Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-87544242x+295403749416\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-87544242xz^2+295403749416z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1400707875x+18904439254750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4035, 86556)$ | $2.6539327260166198725885314182$ | $\infty$ |
| $(-42909/4, 42909/8)$ | $0$ | $2$ |
Integral points
\( \left(4035, 86556\right) \), \( \left(4035, -90591\right) \), \( \left(6729, 101373\right) \), \( \left(6729, -108102\right) \), \( \left(39673, 7678493\right) \), \( \left(39673, -7718166\right) \)
Invariants
| Conductor: | $N$ | = | \( 418950 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $5247876578884009878937500$ | = | $2^{2} \cdot 3^{24} \cdot 5^{6} \cdot 7^{7} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{55369510069623625}{3916046302812} \) | = | $2^{-2} \cdot 3^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 19^{-2} \cdot 31^{3} \cdot 2459^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4906011224105840002635267634$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1636209473318223147128481066$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9950582420479482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.135107361732046$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6539327260166198725885314182$ |
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| Real period: | $\Omega$ | ≈ | $0.074970123959694345793653750289$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\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)$ | ≈ | $3.1834506472024883706735145210 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.183450647 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.074970 \cdot 2.653933 \cdot 64}{2^2} \\ & \approx 3.183450647\end{aligned}$$
Modular invariants
Modular form 418950.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 127401984 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{18}^{*}$ | additive | -1 | 2 | 24 | 18 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $19$ | $2$ | $I_{2}$ | 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 | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 23940 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 18619 & 9540 \\ 18610 & 23579 \end{array}\right),\left(\begin{array}{rr} 14363 & 0 \\ 0 & 23939 \end{array}\right),\left(\begin{array}{rr} 9586 & 4815 \\ 16015 & 17146 \end{array}\right),\left(\begin{array}{rr} 21884 & 14355 \\ 10865 & 18974 \end{array}\right),\left(\begin{array}{rr} 10861 & 14400 \\ 15150 & 9841 \end{array}\right),\left(\begin{array}{rr} 23905 & 36 \\ 23904 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[23940])$ is a degree-$51468809011200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/23940\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 11025 = 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 46550 = 2 \cdot 5^{2} \cdot 7^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 16758 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 19 \) |
| $7$ | additive | $32$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 22050 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 418950.b
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798.d5, 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.