Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-26311352x+51911228504\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-26311352xz^2+51911228504z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-420981627x+3321897642646\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3072, 6631)$ | $2.3919803619345120517359534988$ | $\infty$ |
| $(12123/4, -12127/8)$ | $0$ | $2$ |
Integral points
\( \left(3072, 6631\right) \), \( \left(3072, -9704\right) \), \( \left(33102, 5937556\right) \), \( \left(33102, -5970659\right) \)
Invariants
| Conductor: | $N$ | = | \( 495495 \) | = | $3^{2} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1912741081202833756875$ | = | $3^{18} \cdot 5^{4} \cdot 7^{3} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{1559802282754777489}{1481059636875} \) | = | $3^{-12} \cdot 5^{-4} \cdot 7^{-3} \cdot 13^{-1} \cdot 23^{3} \cdot 50423^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0054407567391992894364489402$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2571869760059591717078545328$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9782957263783809$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.79437487006351$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3919803619345120517359534988$ |
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| Real period: | $\Omega$ | ≈ | $0.14712239859642040303320737051$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.6306222118934275763806793151 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.630622212 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.147122 \cdot 2.391980 \cdot 64}{2^2} \\ & \approx 5.630622212\end{aligned}$$
Modular invariants
Modular form 495495.2.a.bq
For more coefficients, see the Downloads section to the right.
| Modular degree: | 35389440 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120120 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 25939 & 25938 \\ 12298 & 17755 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 40039 & 10912 \\ 105556 & 43647 \end{array}\right),\left(\begin{array}{rr} 43679 & 0 \\ 0 & 120119 \end{array}\right),\left(\begin{array}{rr} 96724 & 43681 \\ 26543 & 98286 \end{array}\right),\left(\begin{array}{rr} 848 & 10923 \\ 42845 & 43682 \end{array}\right),\left(\begin{array}{rr} 96097 & 109208 \\ 78628 & 76473 \end{array}\right),\left(\begin{array}{rr} 12288 & 94193 \\ 66913 & 34200 \end{array}\right),\left(\begin{array}{rr} 120113 & 8 \\ 120112 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 120114 & 120115 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$514198484287488000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120120\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | good | $2$ | \( 99099 = 3^{2} \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $3$ | additive | $6$ | \( 7865 = 5 \cdot 11^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 99099 = 3^{2} \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 70785 = 3^{2} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 4095 = 3^{2} \cdot 5 \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 38115 = 3^{2} \cdot 5 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 495495.bq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365.b1, its twist by $33$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.