Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-38844005x+93289322247\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-38844005xz^2+93289322247z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-621504075x+5969895119750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(46671/4, 8819875/8)$ | $4.5389493350728454745397686440$ | $\infty$ |
| $(-28789/4, 28785/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 450450 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-7826746695825713906250$ | = | $-1 \cdot 2 \cdot 3^{6} \cdot 5^{7} \cdot 7^{6} \cdot 11^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{569047017391330383361}{687121794969610} \) | = | $-1 \cdot 2^{-1} \cdot 5^{-1} \cdot 7^{-6} \cdot 11^{-2} \cdot 13^{-6} \cdot 29^{3} \cdot 285749^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1116420857397275381601843101$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7576169851886225051621820250$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9532509871939737$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.919409179708719$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.5389493350728454745397686440$ |
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| Real period: | $\Omega$ | ≈ | $0.13115501505362913365334241911$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 1\cdot2\cdot2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.5248954939062239174770689285 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.524895494 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.131155 \cdot 4.538949 \cdot 64}{2^2} \\ & \approx 9.524895494\end{aligned}$$
Modular invariants
Modular form 450450.2.a.fv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49766400 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 6 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
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 $96$, genus $1$, and generators
$\left(\begin{array}{rr} 51481 & 12 \\ 68646 & 73 \end{array}\right),\left(\begin{array}{rr} 120109 & 12 \\ 120108 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 60033 & 120112 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 76441 & 12 \\ 98286 & 73 \end{array}\right),\left(\begin{array}{rr} 25026 & 5017 \\ 85085 & 65066 \end{array}\right),\left(\begin{array}{rr} 40039 & 120108 \\ 100100 & 120119 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 36961 & 12 \\ 101646 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 120070 & 120111 \end{array}\right),\left(\begin{array}{rr} 120110 & 120117 \\ 72099 & 8 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$257099242143744000$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $6$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 18018 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 64350 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 13 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 40950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 34650 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 450450fv
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 10010z4, its twist by $-15$.
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.