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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 121275.bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121275.bt1 | 121275en6 | \([1, -1, 1, -33617430230, -2372429651687728]\) | \(3135316978843283198764801/571725\) | \(766166180136328125\) | \([2]\) | \(70778880\) | \(4.2261\) | |
121275.bt2 | 121275en4 | \([1, -1, 1, -2101089605, -37068811375228]\) | \(765458482133960722801/326869475625\) | \(438036359338442197265625\) | \([2, 2]\) | \(35389440\) | \(3.8795\) | |
121275.bt3 | 121275en5 | \([1, -1, 1, -2090670980, -37454633896228]\) | \(-754127868744065783521/15825714261328125\) | \(-21207970691382712335205078125\) | \([2]\) | \(70778880\) | \(4.2261\) | |
121275.bt4 | 121275en3 | \([1, -1, 1, -280531355, 953779972772]\) | \(1821931919215868881/761147600816295\) | \(1020010581726094620963984375\) | \([2]\) | \(35389440\) | \(3.8795\) | |
121275.bt5 | 121275en2 | \([1, -1, 1, -131969480, -573138978478]\) | \(189674274234120481/3859869269025\) | \(5172593980802807844140625\) | \([2, 2]\) | \(17694720\) | \(3.5330\) | |
121275.bt6 | 121275en1 | \([1, -1, 1, 385645, -26777022478]\) | \(4733169839/231139696095\) | \(-309749299112296835859375\) | \([2]\) | \(8847360\) | \(3.1864\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 121275.bt have rank \(0\).
Complex multiplication
The elliptic curves in class 121275.bt do not have complex multiplication.Modular form 121275.2.a.bt
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.