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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 11025z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
11025.p7 | 11025z1 | \([1, -1, 1, -230, 215772]\) | \(-1/15\) | \(-20101434609375\) | \([2]\) | \(18432\) | \(1.2316\) | \(\Gamma_0(N)\)-optimal |
11025.p6 | 11025z2 | \([1, -1, 1, -55355, 4956522]\) | \(13997521/225\) | \(301521519140625\) | \([2, 2]\) | \(36864\) | \(1.5781\) | |
11025.p5 | 11025z3 | \([1, -1, 1, -110480, -6509478]\) | \(111284641/50625\) | \(67842341806640625\) | \([2, 2]\) | \(73728\) | \(1.9247\) | |
11025.p4 | 11025z4 | \([1, -1, 1, -882230, 319169022]\) | \(56667352321/15\) | \(20101434609375\) | \([2]\) | \(73728\) | \(1.9247\) | |
11025.p2 | 11025z5 | \([1, -1, 1, -1488605, -698328228]\) | \(272223782641/164025\) | \(219809187453515625\) | \([2, 2]\) | \(147456\) | \(2.2713\) | |
11025.p8 | 11025z6 | \([1, -1, 1, 385645, -49176228]\) | \(4733169839/3515625\) | \(-4711273736572265625\) | \([2]\) | \(147456\) | \(2.2713\) | |
11025.p1 | 11025z7 | \([1, -1, 1, -23814230, -44724460728]\) | \(1114544804970241/405\) | \(542738734453125\) | \([2]\) | \(294912\) | \(2.6179\) | |
11025.p3 | 11025z8 | \([1, -1, 1, -1212980, -965133228]\) | \(-147281603041/215233605\) | \(-288433615776503203125\) | \([2]\) | \(294912\) | \(2.6179\) |
Rank
sage: E.rank()
The elliptic curves in class 11025z have rank \(1\).
Complex multiplication
The elliptic curves in class 11025z do not have complex multiplication.Modular form 11025.2.a.z
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.