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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 960.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
960.a1 | 960i7 | \([0, -1, 0, -138241, 19829665]\) | \(1114544804970241/405\) | \(106168320\) | \([2]\) | \(2048\) | \(1.3306\) | |
960.a2 | 960i5 | \([0, -1, 0, -8641, 311905]\) | \(272223782641/164025\) | \(42998169600\) | \([2, 2]\) | \(1024\) | \(0.98402\) | |
960.a3 | 960i8 | \([0, -1, 0, -7041, 429345]\) | \(-147281603041/215233605\) | \(-56422198149120\) | \([2]\) | \(2048\) | \(1.3306\) | |
960.a4 | 960i3 | \([0, -1, 0, -5121, -139359]\) | \(56667352321/15\) | \(3932160\) | \([2]\) | \(512\) | \(0.63744\) | |
960.a5 | 960i4 | \([0, -1, 0, -641, 3105]\) | \(111284641/50625\) | \(13271040000\) | \([2, 2]\) | \(512\) | \(0.63744\) | |
960.a6 | 960i2 | \([0, -1, 0, -321, -2079]\) | \(13997521/225\) | \(58982400\) | \([2, 2]\) | \(256\) | \(0.29087\) | |
960.a7 | 960i1 | \([0, -1, 0, -1, -95]\) | \(-1/15\) | \(-3932160\) | \([2]\) | \(128\) | \(-0.055704\) | \(\Gamma_0(N)\)-optimal |
960.a8 | 960i6 | \([0, -1, 0, 2239, 20961]\) | \(4733169839/3515625\) | \(-921600000000\) | \([2]\) | \(1024\) | \(0.98402\) |
Rank
sage: E.rank()
The elliptic curves in class 960.a have rank \(0\).
Complex multiplication
The elliptic curves in class 960.a do not have complex multiplication.Modular form 960.2.a.a
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.