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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 188598.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188598.c1 | 188598x5 | \([1, 1, 0, -51298694, 141397567338]\) | \(2361739090258884097/5202\) | \(32883730580898\) | \([2]\) | \(9633792\) | \(2.7277\) | |
| 188598.c2 | 188598x3 | \([1, 1, 0, -3206204, 2208282780]\) | \(576615941610337/27060804\) | \(171061166481831396\) | \([2, 2]\) | \(4816896\) | \(2.3811\) | |
| 188598.c3 | 188598x6 | \([1, 1, 0, -3039794, 2447946462]\) | \(-491411892194497/125563633938\) | \(-793733315873835556962\) | \([2]\) | \(9633792\) | \(2.7277\) | |
| 188598.c4 | 188598x2 | \([1, 1, 0, -210824, 30641520]\) | \(163936758817/30338064\) | \(191777916747797136\) | \([2, 2]\) | \(2408448\) | \(2.0345\) | |
| 188598.c5 | 188598x1 | \([1, 1, 0, -62904, -5658048]\) | \(4354703137/352512\) | \(2228356331129088\) | \([2]\) | \(1204224\) | \(1.6880\) | \(\Gamma_0(N)\)-optimal |
| 188598.c6 | 188598x4 | \([1, 1, 0, 417836, 179131012]\) | \(1276229915423/2927177028\) | \(-18503748702680838372\) | \([2]\) | \(4816896\) | \(2.3811\) |
Rank
sage: E.rank()
The elliptic curves in class 188598.c have rank \(1\).
Complex multiplication
The elliptic curves in class 188598.c do not have complex multiplication.Modular form 188598.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
