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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 63525r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 63525.p6 | 63525r1 | \([1, 1, 1, 105812, 3848450156]\) | \(4733169839/231139696095\) | \(-6398094861777410859375\) | \([4]\) | \(2764800\) | \(2.8631\) | \(\Gamma_0(N)\)-optimal |
| 63525.p5 | 63525r2 | \([1, 1, 1, -36209313, 82361750406]\) | \(189674274234120481/3859869269025\) | \(106843654095362469140625\) | \([2, 2]\) | \(5529600\) | \(3.2097\) | |
| 63525.p4 | 63525r3 | \([1, 1, 1, -76971188, -137100184594]\) | \(1821931919215868881/761147600816295\) | \(21069053200776818538984375\) | \([2]\) | \(11059200\) | \(3.5562\) | |
| 63525.p2 | 63525r4 | \([1, 1, 1, -576489438, 5327401203906]\) | \(765458482133960722801/326869475625\) | \(9047956486057822265625\) | \([2, 2]\) | \(11059200\) | \(3.5562\) | |
| 63525.p3 | 63525r5 | \([1, 1, 1, -573630813, 5382852811656]\) | \(-754127868744065783521/15825714261328125\) | \(-438065909101761163330078125\) | \([2]\) | \(22118400\) | \(3.9028\) | |
| 63525.p1 | 63525r6 | \([1, 1, 1, -9223830063, 340965280222656]\) | \(3135316978843283198764801/571725\) | \(15825714261328125\) | \([2]\) | \(22118400\) | \(3.9028\) |
Rank
sage: E.rank()
The elliptic curves in class 63525r have rank \(1\).
Complex multiplication
The elliptic curves in class 63525r do not have complex multiplication.Modular form 63525.2.a.r
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
