Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-24617224x+40880372464\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-24617224xz^2+40880372464z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1993995171x+29795809540770\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1972, 0)$ | $0$ | $2$ |
| $(3673, 0)$ | $0$ | $2$ |
Integral points
\( \left(-5644, 0\right) \), \( \left(1972, 0\right) \), \( \left(3673, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 439824 \) | = | $2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $233095919843539637305344$ | = | $2^{16} \cdot 3^{10} \cdot 7^{6} \cdot 11^{6} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{3423676911662954233}{483711578981136} \) | = | $2^{-4} \cdot 3^{-10} \cdot 7^{3} \cdot 11^{-6} \cdot 17^{-2} \cdot 139^{3} \cdot 1549^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2097946539483191271673355886$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5436923988607171651974270954$ |
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| $abc$ quality: | $Q$ | ≈ | $1.070078751332218$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.822983469179748$ | |||
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: | $\Omega$ | ≈ | $0.095260829877212827905424883472$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.76208663901770262324339906777 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.762086639 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.095261 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 0.762086639\end{aligned}$$
Modular invariants
Modular form 439824.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 53084160 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{8}^{*}$ | additive | -1 | 4 | 16 | 4 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $17$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15708 = 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7853 & 2240 \\ 8974 & 4479 \end{array}\right),\left(\begin{array}{rr} 5237 & 6734 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15705 & 4 \\ 15704 & 5 \end{array}\right),\left(\begin{array}{rr} 7659 & 6734 \\ 13594 & 8975 \end{array}\right),\left(\begin{array}{rr} 7141 & 13468 \\ 5306 & 11229 \end{array}\right),\left(\begin{array}{rr} 6731 & 0 \\ 0 & 15707 \end{array}\right)$.
The torsion field $K:=\Q(E[15708])$ is a degree-$200123036467200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15708\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 13328 = 2^{4} \cdot 7^{2} \cdot 17 \) |
| $5$ | good | $2$ | \( 146608 = 2^{4} \cdot 7^{2} \cdot 11 \cdot 17 \) |
| $7$ | additive | $26$ | \( 8976 = 2^{4} \cdot 3 \cdot 11 \cdot 17 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 39984 = 2^{4} \cdot 3 \cdot 7^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 25872 = 2^{4} \cdot 3 \cdot 7^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 439824.n
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c2, its twist by $28$.
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 $0$.