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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 2205.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2205.i1 | 2205j7 | \([1, -1, 0, -952569, -357605172]\) | \(1114544804970241/405\) | \(34735279005\) | \([2]\) | \(12288\) | \(1.8131\) | |
2205.i2 | 2205j5 | \([1, -1, 0, -59544, -5574717]\) | \(272223782641/164025\) | \(14067787997025\) | \([2, 2]\) | \(6144\) | \(1.4666\) | |
2205.i3 | 2205j8 | \([1, -1, 0, -48519, -7711362]\) | \(-147281603041/215233605\) | \(-18459751409696205\) | \([2]\) | \(12288\) | \(1.8131\) | |
2205.i4 | 2205j4 | \([1, -1, 0, -35289, 2560410]\) | \(56667352321/15\) | \(1286491815\) | \([2]\) | \(3072\) | \(1.1200\) | |
2205.i5 | 2205j3 | \([1, -1, 0, -4419, -51192]\) | \(111284641/50625\) | \(4341909875625\) | \([2, 2]\) | \(3072\) | \(1.1200\) | |
2205.i6 | 2205j2 | \([1, -1, 0, -2214, 40095]\) | \(13997521/225\) | \(19297377225\) | \([2, 2]\) | \(1536\) | \(0.77341\) | |
2205.i7 | 2205j1 | \([1, -1, 0, -9, 1728]\) | \(-1/15\) | \(-1286491815\) | \([2]\) | \(768\) | \(0.42684\) | \(\Gamma_0(N)\)-optimal |
2205.i8 | 2205j6 | \([1, -1, 0, 15426, -396495]\) | \(4733169839/3515625\) | \(-301521519140625\) | \([2]\) | \(6144\) | \(1.4666\) |
Rank
sage: E.rank()
The elliptic curves in class 2205.i have rank \(0\).
Complex multiplication
The elliptic curves in class 2205.i do not have complex multiplication.Modular form 2205.2.a.i
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.