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SageMath
E = EllipticCurve("hb1")
E.isogeny_class()
Elliptic curves in class 219450hb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219450.s8 | 219450hb1 | \([1, 1, 0, 105600, 2230272000]\) | \(8334681620170751/137523678664458240\) | \(-2148807479132160000000\) | \([2]\) | \(11943936\) | \(2.7722\) | \(\Gamma_0(N)\)-optimal |
| 219450.s6 | 219450hb2 | \([1, 1, 0, -24982400, 47162880000]\) | \(110358600993178429667329/2339305154932838400\) | \(36551643045825600000000\) | \([2, 2]\) | \(23887872\) | \(3.1187\) | |
| 219450.s7 | 219450hb3 | \([1, 1, 0, -950400, -60218400000]\) | \(-6076082794014148609/100253882690711904000\) | \(-1566466917042373500000000\) | \([2]\) | \(35831808\) | \(3.3215\) | |
| 219450.s3 | 219450hb4 | \([1, 1, 0, -397662400, 3052081720000]\) | \(445089424735238304524848129/206488340640267840\) | \(3226380322504185000000\) | \([2]\) | \(47775744\) | \(3.4653\) | |
| 219450.s5 | 219450hb5 | \([1, 1, 0, -53710400, -81567288000]\) | \(1096677312076899338462209/450803852032204440000\) | \(7043810188003194375000000\) | \([2]\) | \(47775744\) | \(3.4653\) | |
| 219450.s4 | 219450hb6 | \([1, 1, 0, -236248400, -1378122498000]\) | \(93327647066813251630073089/1506876757438610250000\) | \(23544949334978285156250000\) | \([2, 2]\) | \(71663616\) | \(3.6680\) | |
| 219450.s2 | 219450hb7 | \([1, 1, 0, -472060900, 1830578189500]\) | \(744556702832013561199553089/338208906180283330846500\) | \(5284514159066927044476562500\) | \([2]\) | \(143327232\) | \(4.0146\) | |
| 219450.s1 | 219450hb8 | \([1, 1, 0, -3765203900, -88927979497500]\) | \(377806291534052689568887263169/100912963819335937500\) | \(1576765059677124023437500\) | \([2]\) | \(143327232\) | \(4.0146\) |
Rank
sage: E.rank()
The elliptic curves in class 219450hb have rank \(1\).
Complex multiplication
The elliptic curves in class 219450hb do not have complex multiplication.Modular form 219450.2.a.hb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
