Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-733x+4163\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-733xz^2+4163z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-59400x+3213000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2, 75\right) \) | $1.7745753051930334940728443706$ | $\infty$ |
| \( \left(23, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-2:75:1]\) | $1.7745753051930334940728443706$ | $\infty$ |
| \([23:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-15, 2025\right) \) | $1.7745753051930334940728443706$ | $\infty$ |
| \( \left(210, 0\right) \) | $0$ | $2$ |
Integral points
\((-13,\pm 108)\), \((-2,\pm 75)\), \( \left(23, 0\right) \)
\([-13:\pm 108:1]\), \([-2:\pm 75:1]\), \([23:0:1]\)
\((-13,\pm 108)\), \((-2,\pm 75)\), \( \left(23, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 62400 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $16848000000$ | = | $2^{10} \cdot 3^{4} \cdot 5^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{2725888}{1053} \) | = | $2^{11} \cdot 3^{-4} \cdot 11^{3} \cdot 13^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.66127069043979651090176094645$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72107091624387476757964548804$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9241957238555172$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.844442739447851$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7745753051930334940728443706$ |
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| Real period: | $\Omega$ | ≈ | $1.1243078434407673747564599196$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.9806757376192843397014798726 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.980675738 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.124308 \cdot 1.774575 \cdot 16}{2^2} \\ & \approx 7.980675738\end{aligned}$$
Modular invariants
Modular form 62400.2.a.fs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 32768 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 6 | 10 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 269 & 270 \\ 1430 & 29 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 121 & 120 \\ 910 & 751 \end{array}\right),\left(\begin{array}{rr} 623 & 0 \\ 0 & 1559 \end{array}\right),\left(\begin{array}{rr} 1553 & 8 \\ 1552 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1554 & 1555 \end{array}\right),\left(\begin{array}{rr} 521 & 320 \\ 1460 & 1281 \end{array}\right),\left(\begin{array}{rr} 536 & 315 \\ 725 & 626 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$19322634240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $2$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 20800 = 2^{6} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $14$ | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 62400cq
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 312c1, its twist by $40$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{130}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.197706096640000.29 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.560701440000.40 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | add | ss | ss | split | ord | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 1,1 | 3,1 | 2 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | - | 0,0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.