608\) | \(2.2869\) | |
12600.ck6 | 12600cb6 | \([0, 0, 0, 251925, 107887750]\) | \(75798394558/259416045\) | \(-6051657497760000000\) | \([2]\) | \(196608\) | \(2.2869\) |
Rank
sage: E.rank()
The elliptic curves in class 12600cb have rank \(0\).
Complex multiplication
The elliptic curves in class 12600cb do not have complex multiplication.Modular form 12600.2.a.cb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.