Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-961273408x-11468232029812\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-961273408xz^2-11468232029812z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-77863146075x-8360107560294750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7817447/49, 21409215750/343)$ | $7.3916931559707479723589783159$ | $\infty$ |
| $(-18157, 0)$ | $0$ | $2$ |
Integral points
\( \left(-18157, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 405600 \) | = | $2^{5} \cdot 3 \cdot 5^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $35114532758015841000000000$ | = | $2^{9} \cdot 3^{16} \cdot 5^{9} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{2543984126301795848}{909361981125} \) | = | $2^{3} \cdot 3^{-16} \cdot 5^{-3} \cdot 11^{6} \cdot 13^{-2} \cdot 5641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8709851575004010324635689838$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2639311371326234950735215053$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0847562556631796$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.7046550905754865$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.3916931559707479723589783159$ |
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| Real period: | $\Omega$ | ≈ | $0.027114221472318307595174737686$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 1\cdot2^{4}\cdot2^{2}\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $12.826880338330261231922104856 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 12.826880338 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.027114 \cdot 7.391693 \cdot 256}{2^2} \\ & \approx 12.826880338\end{aligned}$$
Modular invariants
Modular form 405600.2.a.gv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 198180864 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 0 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $13$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 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 | 16.24.0.20 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1040 = 2^{4} \cdot 5 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 794 & 783 \\ 93 & 20 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 754 & 995 \end{array}\right),\left(\begin{array}{rr} 253 & 770 \\ 768 & 187 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1025 & 16 \\ 1024 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 479 & 1024 \\ 712 & 911 \end{array}\right),\left(\begin{array}{rr} 402 & 1037 \\ 739 & 1020 \end{array}\right)$.
The torsion field $K:=\Q(E[1040])$ is a degree-$1610219520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1040\Z)$.
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 | $4$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 135200 = 2^{5} \cdot 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $18$ | \( 16224 = 2^{5} \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 405600gv
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240z2, its twist by $65$.
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.