Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-4889408x-3951230688\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-4889408xz^2-3951230688z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-396042075x-2881635297750\)
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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 |
|---|---|---|
| \( \left(-1382, 12826\right) \) | $4.1197954036538688563775074239$ | $\infty$ |
| \( \left(18946, 2589158\right) \) | $6.8229336105923665785966284558$ | $\infty$ |
| \( \left(-1503, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-1382:12826:1]\) | $4.1197954036538688563775074239$ | $\infty$ |
| \([18946:2589158:1]\) | $6.8229336105923665785966284558$ | $\infty$ |
| \([-1503:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-12441, 346302\right) \) | $4.1197954036538688563775074239$ | $\infty$ |
| \( \left(170511, 69907266\right) \) | $6.8229336105923665785966284558$ | $\infty$ |
| \( \left(-13530, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-1503, 0\right) \), \((-1382,\pm 12826)\), \((-1179,\pm 13158)\), \((14122,\pm 1656250)\), \((18946,\pm 2589158)\)
\([-1503:0:1]\), \([-1382:\pm 12826:1]\), \([-1179:\pm 13158:1]\), \([14122:\pm 1656250:1]\), \([18946:\pm 2589158:1]\)
\( \left(-1503, 0\right) \), \((-1382,\pm 12826)\), \((-1179,\pm 13158)\), \((14122,\pm 1656250)\), \((18946,\pm 2589158)\)
Invariants
| Conductor: | $N$ | = | \( 145200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $730768912500000000000$ | = | $2^{11} \cdot 3 \cdot 5^{14} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{228027144098}{12890625} \) | = | $2 \cdot 3^{-1} \cdot 5^{-8} \cdot 11^{-1} \cdot 13^{3} \cdot 373^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.7580173739123282130997236891$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.11896586578280955346924278883$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9442059163301544$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.864714265619863$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $24.577510813327233581766710043$ |
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| Real period: | $\Omega$ | ≈ | $0.10188594757887495128895951391$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.016411913383563156948725001 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.016411913 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.101886 \cdot 24.577511 \cdot 16}{2^2} \\ & \approx 10.016411913\end{aligned}$$
Modular invariants
Modular form 145200.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8847360 |
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| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_{8}^{*}$ | additive | 1 | 2 | 14 | 8 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 | 8.12.0.11 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 356 & 265 \\ 1255 & 6 \end{array}\right),\left(\begin{array}{rr} 496 & 635 \\ 1225 & 726 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 263 & 0 \\ 0 & 1319 \end{array}\right),\left(\begin{array}{rr} 1313 & 8 \\ 1312 & 9 \end{array}\right),\left(\begin{array}{rr} 29 & 560 \\ 670 & 819 \end{array}\right),\left(\begin{array}{rr} 1004 & 1055 \\ 625 & 1314 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1314 & 1315 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$9732096000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 48400 = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 145200jr
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1320f3, its twist by $220$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-110}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{66})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1381698109440000.16 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.826539500160000.53 | \(\Z/8\Z\) | 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 | nonsplit | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 8 | - | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.