Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-1736456x-881276940\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-1736456xz^2-881276940z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-140652963x-642028930398\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-765, 0)$ | $0$ | $2$ |
Integral points
\( \left(-765, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 414960 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $25991686643220480$ | = | $2^{15} \cdot 3 \cdot 5 \cdot 7^{4} \cdot 13^{2} \cdot 19^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{141369383441705190409}{6345626621880} \) | = | $2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 13^{-2} \cdot 19^{-4} \cdot 473579^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2264770657662475907230891564$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5333298852063022813058570349$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9369267896600582$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.229741712262591$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.13151786729995714320268365130$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot1\cdot1\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $4.2085717535986285824858768417 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 4.208571754 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.131518 \cdot 1.000000 \cdot 128}{2^2} \\ & \approx 4.208571754\end{aligned}$$
Modular invariants
Modular form 414960.2.a.fn
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5898240 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_{4}$ | split 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
$\left(\begin{array}{rr} 1992 & 9967 \\ 1967 & 1920 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 15954 & 15955 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 15953 & 8 \\ 15952 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 13681 & 8 \\ 6844 & 33 \end{array}\right),\left(\begin{array}{rr} 9976 & 2003 \\ 13969 & 13998 \end{array}\right),\left(\begin{array}{rr} 12776 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 5324 & 1 \\ 10663 & 6 \end{array}\right),\left(\begin{array}{rr} 4201 & 8 \\ 844 & 33 \end{array}\right)$.
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 | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | split multiplicative | $4$ | \( 138320 = 2^{4} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 82992 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 59280 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \) |
| $13$ | split multiplicative | $14$ | \( 31920 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 21840 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 414960.fn
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 51870.d1, its twist by $-4$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.