Show commands:
SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 297024.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
297024.r1 | 297024r4 | \([0, -1, 0, -20783363329, -1153238810620031]\) | \(15149254835477430732977727429892/243555039\) | \(15961623035904\) | \([2]\) | \(171442176\) | \(4.0080\) | |
297024.r2 | 297024r2 | \([0, -1, 0, -1298960209, -18019031639471]\) | \(14794194217980649552803358288/59319057022291521\) | \(971883430253224280064\) | \([2, 2]\) | \(85721088\) | \(3.6614\) | |
297024.r3 | 297024r3 | \([0, -1, 0, -1298335969, -18037216125215]\) | \(-3693218893320489942273647332/7406340191177313578283\) | \(-485381910768996422666354688\) | \([2]\) | \(171442176\) | \(4.0080\) | |
297024.r4 | 297024r1 | \([0, -1, 0, -81224029, -281242894355]\) | \(57873179869926460658341888/115712943163950039453\) | \(118490053799884840399872\) | \([2]\) | \(42860544\) | \(3.3148\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 297024.r have rank \(0\).
Complex multiplication
The elliptic curves in class 297024.r do not have complex multiplication.Modular form 297024.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.