Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+22178625x-69043756250\)
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(homogenize, simplify) |
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\(y^2z=x^3+22178625xz^2-69043756250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+22178625x-69043756250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2471, 29106)$ | $3.5894444142125140513729897118$ | $\infty$ |
| $(2450, 0)$ | $0$ | $2$ |
Integral points
\( \left(2450, 0\right) \), \((2471,\pm 29106)\), \((1916075,\pm 2652284250)\)
Invariants
| Conductor: | $N$ | = | \( 485100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-2757567803176840500000000$ | = | $-1 \cdot 2^{8} \cdot 3^{18} \cdot 5^{9} \cdot 7^{6} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{28134667888}{64304361} \) | = | $2^{4} \cdot 3^{-12} \cdot 11^{-2} \cdot 17^{3} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3737718528122862464381362485$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.18233407925170259429244634409$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9736344697696858$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.845602160099542$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.5894444142125140513729897118$ |
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| Real period: | $\Omega$ | ≈ | $0.041818238460658695760918127530$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 3\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.2050036378312773217332516152 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.205003638 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.041818 \cdot 3.589444 \cdot 192}{2^2} \\ & \approx 7.205003638\end{aligned}$$
Modular invariants
Modular form 485100.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 70778880 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 4.6.0.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 725 & 2 \\ 22 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 992 & 333 \\ 661 & 332 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1062 & 19 \\ 521 & 1298 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 881 & 8 \\ 884 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 1312 & 1299 \end{array}\right),\left(\begin{array}{rr} 1 & 338 \\ 330 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$9732096000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 53900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11 \) |
| $5$ | additive | $14$ | \( 19404 = 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $26$ | \( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 44100 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 485100.q
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 3300.o2, its twist by $105$.
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.