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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 190608bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 190608.k4 | 190608bl1 | \([0, -1, 0, -12932584, 17904919408]\) | \(1241361053832817/27447552\) | \(5289141309986045952\) | \([2]\) | \(6635520\) | \(2.7080\) | \(\Gamma_0(N)\)-optimal |
| 190608.k3 | 190608bl2 | \([0, -1, 0, -13394664, 16557124464]\) | \(1379233073341297/183927761424\) | \(35442866489548368052224\) | \([2, 2]\) | \(13271040\) | \(3.0545\) | |
| 190608.k5 | 190608bl3 | \([0, -1, 0, 20914776, 87426703728]\) | \(5250513632788943/20176472892708\) | \(-3888004885340431711223808\) | \([2]\) | \(26542080\) | \(3.4011\) | |
| 190608.k2 | 190608bl4 | \([0, -1, 0, -55097384, -140578724496]\) | \(95992014075197617/11235515171364\) | \(2165082971036407200497664\) | \([2, 2]\) | \(26542080\) | \(3.4011\) | |
| 190608.k6 | 190608bl5 | \([0, -1, 0, 77692856, -714020096912]\) | \(269144439804255023/1298611008739638\) | \(-250242248632135554380095488\) | \([2]\) | \(53084160\) | \(3.7477\) | |
| 190608.k1 | 190608bl6 | \([0, -1, 0, -855131144, -9624498929040]\) | \(358872624127382648977/5938169721462\) | \(1144284881197893214298112\) | \([2]\) | \(53084160\) | \(3.7477\) |
Rank
sage: E.rank()
The elliptic curves in class 190608bl have rank \(0\).
Complex multiplication
The elliptic curves in class 190608bl do not have complex multiplication.Modular form 190608.2.a.bl
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.
