Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1035075x-404414750\)
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(homogenize, simplify) |
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\(y^2z=x^3-1035075xz^2-404414750z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1035075x-404414750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-595, 900)$ | $1.0501890217489404400455100186$ | $\infty$ |
| $(1205, 9900)$ | $2.8930705921042124854833186905$ | $\infty$ |
| $(-610, 0)$ | $0$ | $2$ |
Integral points
\( \left(-610, 0\right) \), \((-595,\pm 900)\), \((-585,\pm 950)\), \((-574,\pm 774)\), \((1205,\pm 9900)\), \((1415,\pm 31050)\), \((2639,\pm 123462)\), \((4805,\pm 324900)\), \((35405,\pm 6659100)\)
Invariants
| Conductor: | $N$ | = | \( 478800 \) | = | $2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $319213470240000000$ | = | $2^{11} \cdot 3^{7} \cdot 5^{7} \cdot 7 \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{5257286722802}{13683705} \) | = | $2 \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 19^{-4} \cdot 37^{3} \cdot 373^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2342579693541902708118044133$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.24484795328980203751467268356$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8903179969915103$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.064790953521386$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.9481448612489138576247034878$ |
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| Real period: | $\Omega$ | ≈ | $0.14970094262313511079146703067$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{2}\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $14.122882070992456822621060095 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 14.122882071 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.149701 \cdot 2.948145 \cdot 128}{2^2} \\ & \approx 14.122882071\end{aligned}$$
Modular invariants
Modular form 478800.2.a.z
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7077888 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 15954 & 15955 \end{array}\right),\left(\begin{array}{rr} 13957 & 13964 \\ 13918 & 5979 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15953 & 8 \\ 15952 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 13968 & 5993 \\ 1987 & 1974 \end{array}\right),\left(\begin{array}{rr} 9572 & 15959 \\ 3169 & 15954 \end{array}\right),\left(\begin{array}{rr} 5312 & 15957 \\ 5315 & 15958 \end{array}\right),\left(\begin{array}{rr} 4201 & 8 \\ 844 & 33 \end{array}\right),\left(\begin{array}{rr} 13688 & 3 \\ 4565 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$183000209817600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $8$ | \( 53200 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 19 \) |
| $5$ | additive | $18$ | \( 19152 = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 68400 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 25200 = 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 478800.z
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 15960.m1, its twist by $60$.
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.